Commentary on Hegel\'s Logic 6: Quantum

1 Second chapter. Quantum. We have begun with a state of quantity, that is, with a state consisting of some units – situations or objects – such that a) these units could be repulsed or multiplied – we could say that we had then divided the quantity further – and that b) all these units could be seen as aspects of one unity – this is why the quantity in case could be taken as a unity. Then we noticed that either of these aspects – the discreteness and continuity – could be taken as the actual state. The result was a division of two interpretations of quantity. The possibility of the two interpretations or divisions of quantity contained a possibility to compare one state of quantity with another state of quantity – state of two units according to a division to a state of one unit according to same division, or a whole to a half. Hegel’s construction is based on the possibility of dividing a given quantity – if you have a quantity, then you can relate it to its part. Thus, we could work all mathematics within one quantity: if you don’t have enough units to work with, just divide the quantities until you have the necessary amount of units. This corresponds with Hegel’s constructivism: number system is never finished, because we could always produce new numbers through a further division. The question is whether all quantities are divisible. Hegelian quantities undoubtedly are defined as implying the infinite divisibility of quantities to quantities: yet, we may ask whether there truly are any Hegelian quantities. Hegel’s construction of a state of quantity was based on the possibility of indefinite repulsion, which already seems to presuppose the possibility of infinite divisibility or other neverending source of new units. Of course, the states and objects constructed through repulsion might be merely virtual, like many previous states and situations of Logic could have been. Indeed, this is even implicit in Hegel's paradigmatic notion of repulsion: add a ”void” or situation of an object as a new object to your ontology. Yet, because of the connection of Hegelian thought with language, we should at least be able to refer to these new objects with signs: that is, we should be capable of making indefinitely large amount of signs. We should then either be able of making our signs ever smaller – which already presupposes infinite divisibility of matter – or then we should be capable of finding indefinitely many places where to produce our signs – which presupposes infinity of space and matter in it. If these presuppositions fail, then it seems that Hegel must either revert to a very strict finitism or accept the possibility of purely virtual mathematics or mathematics working with pure thoughts. I shall suppose that Hegel’s construction is possible, although the problem is truly central for Hegel’s philosophy and has repercussions even in Hegel's Philosophy of right, in which Hegel appears to assume, almost without any argument, that raw materials are indefinitely renewable. The division of this chapter is once more far from satisfactory. The first section deals with basic properties of what is apparently only one sort – although a basic sort – of quanta or

2 quantities related to other quantities. Then the second section gives a division of related quantities – and then in a Hegelian fashion interprets this division as a division between interpretations – but also, following the example of the second chapter of the first division, speaks of the finity of the states of quantities. The third section then considers what we could call infinity in the realm of quantities – note the continuation of the parallel – and ends with a construction of what could be called functions or mappings in the realm of quantities.

1./344. A) A determined or limited state of quantity fixed to a certain division is a number.

We begin our journey with the structure of quantities related to one another. These quantities are said to be determined perfectly: apparently it means that we have fixed one division and one reference unit, which then creates a context, in light of which we measure and compare all the quantities involved. It is only here that it makes sense to speak of numbers and number systems – or partial number systems, that is, potentially ever-growing systems of numbers – because it is only with a fixed unit that we can count: it is hardly appropriate to talk of counting when the unit is constantly changing into new unit. Consequently, Hegel’s discussion of common arithmetic starts actually here. 2./345. B) We can separate two kinds of quanta: an extensive quantum contains a fixed plurality of different units; these units can be taken as merely ideal and then we have an intensive quantum or a degree. An intensive quantum is in one sense independent of other quanta, but in another sense it is determined only by its place among other quanta.

If the section on numbers is concerned with constructing the common arithmetic, this section tries to explain the difference between the so-called extensive and intensive quanta. Kant had already interpreted these terms within the context of his transcendental philosophy, but Hegel’s interpretation differs in being not dependent on the way how these quantities are experienced. Extensive quantum in fact seems to mean much the same thing as number for Hegel – both are states of quantities numerated in comparison to some fixed unit – or at least the difference is negligible. An extensive state of quantity seems then ultimately reducible to what is called set: it is a collection of different units. Now, these units could be seen as aspects of one whole: there would be one unit in different places. Then the quantity in question would not be extensive, but intensive. Such an intensive state of quantity would be independent of other quantities in some sense: one intensive quantity couldn’t be part of another, for instance. Yet, an intensive quantity is still necessarily related to other quantities in another sense: an intensive quantity is a quantity only because it can be given a place in a certain order of quantities, for instance, according to when it arises in the order of

3 construction. The place of an intensive quantity in the number system is then completely arbitrary, depending here on the contingent starting point of the construction.

3./346. C) Because the place of the intensive quantity is arbitrary, it can be taken to be as large as we wish: in this sense there can be quantitative states of infinity.

It has already become clear that Hegel cannot tolerate any actual quantitative infinities within his philosophy. Thus, when he announces that in the final section we construct an example of a quantitative infinity this can at most mean a potential infinity. Indeed, Hegel’s short description of the transition seems to be in favour of this idea. The important fact in the transition is the externality of intensive quantities: the value of a grade is determined by its place in the scale of grades. Now, this place can be arbitrarily varied in the sense that we could start the ordering from any grade at our will. Thus, with a suitable placement of the reference quantity, we can make our given quantity as high or as low in the ordering as we like: this is the potential infinity Hegel here speaks of.

A. Number

Although no construction of new structures occurs at this section and the whole consists mostly of an analysis, it is still an interesting attempt to interpret arithmetic in the Hegelian framework. If we are following Hegel's lead literally, number system could only be presented partially because of the finity of the number of quantities involved: if we could always represent only finite sets of units, we could also represent only finite sets of numbers. Still, because new numbers could be constructed out of old one, we could always achieve the required level of perfection. Yet, the section itself gives only the basics for the reconstruction of arithmetic, presenting nothing more than an analysis of what number is supposed to be. The account of calculation is left for the Remark, because it uses concepts not yet available at this stage: a further reason to suggest that the place of the division of quantities should be moved a bit later in the whole order of Logic.

1./347. A state of quantity is now a quantum or related to and limited by other states of quantity. The difference between continuous and discrete interpretations is without a meaning.

The beginning of the section reminds us of the position we have just constructed. We have related states of quantities to one another by fixing certain quantity as unit to which all other quantities should be related. After this fixing it does not matter whether we interpret a state of quantity as continuous or as discrete. It is of no concern how the quantities in question are intuited – whether they are lengths, times, masses or some other quantities – and we could as well be investigating

4 some abstract realm of sets or other similar virtual objects. Thus, there is no reason why we should prefer one interpretation over another: continuity or discreteness. The important matter is how large a quantity is compared to the reference quantity.

2./348. As a state of quantity contains an aspect of independency only as ideal, every fixation of division is in some sense arbitrary for it: on another sense, it must have fixed limit in order to be a quantum – a state of quantity has the potential of its unit being fixed and thus having a possible aspect of its continuity being limited.

We know from the beginning that a state of quantity can be described only by a series of situations which we have called divisions: the mention of the state of being-related-to-itself refers to the fact that in that chapter we discussed objects that were seen as independent units, while in the state of quantity all such apparently independent units are always further divisible and thus not the final level of entities. We have counted so and so many units in our state of quantity, but we could discern more units within it: one meter is hundred centimeters. But this arbitrary nature of all states of divisions is just one aspect of quantities: this is familiar from the discussion of the infinite divisibility of matter. On the other hand, going further in dividing a quantity leads to nothing qualitatively new, but only to a further division. Thus, although a unit can be turned into a combination of parts, these parts are then only new units and the whole process could be began anew.

3./349. Quantum consists of units, but these units are quantitative: 1) all units of quantum are in some sense copies of the same unit; 2) in another sense, there are many units, which are either potential (in continuous interpretation) or constructed (in discrete interpretation); 3) finally the units of this quantum always differ from units of other quanta.

After so few preliminary explanations Hegel is ready to give an analysis of the aspects of numbers: the task is actually quite easy, because they should be quite familiar by now. Every limited state of quantity or quantum must have some units – object or situations – but because this is a state of quantity, we can always interpret these units as mere versions of one unit: here we see the word ”unit” in the meaning of meter, kilogram or any arbitrary unit of measurement. All units of number must be identifiable with one another: otherwise, we wouldn’t be able to speak of a number of them, like five liters. A unit within a number is in some sense related only to itself, that is, to versions or copies of itself. A fixed unit quantity – a state of quantity with only one unit – forms a reference point with which all quantum states are compared. The second aspect of quanta is also one shared by all states of quantity: any quantity can be interpreted as consisting of many units. In some cases this plurality of units is obvious, that is, when the quantity in question is interpreted as discrete or the fixed division of units shows the quantity as

5 a plurality. Yet, an apparently continuous quantum – like any unit quantum of a given division should be – can still be divided and thus seen as a plurality of units: a meter is ten decimeters. Thus, there are many units enclosed within a quantum. The first two aspects of a quantum are shared by all quantities, but the final differentiates quanta from other quantities: or should I say, quantities in a context where they are quanta from quantities in other contexts, because a quantity can always be seen as a quantum, as we have seen. A quantum is nothing but a quantity related to other quantities. It may be at first related only to a part of itself, and then the units of the part would be also units of the whole. Yet, we could interpret the units belonging to the part as differing from the units belonging to the whole: we could abstract from the fact that they are same units in different situations. Thus, we could think of the quantities related as having completely different units. Undoubtedly they must all be versions of the same unit in a wider sense – a kilogram of one quantity should be similar to a kilogram in another. Yet, we have now fixed a context according to which the difference of the units is more essential than their identity: units of other quantities are excluded from being units of this quantity.

4./350. When quantum has been constructed as having all these aspects, it is a number: it is constructed in such a way, when it has been fixed as a certain plurality in relation to a certain unit – thus, the number is in one sense discrete, although it has a continuous aspect because its units can be identified. Quantum is fully determined, because it has this fixed plurality of units: continuity without any fixed unit is still undetermined.

The aspects of number have been exposed, and only thing left is to define an object having these aspects as a number. Hegel uses the curious expression that the number must be perfectly posited in these determinations: he then explains that this mean that we have constructed a situation in which the quantum in question is a certain plurality of units – thus, we have to choose such a division that the quantum is seen as many units, or otherwise it is only implicitly a number, that is, number one is not truly a number in Hegel’s sense of the word. An explicit number must then be interpreted as discrete, although it has also the aspect of being possibly seen as continuous, because its units are all of the same variety. In a number, Hegel continues, the quantum is fully determined, that is, we have picked up a certain division where there clearly is a plurality of independent units or “ones”: if the quantum would be taken as continuous, on the other hand, the different units would be discernible only as implicit units.

5./351. Quantum means limited state of quantity, but in a number the quantity is constructed as having many units. This collection of units is determined – we can say which units belong to it and which don’t – and number thus has an amount as aspect: the other aspect is the unit which repeats itself or is continuous within the number.

6 Interestingly Hegel enumerates here only two aspects of number although he just a few paragraphs ago spoke of three aspects. Yet, there is no true contradiction, because the previous third aspect – separation of different sets of units into different quanta – has been incorporated within what was the second aspect of the number there and what is here the first aspect. Previously this aspect consisted only of the fact that for every unit of the quantum it is possible to find another unit – that there is a plurality of units within the quantum. Now this aspect also contains a facet that we can separate units that do not belong to this quantum from units that do belong in it: the former third aspect. Thus, the first aspect of number, according to this paragraph, is then the determinate plurality or amount of units contained within the quantum. This determinate plurality is once again what discreteness was to any quantity. The aspect of continuity is once more represented by the choice of the unit with which the quantum is measured – or the reference quantum, to which other quanta are compared – because this unit can be seen as merely copied into different versions within this quantum.

6./352. A plurality of units in a quantum consists of units, because the units are interpreted as independent: the fact of being-limited-to-this-quantum is irrelevant to units, although the units are not irrelevant to this quantum. In a state of being-here we could separate the aspect of being from the aspect of being-limited, like here it is irrelevant to the units that they are gathered into these particular quanta: in another sense, the two aspects of a state of being-here were necessarily combined or the object that was here was also limited in this particular way. We might think that one unit limits a set of hundred units from others: because any unity is as good as the others, the limiting unit could also be anyone of them – thus, we can always take the plurality of units as a unified plurality.

Because the numbers are primarily taken according to their discrete aspect, the unities within numbers are independent of each other: although they are “of the same unit” we interpret them as different objects. Now, we have collected, say, a set of ten units into a number: all the rest of the units beyond these ten belong to other sets or numbers. Hegel compares this relationship of units and the number they are currently limited in with the relationship between an object in some state of being-here and the quality of the state of being-here. In some sense the object is also free of the quality or its position as related to objects with other qualities: we could abstract the object from this relationship and regard it as something “in itself” or without relations to anything. Yet, the analogy is not perfect, because in another sense an object is always attached to its quality: when we begin the comparison with other objects, it must be found in the same spot of the quality space, or otherwise it has changed to another object. This disanalogy is expressed in a popular fashion by saying that only one unit – or actually, the stopping of the counting to that final unit – is the only thing that unites all these units into the contingent whole called number: we could as well have continued to take one unit more into the number. Yet, in another sense this popular comparison is

7 faulty. There is no final unit, because there is no discernible difference between the units: one unit could as well take the place of the others. It is this lack of difference between units and our possibility to see them as versions of the same unit which makes the number into a unity: it is not ten whatevers but ten copies of the same unit or the same unit repeated ten times. True, the arbitrariness of the number is still in effect, but it is not the arbitrariness of a last unit included in the number: instead, it is arbitrary how many times this one unit was copied – we could have given one more version of it.

7./353. Certain division of units separates numbers, but this separation is only an external comparison because of its quantitative nature: number is primarily abstracted from other numbers, and this being-abstracted is an intrinsic property of all numbers. Yet, it makes number also completely contingent [because any other number could replace it]. Number is a given that in some sense does not need a relation to another number: it has also an aspect of being-acertain-amount-related-to-some-fixed-unity – in this sense it is determined only in relation to another number. These two aspects of number or quantum – independency and relatedness – form its defining quality.

The main interest for Hegel in all sorts of quanta is the double-aspect of being-independent-andnecessarily-related: yet another apparent contradiction of the Hegelian kind. Thus, it is no wonder that Hegel’s analysis of numbers ends with the discovery of this double-aspect: we shall see it in a more explicit manner in some other forms of quanta later on. Every number is independent of other in one sense. We can abstract, say, number two from other numbers: two has apparently no intrinsic relation to four or five – a pair of aces in my sleeve is completely independent of the quartet of aces in the pack. In fact, it is completely conventional why we separate the two aces to form a group of their own different from the groups of four aces, at least if we put them all into the desk: this is a question of a mere arbitrary division of objects or situations according to no qualitative principles. Now the collections of cards are not quite proper examples of quanta or numbers of Hegelian variety, because cards are not divisible to further cards. Among true quanta and numbers – which are infinitely divisible – the selection of units is also arbitrary, because we might as well have taken some smaller or larger units. It is in this sense that quanta or numbers are then also necessarily related to other quanta and numbers. We have here a rod of four meters – but it is a rod of four units only because we have picked the unit in a certain way, i.e. to be a meter long. A number consists of a certain amount of units: these were the aspects or constituents of a number. Thus, they are determinately defined only in relation to some preselected unit number.

Remark 1

Hegel has not said very much of numbers, beyond constructing a structure type that should be

8 defined as number in Logic. It is the primary task of this remark to make Hegel’s rather sparse words into something concrete, that is, showing how a system of arithmetic could be built on basis of constructions of Logic. It is clear how Hegelian arithmetic would differ from, say, the so-called Peano arithmetic. Whereas a Peano arithmetic consists of a determined set of entities called numbers and a number of relationships between these numbers – a number being successor of other is a primary relation and i.e. that of being a sum of two numbers is a relation derived recursively from it – a Hegelian arithmetic, on the other hand, cannot have a predetermined set of entities, because the numbers are constructed in the course of arithmetical operations, which take the place of the relations of Peano arithmetic. Thus, while in a Peano arithmetic 1 + 3 = 4 should be interpreted as a relationship between numbers 1, 3 and 4, in Hegelian arithmetic it should be seen as a process of constructing number 4 from 1 and 3.

1./354. Usually spatial and numerical magnitudes are characterized as continuous and discrete magnitudes: geometry would then investigate continuous quanta and arithmetic discrete quanta. Then geometry would not investigate as determined quanta as arithmetic: it could merely use comparison to indicate which lines etc. are equal. Circle can still be defined in a purely geometrical fashion by the equality of the radius, but even triangles etc. require the determination of the number of the angles: angle as a unit cannot be found through comparison. Points are in some sense similar to units, but in space they can merely be used to determine lines – continuous quantities where we have abstracted from all points or units. If we would want to truly determine i.e. a line, we would have to determine one length of line as a unit to which others would be related as numbers.

Hegel describes as a common opinion that geometry and arithmetic differ in such way that geometry studies continuous, while arithmetic discrete quantities. Because continuity and discreteness are not so much characteristics of different species, but of different interpretations of quantities, Hegel’s conclusion – that geometry and arithmetic must then work at different levels – is obvious. Yet, as Hegel notes, this would limit geometry into a very restricted field. The most we could say of the sizes of line, figures etc. would be that some of them were equal. It is unclear whether we could even say that e.g. one line would be longer than another, because we wouldn’t have any way to measure them: perhaps we could say that a part of one line would equal whole of another line. In such a geometry we couldn’t even differentiate between triangles and quadrangles, Hegel says, because their difference is essentially one of a number: of angles and sides. Even more so is all measurement of lines etc. prohibited, because measurement would require fixing of a unit and thus essentially move to what was called arithmetic. Notice that the unit to be fixed should be a line: only a line can be used to measure other lines. Hegel mentions that a point is in some sense a unit: at least it is indivisible. Yet, a point cannot be used to measure lines: in Hegelian mathematics a line couldn’t consist of points, but point could be only a limit of a line. We perhaps have in a

9 geometry a construction by which to construct a new point from a given point, but these would only determine a line as its endpoints, and the line so determined couldn’t be measured in any way as a sum of those lines: the line constructed would be continuous in the Hegelian sense of the word.

2./355. Arithmetic does not investigate numbers, but operates with them: number in itself is indifferent and it must be enlivened by connecting it to something outside it – this is calculation. Different species of calculation can be arranged systematically according to the characteristics of numbers involved, although this is not usually emphasized in books on mathematics.

Arithmetic does not analyze numbers, because there is nothing intrinsic in numbers to analyze: numbers are determined only by their relation to the unit. What we can do is to operate with numbers, that is, given some numbers, we can use some constructions to find or make new numbers from them. Note that Hegel is quite serious with the word operation: it is not like we would just point out some relationships that were already there – instead, these relationships could be said to be created with the calculation. Numbers themselves do not e.g. add up into new numbers: it is some external observer who has to come and reorganize the numbers in a new fashion. Calculations come in many forms, that is, there are number of constructions one could use to manipulate numbers. In the following paragraphs Hegel gives an ordering of the forms of calculations: some numeric manipulations are simpler, while others are usually defined through the simpler manipulations – e.g. multiplication is a development of addition.

3./356. Numbers are arbitrary collections of units: a number is an analytical figure with no internal connection between units. Counting is external production of new numbers: the difference of calculations can be merely external.

Hegel uses a curious expression that numbers are analytical figures. As we shall see later, analytical for Hegel means something with no transitions, except perhaps abstraction – something unmodal compared to synthetic as modal. A number is analytic because its constituents are independent units: these units are not connected by any necessity – it is not a law that given this unit we should be able to find precisely that unit – but only by our arbitrary decision of putting these units into one number. As external and arbitrary as the constitution of numbers are the constructions by which they are created and calculated. There is no great change involved in adding units together – just arbitrary change in the way how they are collected together. Although the operations or calculations can be ordered in some fashion, this ordering itself can be only as arbitrary as the operations themselves: it is mainly the external reasons – needs of common life and science – that determine which calculations should be pursued.

10 4./357. Number has unit and amount as its aspects, to which all qualitative differences in counting should be reduced: furthermore, as quanta, numbers differ in being either equal or unequal – identical or different according to magnitude – and this difference of essence is another differentiating facet of calculations.

The species of calculations can be separated according to two things. Firstly, every number consists of two aspects or facets, the unit to which it is related and an amount of these units. This is a structural difference within numbers and separates calculations using this difference – e.g. multiplication and exponentiation – from calculations not using this difference – e.g. addition. It is only the former sort of calculations which have some interest for Logic: indeed, they come up in the next chapter on quantitative relations. A second dividing factor is whether the numbers from which the calculation is to proceed are supposed to be equal or not: more precisely, it is the question of whether an inequality of the numbers used in construction is allowed. Note that Hegel does not mean by number, what could be called, a type of “1”, “2” etc., but a number token – a set of given amount of units, we could say: hence, it is possible that there are two numbers with same amount of similar units and that we e.g. add two numbers of same amount together – in “2 + 2 = 4” the two 2s are supposed to refer to different numbers. Note also that because the structures of equality and inequality – or similarity and dissimilarity according to quantitative determinations – are introduced only afterwards, in the study of essence, it seems that a true account of arithmetic should not precede the book on essence – at least we couldn’t organize the different calculations according to the criteria of equality and inequality. Still, we get the following division of calculations: a) calculations where the inner structure of numbers has not been taken into account and where thus a complete inequality of the numbers used is allowed (addition); b) calculations where the inner structure of number matters – where numbers are to be used as aspects of a new number – but where the numbers to be used can be unequal (multiplication); c) calculations where the inner structure matters and where numbers to be used must be equal (exponentiation).

5./358. Numbers can be found either by combination or by separation: thus, we can speak of positive and negative calculations.

There is yet one more dividing characteristic to mention: constructions can go in both ways, either from simpler numbers to more complex or the other way around – that is, we could either combine or separate numbers. With this modification we get a total of six possible types of calculations. It may seem strange that Hegel downplays the last criterion. True, in one sense it is just a question of mirror images of the same construction. Yet, in another sense the so-called negative calculations are far more complicated than positive ones. A positive calculation is a simple operation which always has a result, while negative calculations have more or a nature of finding an unknown – with what

11 number must this be combined in order to get that result – and might not always have a solution, at least if the numbers are restricted in some way. But we must notice that Hegel says negative calculation should start from previous combinations of numbers: the result of a negative calculation should then be always possible to find.

6./359. The first production of numbers is combination of many units into a number: as units are independent of one another, they must be presented through senses – we count with fingers, beads etc. The break in counting units is arbitrary. The arbitrary decision of what amount of units is taken as a new unit determines the numeric system (dyadic, decadic etc.)

Before we can operate with numbers we must produce numbers: this production was essentially what we did in the section on numbers and it is an essential presupposition of all calculation. What we need in order to make numbers is some way to produce units – actually the only interesting bit in the whole proceeding, but the arithmetic does not investigate it. The production of units is completely formal: we may use our fingers or beads to count, but we can also produce some meaningless signs, like points or lines, in order to get the required units. Next phase consists then of pulling some of these units together into a number or a set of units: Hegel describes this as a break in the counting. At least it should be completely irrelevant to the units produced, that is, we could have continued counting a little while longer: this possibility obviously presupposes the possibility of indefinitely large set of signs, either through infinite resources of matter or then through its infinite divisibility. Hegel notes also the fact that we can take some number as a new unity with which to continue counting anew: that is, after getting ten given units produced, we may treat every such decade as a new sort of unit. This possibility is clearly based on the fact that the state of division by which the unit is chosen is arbitrary, thus making Hegelian arithmetic a bit more flexible than a constructivism where we would have to at least in principle reduce all numeric expressions to units – at least such a reduced expression should be more essential: in Hegelian context there cannot be any such privileged level of units. 7./360. Produced numbers can be combined into further numbers: when it doesn’t matter whether the combined numbers are equal, the combination is addition. “7+5=12” or “7x5=35” is learned by counting with fingers etc., and then we have to memories it: this is made easier by arithmetic tables.

The first operation with numbers or addition is actually just a slight modification of the construction of numbers: perhaps this is also a reason why it is not mentioned outside this remark, as the making of numbers was already investigated in the previous section. The addition is characterized by the inequality of the numbers operated: or actually, by the possibility of the numbers being unequal. We

12 are given a set with units – say, 7 – and another set with units – say, 5 – and a task to create a set with all these units. Note that we must always think of the units of different sets as dissimilar, no matter how they are produced, i.e. although we would have originally introduced one number as a part of the other – in addition we abstract from the possibility that units in different numbers would be same units. How then the task of addition is completed, for instance, task of adding 7 to 5? Simply by counting seven units and then counting five more and stopping there. The operation of adding is already indicated in the problem and there is nothing problematic in it: we just have to do what we are asked to do. We can, undoubtedly, make the task easier through some gadgets. In the previous paragraph, for instance, we saw how the presentation of numbers could be shortened by taking some given number of units always as one sort of unit. Further way of simplifying arithmetical tasks is to learn the results of some basic operations: in that way we don’t have to count the result of some operation anew every time, but we can trust our memory – or a book of tables – to provide us at least a partial answer to the task.

8./361. Kant suggests that calculation is synthetic because we cannot by thinking resolve e.g. the sum of seven and five, but we must make use of intuitions. In one sense addition and making numbers in generally is synthesizing, but because the arbitrary task wholly determines the result it is in another sense analytic.

As always, it is difficult to decide whether Hegel is with or against Kant. Furthermore, it is particularly difficult to say whether his possible criticism of Kant is justified when we remember the particular nature of Hegelian arithmetic. We will see later that Hegel associates analysis mainly with what could be called unmodal investigation of situations and objects: analysis may show that a given situation consists of some further subsituations, but it cannot connect it to any further situations outside it, that is, it cannot produce any new possibilities. Synthetic method should then be characterized as essentially modal: it shows connections between different situations and contexts. We might also say that synthetic method changes the situation or context, while analytic merely studies it. Hegelian concepts of synthesis and analysis differ then from what Kant calls synthetic and analytic in relation with judgments. Yet, it seems that Hegel should then have admitted that arithmetic also used synthetic method: in calculation we change the situation, first, by creating new units, then, by rearranging them into different sets. We may forget the first problem, because creation of the units is clearly not a headache of arithmetic: arithmetician just presupposes that she has the units she needs in her calculations. Second question seems more problematic, and Hegel indeed accepts that in this sense arithmetic does some synthesizing: it takes units and makes numbers from them. Yet, this making of new sets is not the problem of arithmetic, we could say. Arithmetician works with so clearly defined

13 operations that she does not need to wonder how to unite seven and five into a new set: the leap from 7 and 5 to a set connecting their units is not the problem. What is the problem then is to find out how much units there are in the result created by this mechanic operation. This investigation is not synthetic, but consists of mere analyzing or cutting up the situation involved: we count the units in the situation and find out there are twelve of them. Thus, Hegel is correct in saying that arithmetic is in some sense analytic.

9./362. The apriority is also insufficient determination for counting: for Kant, pure aposteriority belongs only to sensations, while counting involves abstract intuition of structures with mere units. Apriority and aposteriority are relative determinations: feelings are in some sense a priori – they are determined by our drives and constitution of our senses – while space and time as structures of objects are in some sense a posteriori.

Hegel is here perhaps a bit unfair towards Kant, who might have admitted that a priori and a posteriori are in most cases only abstractions out of concrete cognitions, although these abstract elements can be presented in separate judgments: in every concrete cognition, a posteriori and a priori elements are combined. Yet, there are elements in Kant’s account Hegel would not appreciate. For instance, the idea of assigning some elements of cognition to objects and some to the faculty of cognition seems for Hegel a doubtful explanation of the necessary connections found in experience as a fact. Furthermore, Hegel probably wouldn’t put so much value to what could be called pure cognitions – that is, cognitions of pure space and time – as Kant does: mathematics does not stand high in the list of Hegel’s favourite sciences. In Kant’s model of cognition, the objects provide only the matter of the cognition – the sensory input – while the faculty of cognition provides all formal elements, beginning from space and time and ending with categories or universal structures. Hegel – who is not interested in assigning these elements to object or subject at all – sees here a more of a continuum of different levels. Now, arithmetic fits somewhere in the middle of this continuum. It abstracts from all sensory inputs: or it is indifferent how the units feel, as long as we can separate them from one another. But arithmetic abstracts also from quite a number of categories: it is indifferent how the units are produced, for instance. All that is left is the fact that they are units: individual entities, at least contextually independent of each other. In fact, Hegel continues, most of our cognitions fall somewhere within this continuum. Feelings in the wide sense are perhaps nearest to what could be called a posteriori, but they also are determined by the constitution of the subject: we feel or sense things differently depending on our health, expectations, desires etc. Kant would undoubtedly say that these are not truly a priori elements, because they are not necessary, yet we could perhaps call the subjectively or contextually necessary. Furthermore, space and time are ranked as a priori by Kant: yet, as Kant himself would admit, space and time are cognized only as being filled by objects,

14 thus, a spatial and temporal cognition always requires something a posteriori.

10./363. Kant also insists that some basic geometric propositions, like that a straight line is the shortest distance between two points, are synthetic: straightness as such does not imply anything quantitative. But straightness means nothing but simplicity; in quantities, likes lines, this means the least possible. Geometry should leave the proof of its axioms to other sciences.

Kant’s choice of an example for a synthetic a priori axiom of geometry is admittedly poor one. Perhaps he thought straightness to be a sort of unanalyzable mark by which to recognize e.g. straight lines: then the connection between this mark and the criterion for being the shortest line would be new information, not implicit in the concept of straightness, and thus synthetic in Kant’s sense. Yet, we may question the assumption that straightness is unanalyzable qualitative mark of lines: we might imagine a person incapable of separating straight line from curved. Then the question arises how to define a straight line: although neither Hegel nor Kant could have known, this question of definition of straight lines is even more important in non-Euclidean spaces. Hegel’s first suggestion that it means simplicity of direction is not very helpful: the question of simple direction is as vague as the question of simplicity. Hegel’s second suggestion that straight line is determined only by two points through which it moves is also too indefinite: we still need some rule by which to determine a straight line given two points. Hegel’s final solution actually seems to be that the judgment “straight line is the shortest distance between two points” actually is or at least is involved in the definition of the straight line: ”find the simplest line determined by this quantity” means ”find the shortest line determined by this quantity”, similarly as the task of finding simplest number of a certain type means finding smallest number of that kind. If straight line is then defined as the shortest distance, Kant’s example has actually been shown to be analytic. If Kant had suggested the parallel axiom as an example of synthetic a priori judgment of geometry, Hegel would have had more difficulties. It is unclear how serious Hegel was in his demand that Euclid’s axiom should have a proof and what sort of proof he envisioned for it: at least we know it cannot be a mathematical proof. Perhaps he wanted not a true proof, but a construction of an example of parallels with the wanted property – that is, an example of Euclidean space. In this case, Hegel’s “proof” would be merely an acknowledgment that we happen to live in an Euclidean space.

11./364. Synthetic a priori judgement could be defined as a connection of contexts of identity and difference, which can be found in any intuition. In calculation, there is only a repetition of a single unit involved in mere external differences: similarly, a straight line as the shortest line between two points consists only of one context.

15 Hegel’s definition of Kantian synthetic a priori judgment would not be accepted by Kant himself: something similar we saw in the paragraph 130 where Hegel used too strong concept of Kantian synthetic unity of self-consciousness. Kant could accept the other side of the Hegelian definition – that synthetic a priori judgments connect things which appear to be manifold – but not the other aspect – that they derive many aspects from one apparently unified and not manifold object. Hegel’s definition coincides more with his own Logical method that not just unifies manifolds, but also creates them: how this is done should be familiar to us by now. Now, it is this second aspect of his method that Hegel finds unsatisfied in Kant’s examples: they do not create enough diversity. Counting is in some sense creation, namely, creation of new units – it is a synthesis in this sense – but this production of units is actually more of a multiplication of one unit in many different situations. Furthermore, the combination of the units happens quite arbitrarily, merely by collecting some of them into one set. In addition, the discovery that a straight line is the smallest line between two points is even further from a true synthesis, Hegel says: it is merely a piece of analyzing what is meant by straight line. A bit of synthesizing would be involved in the construction of the line, although even more would be evident in a construction of a curve, which is not limited to one straight line or direction.

12./365. Addition as a positive operation corresponds with subtraction as negative: it is separation of a number into possibly unequal numbers.

After the tour to synthetic a priori judgments, Hegel abruptly returns to the classification of arithmetic operations. In addition we combine two or more numbers or sets of units to produce a new number: it is indifferent whether the numbers in question are equal or unequal. Now, for every such addition there is a corresponding operation of subtraction – or actually there are many of them. In subtraction we are given a set produced by addition – thus we do not need to worry of negative numbers yet – and we are given the task to separate certain smaller set from it, the result being a set which together with the separated set produces the original set.

13./366. 2. Of multiplication. Next we could assume that the numbers to be added are all similar: then they are of the same unit, which differs from their amount. Multiplying is an operation of counting together an amount of units, which are amounts in another context: it is indifferent which of the two numbers is taken as amount and which as unit. Originally results of multiplication have been found by fingers etc. and then the result has been memorized.

Out of the blue appears number two without any preceding one: undoubtedly it should have contained the operation of addition, as the number two contains multiplication. While subtracting was merely an addition backwards, multiplication is a further modification of addition. At first we

16 decide that the numbers to be added should be all of the same size: e.g. they are all fours. Now, we can take these fours as a new unit, and there is a certain amount of these units: say, three fours. The task in multiplication is to connect these three fours into one number: we could actually just make a set of three fours, but usually the task of multiplication ends with returning the division of the quantities to the original units – in this case, to units which are twelve in this quantity. In multiplication, different numbers play different roles: one is the unit and one the amount – although the roles can clearly be changed without a change in the result. Although the multiplication is a bit more complex operation than addition, it is still fairly mechanical: one needs only to add the given units to themselves for a given amount of times. The result of a particular multiplication can then be memorized, in order to simplify future calculations.

14./367. The corresponding negative calculation is division: both the divisor and the quotient can be taken either as a unit or as amount.

The operation of multiplication can be followed also the other way around – that is, we can take some given result of a multiplication and find the other factor of it, when the other is given. This counteroperation for the multiplication is of course division. Because the operation begins with results of a multiplication, we need not yet worry of the possibility of fractions. In the multiplication we could give different roles to different factors – one of them being unit and one amount of these units. Now the same roles can be assigned, firstly, to the factor given in the task – the divisor – and the result of the task – the quotient. As the assignment of the roles was arbitrary in the multiplication, so it is also in the division: we can ask either what amount of this unit makes the number or what unit is needed in that amount to make this number.

15./368. 3. Exponentiation. In multiplication, the unit and the amount can be unequal, but if they must be equal, then the multiplication is exponentiation (its counteroperation is finding a root), especially squaring: further potencies are mere formal repetition of factors, and when using odd exponents, results in further inequalities. Because everything of importance is included within squaring, all polynomial equations of higher degree must be reduced to quadratic equations: in equations of odd degree, the reduction might involve imaginary magnitudes – similarly all geometrical figures must be reduced to right-angled triangles.

Final modification of multiplication happens when we declare that the amount of units to be added shall be same as the amount of a unit, i.e. that the factors of the multiplication shall be equal. This construction is obviously squaring of a number. Its counteroperation – finding a square root – is only briefly mentioned: here we begin from some square – thus, no fear of irrationals here – and discover the number of which it is a square. One may ask why Hegel does not mention logarithm as

17 another possible counteroperation. In a mere case of squaring there, of course, would be no need of further logarithms: the answer could be only number two. Furthermore, even with higher exponents involved, the logarithm wouldn’t be a mere operation resulting to ordinary numbers in Hegel’s sense: the exponent found with the operation of logarithm would describe more a level of operation than a number of units. Squaring is the final phase in the modification of addition: here we have a difference of unit and amount, but in such a way that the quantity of the unit and amount are same. Further modifications could be made, Hegel allows, but they would not bring out anything new to this classification: further exponentiations would merely add the amount of the factors of multiplication and there would be no rule to determine this amount. Hegel’s attempt to connect the primary nature of squares in his classification with certain properties of polynomial equations is of mostly speculative nature. Hegel knows that as yet no one had been able to provide a general solution to polynomial equations of degree higher than 4 – in fact, it had already been shown that such a general result did not exist – but that in some cases the equation could be returned to equations of lower degree. Furthermore, he is aware that the general solution for cubic equations required sometimes the use of imaginary numbers or squares of negative numbers. Yet, on these flimsy grounds it is perhaps too hasty to conclude, like Hegel wants to do, that the quadratic equation is a general model for all equations of larger degree.

16./369. In some sense it would be correct to investigate numeric relations before exponentiation, because like multiplication, numeric relations depend on the difference of unit and amount: yet, in numeric relations the quantities are not just independent, but related to other numbers.

The operations considered here have been restricted to the realm of what could be called positive natural numbers – with the exceptions, firstly, that there is never a question of our having contact with the whole infinite set of these numbers, and secondly, that there is always the possibility that another, smaller unit would be selected and the number system so changed. If we ignore the negative numbers and the zero – which would be particularly hard to add into the Hegelian system of numbers – we could have introduced fractions as indicators of quantitative relations at least when we arrived at multiplication and division. Yet, undoubtedly these fractions would be something different than regular numbers – they could not be described as sets of units – and thus they could not be officially introduced at this stage of construction of mathematics.

17./370. The preceding discussion is not a philosophy of calculation, because it has been reigned by arbitrary classification: philosophy must know where it cannot be applied and where we must use such external guidelines as equality and inequality – matters would be confused if ideas were forced to things they cannot apprehend. In arithmetic

18 we need only external classification: therefore it is fit for elementary studies.

Hegel ends the remark with a note that the classification of calculations has not been a proper part of philosophy. Now, the difference between proper philosophical discussion and the discussion of calculations can be somewhat difficult to understand, because there are apparently not so many differences between them. In the main section preceding the remark Hegel constructed – or reconstructed – a model for a potentially infinite number system, while in the remark he has similarly constructed a set of methods that model the arithmetical calculations in a fairly accurate fashion. Hegel’s remark that the classification of calculation would not be immanent is rather unexplanatory, because parts of Logic itself suffer from the same guilt of arbitrariness. For now, the difference seems then to be more of a degree. The construction of calculations seems more arbitrary than the construction of numbers: the former does not bring anything ontologically new to mathematics, but merely new methods by which to manipulate mathematical entities – thus, Hegel says, a student can easily learn arithmetic, because in it she merely has to learn some set of operations or things to do, while the question of the ontological status of numbers would be too hard for a youngster to conceive. Consequently, Hegel will accept the construction of quantitative relations as a part of his Logic: these relations form another class of entities altogether. Generally, it is the introduction of quantities that Hegel sees as the final step into the arbitrariness that doesn’t belong to philosophy. Of course, we could model situations of different sizes with the method of Logic, but such an addition of objects of already familiar kind would not be ontologically interesting.

Remark 2.

The final paragraph of the previous remark already paved the way for the subject matter of this remark by noting how calculations couldn’t anymore be considered as part of philosophy proper. This specific note is now generalized to comprehend all mathematical notions. The question of the relationship of mathematics and philosophy has, undoubtedly, been heated from the beginning of the philosophy, but it was especially the mathematical speculations in the philosophy of nature in German philosophy of his times that perhaps occasioned this remark. Hegel’s attitude can be easily discerned even from the just mentioned paragraph: philosophy is no place for mathematical notions.

1./371. Pythagoras used numbers to express abstract structures and mathematical means, like exponents, have been used in this manner also in modern times. Pedagogically numbers have been seen as a best example of matters of spirit.

It is doubtful whether Pythagoras truly was the person who instituted the idea of numbers as

19 ontological structures to Greek philosophy – he was perhaps only a fitting sage who was later hired as the spokesman and symbol of this philosophy of the so-called Pythagoreans, as Aristotle called them. Yet, in Hegel’s time it was still almost a tradition that Pythagoras based his philosophy on numbers – that the world was supposed to be constituted by numbers: apparently this was meant to say that numbers expressed some basic structures of all things. Plato and the Platonists, of course, took this Pythagorean view seriously, and a trend of Platonist mingling of mathematics and metaphysics can be discerned more faintly or more clearly in the philosophy of modern times, at least up to Hegel himself. Indeed, it is the so-called Schellingian school of thought to which Hegel himself at one time adhered that is the attempted target of this remark. This is clear from the mentioning of potencies: Schelling used this expression to indicate difference of levels in the constitution of the world: similarly as squares and cubes – or planes and solids – so also inorganic and organic can be differentiated. The value of mathematics in learning has also been a common element in philosophy for a very long time. Plato, once again, began the trend by noting that mathematical entities are the first abstractions human consciousness is able to handle. In modern times, we can also see a similar trend in the so-called rationalist philosophy, which took the mathematical method to be the primary example to follow in the study of philosophy also. Even Kant, who clearly knew that philosophy and mathematics had different aims and methods, admitted that mathematics was at the same time most certain and most informative of all sciences, being thus far in advance of philosophy. Hence, it seems a poor choice for Hegel to go against the trend of philosophy so far and downplay the importance of mathematics both for the philosophy and for higher education: we shall see in a moment how he justifies his view.

2./372. Number was the most determined state of quantities, but it was completely arbitrary how a quantity was numbered [, that is, to which unit it was compared]. Arithmetic is analytical, because it investigates a mere external construct which has no nature of its own: arithmetically viewed objects have only arbitrary and relative characteristics.

Hegel finally enlightens more his previously somewhat unclear idea that mathematics was incapable of philosophical presentation. Number system was the most determined form in which quanta appeared: in it, all the quanta had a clear value which separated it from other quanta. Now, this determinateness of a number system was still arbitrary or context dependent: the value of a quantum depends on what is chosen as the unit. Thus, all properties of numbers as mere numbers are in a sense constructed by us: a certain quantum wouldn’t be e.g. two meters if we hadn’t chosen an appropriate unit of meter. Numbers as mere numbers have no properties of their own that we could investigate or model with the method of Logic, and this lack of essential properties is inherited by all studies of merely numerical fashion, e.g. counting distances between different planets etc. Note

20 that this lack of philosophical interest does not concern ratios between different quanta, which could well be characterized by the nature of objects themselves. Hegel repeats the judgment that arithmetic is merely analytic: after we have done the arbitrary assignment of units and thus created a fixed model of numbers, arithmetic has no other task, but to investigate this one model which has no indefinite plurality of aspects to be synthesized, as every concrete object has.

3./373. Structure of number is, as a general structure, an abstraction in comparison with any sensuous experience: it shows nothing but the form of relativity essential to sensuous experience – on the other hand, it can be seen as an indefinite repetition of method.

The previous paragraph demarcated the category of number – that is, the general structure for all numbers – from all other categories. Number is something that is determined only by an arbitrary choice of a unit or reference point. It is supposed to be externalization of thought. Thought here refers obviously to the method of Logic, but what of externalization? In Hegelian phraseology, externalization (Entäusserung) refers often to a movement by which something is created in the objective world, and occasionally, in which one gets lost to these created objects: for instance, work is generally an example of externalization of consciousness, and the second aspect is present when the work is related to an incapability of recognizing oneself in what was achieved by the work. Here, the exemplary system of numbers is obviously a construct of the method in Logic: it is constructed by taking together units created by the method. Furthermore, this construct is in a sense unfamiliar to the method, because it is merely relative: there is no right or wrong interpretation, but an indefinite number of possible numeric expressions for one quantum. An exemplary number is created by repeating the method of creating units, but there is no reason why we should stop the repetition at any particular time. Now, the structure of number shares this externality or relativity with sensuous experience. We are accustomed to measure things in relation to our own measures: a mountain looks big, but a molehill small. Yet, nothing actually prevents us from choosing another reference point, and indeed, this is done when it happens to be convenient. For instance, for measuring the length of a sofa, a kilometer is way too long unit. On the other hand, in galactic measurements it is usually too small and insignificant. The most appropriate place of application the kilometer has in smaller distances, e.g. the distance between home and office. This sort of relativity is common to numbers and sensuous experience, but the experience has also some other relative properties: there is no reason to choose one spatial or temporal reference point over another. Despite this similarity there are also considerable differences between the category of number and sensuous experiences. Foremost is the fact that the category or general structure is general: categories and experienced objects occupy different levels. An experienced object has much more characteristics than mere relativity: it is

21 green, round etc. It is only when we abstract from such other qualities that we receive an idea of what number is.

4./374. Even the ancient philosophers said that numbers formed a third realm between thoughts and sensuous experience. They used numbers in philosophy only as the final reserve, and even Pythagoreans tried to change their language to non-numerical. Thus, it is sad when nowadays mathematical – and even poor mathematical – expressions are used in philosophy.

Once again, it is unclear whether there is much historically accurate in Hegel’s account of Pythagorean’s teachings: the earliest references of Hegel are only from the time of the NeoPythagoreans or Neo-Platonists. At least the sequence of philosophers in one lecture series of history of philosophy – that philosophy began with sensuous experience in Ionians, developed through study of numbers in Pythagoreans, until it finally found its own area in ontological studies of Parmenides and Heraclitus – has no base in reality, because most of the aforementioned thinkers lived about the same time. But the Pythagoreans – or more generally, ancient philosophers – are here only a helpful support, firstly, for Hegel’s idea that numbers share the properties of both sensuous things and categories, and secondly, for his criticism of modern, especially Schellingian philosophy and its use of mathematical determinations. It is particularly the latter agenda that controls Hegel’s discussion in this remark. Even ancient philosophers were not so dim-witted as to think that numbers or mathematics in general could express philosophic truths or concepts: even Pythagoreans understood to use them only as mere symbols of true ontological structures. As a modern equivalent of Pythagorean and Schellingian heresies, we could perhaps mention the three categories of Peirce, aptly named firstness, secondness and thirdness: the supposed origin in the theory of relations is not one whit better than Schelling’s idea of potencies.

5./375. We can relate numbers to multiplicity, because multiplicity is also a simple category belonging to sensuous experience: more concrete thoughts cannot be meaningfully related to numbers.

Undoubtedly numbers could instantiate some basic structures: even Hegel admits as much. We can, for instance, take any number or a set of units as an example of a multiplicity. Furthermore, a unit number could instantiate e.g. unity. The problem is that not all structures can be expressed through numbers. What Kant had called dynamical categories are a perfect example. Let us take causality, for instance. We could not say that in or between numbers there would be some causal or even explanatory relations: although a set of two units contains a set of one unit, the latter does not explain the existence of the former. At most we could say that a set of one unit is a condition for a set of two units. We still need something to produce a set of two units from sets of one unit, namely,

22 the method or operation of adding two numbers together: it is only this method that we can perhaps say to explain the existence of certain numbers in a number system. We could say something like “three symbolizes causality”, because a causal chain needs at least three elements: a beginning state, an end state and an operation or regularity leading from one to the other. Then again, we might take three as a symbol of a traditional family, because there we have three roles of a husband, a wife and a child; or three could be a symbol for triangles etc. But here the relation of a number and a ontological or concrete structure would be reversed from the relation of a number and multiplicity: the three is instantiated in all of these structures, because they are structures with three elements. Because such structures or situations abound, three could be used to symbolize quite a lot of things.

6./376. It makes thinking difficult when its structures have been designated as numbers: it is hard to understand that one thing can be identified with three different things. In common life we may take this as a proof of the contradictoriness of speculative truths: in fact, such a proof concentrates on inessential questions.

The choice of what to count as a unit is arbitrary: this is part of larger choice of what to take as the criterion of identity. Now, there are two choices. We can remain in our arbitrary, but determinate choice and investigate what sort of entities we find in that context. In this sort of study, one form of which is mathematics, the numeric values of quantitative things cannot change: if there is a unity, then it cannot be replaced by a trio. This definiteness is what the common reasoning upholds against the speculations of e.g. trinity: a unified entity with three persons. But there is another possibility. We can change our arbitrary choice of unit or identity criterion and determine the basic objects anew: instead of e.g. book we may count pages. Thus, a unity can be seen as a trio – what was earlier seen as aspects of one person can be seen as three independent individuals – but this is not a very interesting connection: it is a particular instance of a relation between unity and multiplicity.

7./377. It is harmless to use numbers and figures as symbols, but any wisdom in them must be presented also in clear language.

Hegel’s attitude towards symbols in general is non-appreciating, thus, it is no wonder that he does not value mathematical symbolism very highly either. A symbol is a mere indicator for its meaning and usually depends on some similarity between the structures of the symbol and the symbolized: of course, one may fail to understand what characteristic of the symbol should be found in the symbolized object or meaning. The passage from symbolism to thinking requires two steps. Firstly, we should translate the symbolism into clear language. Words do also refer to meanings, but the relation between a word and a meaning is not based on some supposed similarity, but to a mere arbitrary convention. Secondly, we should be able to ourselves find or construct an instance of the

23 meaning presented in the symbols. For instance, the eternity symbolized in a circle or snake eating its own tail is understood better when we note that we can interpret every moment as a sort of eternity; or, the possibility of three persons being a unity is better expressed by our capability of changing identity criterions than by an unclear symbol of a triangle.

8./378. Mathematics or other applied sciences should not be used as a method for logic, but logic should precede these sciences.

It is places like these that raise skepticism against the possibility of mathematizing Hegel’s Logic: if the master himself denied it, why should we attempt it? We have already spoken something of the possibility to “formalize” Logic and found nothing wrong with the idea: indeed, we could present it as an abstract game of making models for situations with strict rules of how to change situations. Now, what Hegel is against here is taking mathematics as the first in order in relation to philosophy – and mathematics, for that matter, which includes only some simple manipulations of quantities: it would be futile to say that e.g. construction of certain abstract structures could be reduced to numeric constructions. Such mathematics clearly is no study of basic methods: it is different in the modern mathematics which includes such areas of interest as game theory, for instance. Logic undoubtedly shouldn’t lend anything from other sciences: otherwise, it wouldn’t be the first science giving the basic structures of all entities. Instead, in it we should be able to construct everything from any given information – or from no given information, whichever description one prefers. Yet, its independence shouldn’t prevent it being generally scientific in nature, that is, using some strict methodology, capable of being expressed in clear rules, and even mathematical in the sense of making its constructions mostly at the level of meaningless signs capable of clear manipulation.

9./379. Numbers are a good way to introduce youth to the study of non-sensuous abstractions; yet, it also uses only mechanical thinking and makes men into machines.

Hegel’s attitude towards mathematics as a pedagogical device seems non-appreciatory. Mathematics is a good tool in elementary schooling in that it helps the student to take her mind away from concrete instances to generalities: for instance, it is not the five apples and seven apples that matter, but only the five and seven instantiated in this case. Yet, mere learning of mathematics does not suffice: an engineer who knows nothing but some calculatory routines has not developed her abilities as far as she could. The engineer can only repeat what has been taught to her, but she can’t create anything new. Thus, a full education of youth should also contain study of culture and history, which help one to grow into a fully rounded human being. Note that the mathematics Hegel is talking of here does not mean what we would or should nowadays call mathematics. It is more of

24 mathematics of engineers and shopkeepers than true scientific study of mathematics: it is applied and not pure mathematics. Hegel is thus not making a demarcation between mathematical and humanistic studies, but between merely practical learning of applied operations and theoretical learning of pure study: modern mathematics would undoubtedly belong to the required part of the study, as being helpful in understanding e.g. wider connections between areas that Hegel calls mathematical.

B. Extensive and intensive quanta After the relatively clear structure of the section on numbers – which reconstructed numbers in base of Hegel’s own Logic – Hegel now presents us with a bundle of different issues wrapped in one package. The title would suggest that the main issue would be to describe basic species – or interpretations, as we will later see – of quanta, and indeed, two-thirds of the section work on this subject matter. This two-thirds in itself show us a familiar structure: first two concepts are introduced and then they are shown to be connected – difference being that this movement now happens in two instead of three sections. After these two sub-sections Hegel has added a third that works as a sort of analogy for the similar section in a previous division in the chapter on Dasein: they both make a transition to infinities, there to qualitative, here to quantitative. Thus, the task of this section is not just to describe extensive and intensive quanta, but also to show that in some sense we can find contexts in which it makes sense to speak of infinite quanta.

a. Their difference

The first subsection introduced the concepts of extensive and intensive quantum, which played some role already in the philosophy of Kant – form of intuition being extensive and matter of intuition being intensive quantum: of course, Hegel must characterize them in somewhat different manner, because he cannot base them on the cognitional framework of human reason. Like the whole section in which this subsection is imbedded, the subsection is of somewhat disorganized nature: which is, of course, good reason to stop taking the trivisions of Hegel seriously. It has its own subtrivision, which is of a highly peculiar nature. First, extensive quanta are introduced – or actually we are merely told that numbers already are extensive quanta. Then we are shown how intensive quanta can be constructed from extensive quanta: although this would strictly speaking belong to the next subsection. But what then can the third step do? We would expect a construction of extensive from intensive quanta, but this is already reserved for the next subsection. Thus, the third step seems to consist of analyzing relations between different intensive quanta or degrees, while the

25 first step analyzed degrees in separation.

1./380. 1. Extensive quanta. Quanta are determined by the amount of the units: quantum is a plurality that cannot be separated from certain division of units – an extensive quantum.

We began this chapter by a quantitative state of being-related-to-other-quantitative-situations: then we fixed some division of units – a reference quantity, so to speak – and then we could fully determine what number of units this quantitative state had: we are speaking of meters, thus, there are five meters. Now, nothing new is actually added to this idea of number here: the quantum is taken as a situation or even set with certain number of units and this number of units determines the quantum completely. We are merely given a new name: this number could also be called an extensive quantum. In Kant’s transcendental philosophy extensive quanta were defined through the nature of human cognition: extensive quanta required space or time in order to be constituted in our experience and belonged thus to the form of intuition. Thus, an extensive quantum for Kant was something that could always be divided into pieces. Hegel’s definition leaves out the epistemological or transcendental side of Kant’s definition, but leaves the idea that extensive quanta consist of smaller pieces: extensive quanta are sets or collections of units.

2./381. The difference of extensive and intensive quanta is not the difference of continuous and discrete magnitudes: the former concerns the difference of related quantities, the latter only the difference of quantities in general. Continuous quantity has not yet been divided, while in a merely discrete quantity the continuous unit has not been determined: they are combined as aspects in number. Numbers are extensive quanta, although they are more determinately plural.

It seems remarkable that Hegel would even think that someone would suggest identifying extensive quanta with continuous quantities: extensive quanta are sets of units and thus a discrete plurality, while intensive quanta, as we shall see, are more of undivided unities. Yet, we should remember that Kant connected extensive quanta with space and time: they were, for Kant, especially geometric magnitudes. Similarly, we have seen that a close connection between geometric and continuous quantities had been suggested, thus making it natural to identify extensive and continuous magnitudes. Hegel at once notices that this identification cannot hold, because extensive magnitudes are also explicitly discrete: they can be counted. Thus, in the case of geometry, extensive magnitudes are measured lines etc., while continuous magnitudes are still unmeasured. Similarly, extensive quanta are not merely discrete, because we explicitly know that it has a single unit of measure that is merely repeated to produce some given length. Generally, continuity and discreteness are mere ways to interpret quantities, even when they are not related to other quantities: whether they are taken as a unity or as plurality. Extensive and intensive, on the other hand, are

26 ways to interpret quantum’s relation to other quanta. Hegel makes a puzzling note that numbers differ from extensive quanta in being more of a plurality than them. It seems likely that the difference of extensive quantum and a number is meant to reproduce the difference of continuous and discrete quantities: an extensive quantity can be measured, but it is primarily taken as a unity, while a number is primarily taken as a discrete multiplicity. This difference is, at this stage, undoubtedly only one of viewpoint or context: it is not a strict division of quanta, but a moderate scaling of them.

3./382. 2. Intensive quanta in themselves. [In a context of a number system] we don’t need other quantity to determine the magnitude of this quantity: in a number this independency is posited as an independent set of units. All of these units are copies of one unit: they can thus be taken merely as aspects of one continuous unit – set of units can be abstracted, while the numerical determination of the quantity remains.

Hegel’s first statement must be taken with a grain of salt: when we have determined a certain number system by fixing the unit, then we can determine a quantum without any reference to other

27 quanta – of course, there is the implicit reference that a number is always of a certain size only in some particular number system or division of units. Thus, although set of units is independent of other units – we don’t need to look at other lengths to measure this length – it is still possible to divide the units further and thus in a sense “multiply” the size of a certain number – we still need the measuring stick of a certain size. The set of units contained in an extensive quantum consists, now, of similar units: no unit differs qualitatively from others, or at least such differences are viewed as inessential. Because of this similarity, we can then, Hegel says, take the apparently differing units as merely one unit in different situations: they are mere aspects of this one unit. Although this construction or change of interpretation should be familiar by now, it never strikes to amaze reader. In this case, we take a quantity arranged in what could be called spatial manner – like a meter, it consists of independent parts – and interpret it in what could be allegorically called a temporal manner – like a change of some underlying object, one thing appears in different situations. Yet, it is based only on the fact that in some context these objects cannot be differentiated: they just seem like the same object. Instead of five different units, we now have one unit repeated five times – and this is explicitly the fifth repetition. The order has taken place of amount – an important aspect of the transition, as we shall see.

4./383. Extensive quantum was determined by a certain set of units, but now its determination has been changed into something unitary: quantum has been made intensive and its limit has been constructed as a simple degree.

Quantum or its limit – remember that, as there is yet possibly no objects to speak about, these mean actually the same thing, or at least are aspects of the same thing – have been interpreted anew. What was once a set of independent objects or states of being is now a simple situation with only aspectual differentiation. Hegel calls quantum according this new interpretation an intensive quantum or degree. The former name especially reminds us, once again, of Kant. When extensive quantum was for Kant the quantitative structure of the form of perception, the matter of perception or sensation is characterized as an intensive quantum: it is not something spatial or temporal, but only something within space and time, that is, something possibly momentary. Hegel, of course, cannot explicate intensive quantities in a similar manner, thus his preference of the second name, the grade: grade is something indivisible with a certain place in a continuous scale of grades, as we shall soon see. 5./384. Degree is a determined quantity or quantum –it can be expressed by a number – but it is not a set of independent units: “ten grades” refers to the tenth grade and not to a set of ten grades – independent units have been integrated into a

28 unity.

If an intensive quantity cannot be characterized by its role in our cognitive framework – that it is, as Kant says, the quantitative structure of momentary sensations instead of spatio-temporal intuitions – then its characterization must be taken elsewhere. Now, an extensive quantity was characterized as a set of independent units: as a number, it would be a cardinal number. Intensive quantity or degree, on the other hand, should not be understood as a plurality: if I speak of twenty degrees of Celsius, there is no literal set of Celsius-degrees to which I could refer. In this case, there is actually only one grade of heat that happens to be twentieth in a certain scale. Intensive quantity, as Hegel sees it, expresses thus a place in some ordering: an ordinal number, we could well say.

6./385. 3. Scale of degrees. As a number quantum is constructed as determined: as a degree it is constructed according to its essence, because degree is in one sense independent, but in another merely relative. Number is a set of independent units and thus dependent of something else; in degree the independent units have been abstracted of, and still, quantum is determined only by some amount of units. Thus, the units must be outside the degree: in one sense degree is independent of everything external, in another it is not.

Hegel makes an interesting note that extensive quanta or numbers are quanta in their determined form, while intensive quanta or degree are quanta as they are in themselves. We have already heard the statement that numbers are quanta in their determined form – in a number, we have chosen some particular state of division and thus we are capable of expressing the quanta as a set of units. Now, according to its concept quantum is something that is, in one sense, an independently determined structure, and in another sense, dependent in its determination on other structures or entities. As a set of units, quantum gets its determination undoubtedly only from the connection to the unit: it has its “externality in itself”, that is, its structure already belies its reliance on something else – a set of units is merely an external collection of independent units. When we have interpreted quantum as intensive and abstracted from its possible division to units, it seems more of a merely self-related object or structure: a temperature does not consist of anything nor is it an arbitrary collection of anything. Despite this independency, a degree is also in another sense connected to something else beyond it – thus it realizes the concept of a quantum. This is the fourth degree of temperature: this sentence determines the degree completely. Yet, this determination requires an implicit reference to an external framework stretching beyond this particular degree: there must be some degree in relation to which this degree is the fourth. In fact there must be at least a series of four degrees of which this degree is the last. It is only when we have constructed such a scale of degrees that we can assign definite places to the individual degrees: the independency of a degree is always

29 connected to an arbitrarily assigned relational framework.

7./386. In this way quantum has realized its concept: its determination is indifferent or only relative. A degree is one of many in one sense independent degrees which are in another sense essentially connected to one another: given any degree, we could always find a larger or smaller degree.

This paragraph is mostly a repetition of the previous: a degree is on hand independent of all other degrees, on another hand it is related to others and its determination is only relative to the whole framework of degrees. What is new is the emphasis on the so-called change of degrees: as usual, we shouldn’t think merely of temporal changes, but also of a change of colour from one end of the rod to the other etc. The word “change” refers to two things: to the relatedness of degrees and to the relativity of degrees. Of the second characteristic we shall speak more later on, so we can concentrate on the first. Every degree is related to another degree, that is, when we are given a degree we can always find another degree, either smaller or larger. Every degree represents a place in some ordering – it is the fourth etc. Clearly then we must be able to find the previous degrees, in order to say that this is the fourth degrees. Furthermore, because this is a quantitative progression we should have the ability to find even further members of the series: “larger” ordinals, as it were.

b. Identity of extensive and intensive magnitude After introduction of numbers – sets of independent, but indiscernible units – we met extensive quanta, which apparently meant almost the same thing as numbers, except that in them the emphasis was laid on the continuity or indiscernibility of units: thus, while a set of gold nuggets would be a number, a length of a rope would be an extensive quantum. In practice this difference is only one of viewpoint. Then we were shown how these extensive quanta can be interpreted as intensive quanta or degrees: we could abstract from the fact that an extensive quanta consists (at least potentially) of some units and concentrate on the relation that the quantum has to the framework of quanta. The task of this section then would be to show that from an intensive quantum one could construct an example of an extensive quantum. Note the more relaxed nature of the transition to this direction, where the quantum in the beginning and the quantum resulting are not necessarily identical: or at least we must interpret and structurize the degree anew if we want to truly make it an extensive quantum.

1./387. A degree does not consist of independent and external units, but it is also not a mere undetermined unit: a degree is one of a plurality of degrees – it is determined when the contexts of oneness and plurality are related. Thus, we must be able to construct some amount from a degree, although this amount is external to it: a degree is twentieth, because

30 there are twenty degrees.

Hegel begins by reminding us of the two sides a degree should have. Firstly, a degree is not something external within itself: that is, it is not a mere arbitrary collection of independent units, like a number would be, and it even couldn’t be taken as such, like an extensive quantum could, because it is indivisible. Yet, it has a necessary relation to some amount, or we can construct an amount of units from it. Degree is defined by its place in some ordering of quanta: the measure of temperature is defined by its relation to other possible temperatures. Consequently, if the temperature is now, say, twenty degrees Celsius, then we must have at least twenty possible stages in the ordering from a zero degree to the current temperature. Note that the ordering of the degrees is also based on the division of the scale: if we choose a more fine-grained division, the placing of this degree rises numerically with the addition of new degrees, although it is, of course, not experienced as a higher temperature.

2./388. In one sense, a degree is determined by its relation to an external amount which it excludes from itself; in another sense, this amount can be interpreted as belonging to the degree – in this interpretation it is an extensive quantum.

It would actually be very easy to construct an example of extensive quantum, once an intensive quantum just described would be given. Because from a degree – like twentieth – we could always find a corresponding number of degrees – twenty degrees from zero to this degree – we could similarly found an instance of extensive quanta. Yet Hegel wants something more, that is, he wants to show that we can interpret this intensive quantum also as an extensive quantum: the amount of units should not be just something external to the degree. Hegel suggests that we could identify the intensive quantum with the amount of steps necessary to reach it from point zero. In this manner, the intensive quantum could thus be identified with a certain extensive quantum. For instance, suppose a certain degree of temperature, like 200 Kelvins. This temperature is a certain place in the Kelvin scale of degrees, but it could also be interpreted as the number of steps from the absolute zero point of the temperature. If this reinterpretation is accepted, then Hegel has succeeded in changing degrees into numbers: in the other case he must accept the weaker conclusion.

3./389. Extensive and intensive quanta are different aspects of quanta in general: in the one the amount of units is interpreted as belonging within the quantum, in the other it is seen as belonging to the framework of quanta. The plurality of units can be abstracted from the extensive quantum – it can be made intensive; intensive quantum can be interpreted as extensive.

31 Extensive quanta could be interpreted as intensive. Take some extensive quantum, like four meters. We could abstract from the fact that it can be divided to smaller lengths and still retain it as a sort of measure: we could compare it with other lengths and say that it is the fourth in the series of lengths with meter-units. Thus, we would have discarded what made the length into extensive – its relation of consisting or being divisible to, in other words, the length as a potential plurality of units. We have taken the length as an indivisible unit, and what remains are its relations to other such units – it is the fourth in the order of length degrees. Furthermore, intensive quanta could be interpreted as extensive. If we take any degree, we could identify it with a set of steps from the beginning of the scale to this degree: a fourth degree would be taken as a set of four steps. Thus, we would have taken the amount of units outside the degree and reinterpreted it as belonging “within” the degree – result being an extensive quantum. We thus have two possible interpretations of one quantum: if it can be seen as extensive, it can be seen as intensive, and vice versa. Thus, like discreteness and continuity were revealed as two aspects of all quantities, so extensiveness and intensiveness have been revealed as two aspects of all quanta: it is just the question of where we place the amount of units that determines quantum: within it or outside it.

4./390. Extensive an intensive interpretations are aspects of the same object, which could be arbitrarily taken according to any one of the two: because an object is something qualitative, its quantitative determination is only external to it. We could talk of quanta without mention of any substrate: yet, we have constructed an example of an object with a quantitative determination and with aspects of unity and amount.

With this identity arises the qualitative something: the identity here refers to the fact that a quantitative structure or situation could be interpreted as extensive and as intensive – extensive and intensive quanta are not necessarily always different. Similar transitions have been made when passing from Dasein and Fürsichsein to Daseinde and Fürsichseiende: we have noticed that same structure could be interpreted in two ways and thus we have gained the right to call it an independent object – something or one. A quantitative structure need not have any concrete object to quantify: for instance, a length of five meters need not be a length of any material object or collection of such objects, but merely a length of a situation or collection of situations – a space between two points. Yet, even if have merely such objectless quantities to talk of, we can still construct examples of objects, that is, we can take the quantities themselves as objects. Thus, we can also have examples of quantities that describe some objects: extensive quantities that are collections of quantities – e.g. a group of lengths – and intensive quantities that describe places of quantities in orderings – e.g. an ordinal place of a length in a system of lengths.

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Place in a number scale can be identified with a set containing unit steps from zero to that position

Remark 1.

In the previous sections Hegel wanted to show that an extensive quantity could be seen as intensive – we could abstract from the fact that an extensive quantity is a set of units – and vice versa – we could interpret an intensive quantity anew as a set of steps to it. In this remark it is mainly the second transition that Hegel is interested to exemplify, or actually, he speaks of a weaker version of that transition. Here it is not so much question of seeing certain intensive quantity as extensive, but of connecting it with a certain extensive quantity: Hegel’s aim is merely to show that we cannot speak of an intensive quantity without having a related extensive quantity by which to measure the intensive quantity. Thus, for instance, Kant’s definition of intensive quantity would in a sense be meaningless, because something could be known as intensive quantity only through the help of spatial or temporal, that is, extensive quantities.

1./391. Usually extensive and intensive quantities are separated as two species of quantities. Furthermore, extensive quantities in e.g. spatiality of matter are transformed into intensive quantities, because latter are supposed to be more dynamic.

The main target of Hegel’s criticism is the so-called dynamic theory of matter, instigated by Kant and upheld by Schellingian philosophers of nature – this theory supposed that matter should not be conceptualized as a set of material parts with pieces of void to explain the different densities, but as a degree of space filling forces: we saw something of this criticism already in the end of the section on Fürsichsein. On a surface level, it seems that Hegel should have no qualms against the idea, because it could well fit in with his own attempt of seeing things of nature more as precursor of

33 living and conscious things than as mere mechanical aggregates. Yet, it is Hegel’s conviction that especially in its mathematical aspect the dynamic theory differs in no essential manner from the mechanist-atomistic explanations of matter and density.

2./392. Firstly, differing independent parts combined into a whole should be replaced with an expression of a force: these structures shall be studied later. Structure of intensive force and its expression is more informative: still, the expression of the force has the structure of an extensive combination of parts.

The difference between dynamic and atomistic interpretations of matter is twofold. Firstly, there is the mathematical difference between intensive and extensive quantities, and secondly, there is the more ontological difference between understanding matter as an expression of forces and as a collection of parts. Now, the second difference shall become subject of discussion later, in the book on essence: thus, Hegel mostly ignores it at this stage. Still, he has something to say of the matter here. Hegel admits that the two views are in some sense different. The atomistic view admits only the external shell of the matter, where all we see is independent parts collected arbitrarily into an aggregate. The dynamism, on the other hand, suggests that we can also see the piece of matter as a unity: as a combination of some basic forces. In this sense dynamism is more informative than atomism: atomism fails to account the unity of material object, while it is explicitly stated as a basic force in the dynamism. Yet, the dynamism cannot rid itself of the seemingly arbitrary aggregation. The unity of a force is necessarily related to a side of what the force expresses, and this expression is just the atomistic world of arbitrary aggregates and collections of units – in this sense dynamism is no advance over atomism.

3./393. Dynamism also replaces extensive quantities with degrees: but we have seen that both are merely different aspects of same quantities.

It is the supposed mathematical difference between dynamism and atomism that we have the means to handle now: or actually, we have the means to show that there really is no difference to consider. Intensive and extensive quantities are merely two ways to interpret or use quanta in general. In atomism we are supposed to speak of matter as a collection of atoms, while in dynamism we should interpret matter as a degree of force in some definite place in an ordering of forces. Yet, in order that we could properly quantify the degrees of forces, we have to change them into extensive quantities: that is, we have to express them in a form of a collection of units. We have to measure the effects that the space-filling force has – for instance, by counting how large a volume a certain mass of matter fills. Hence, we would have landed back to extensive quantities, which the dynamism was supposed to avoid.

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4./394. Every quantum can be interpreted as extensive or intensive. For instance, a number is a collection of units or extensive: but it is also a unified number having a certain place in number system or intensive.

In numbers there seems to be most reason for assuming true identity between extensive and intensive quanta, or cardinal and ordinal numbers, as we should say. True, in modern mathematics we have to separate the two concepts, because the class of infinite ordinals has far more representatives than the class of infinite cardinals: ordering is in this sense more complex relation than mere amount. Yet, in the realm of finite numbers the two can be easily identified. Number two or the general structure of all pairs is the second number in a number system; and vice versa, the third number in a number system is the number three.

5./395. The unit of circle is called a degree, because it is determined by its relation to other parts of the circle, although as a spatial figure it is simply an extensive quantity.

Of all the examples Hegel gives of the so-called identity of extensive and intensive quanta, this seems to be the only one where the transition is made from extensive to the intensive side. Hegel seems to speak of a figure limited by a part of a circumference of a circle and two radii of the same circle. In one sense, this figure is only a figure: it consists of smaller figures and can thus be seen as an extensive quantity or some definite area. Yet, we can also abstract from the fact that it can be divided to smaller areas, and indeed, we can forget everything outside this certain circle: or actually, we can think of this circle as a circle in general. In this sense, the figure is a certain part of the circle and we can say that certain figure of the same type is a unit figure: let us say that there are 360 of such unit figures in the circle. Then the number that the figure has related to the unit figure expresses the degree of the angle between the two radii of the figure. We may indeed think that when the degree of the angle grows, then we are merely moving the ends of the same lines away from each other. If we interpret the number of the degree in this manner, we cannot say that the degree would consist of smaller degrees, although it is related to the framework of such angle degrees.

6./396. More concrete objects are extensive on the surface and intensive within: mass is an amount of pounds, but is also felt as a degree of pressure; and a degree of pressure is measured by seeing how many pounds it can move.

Hegel’s examples are ordered on a scale of increasing complexity. At start we studied a numerical example, then a geometrical. The next step is to speak of concrete objects, albeit only of their mechanical properties: at this stage at least the identity of the extensive and the intensive quanta

35 becomes more of a connection between two related quantities, unless these quantities are interpreted as identical. The quantity of mass or weight – Hegel does not clearly separate between the two – of an object can be interpreted in two familiar ways. Firstly, weight is felt as a sensation of pressure: we feel the weight of the object when we carry it around. This sensation can be larger or smaller – thus, it can be taken as a degree or intensive quantity. Secondly, we can measure this degree of pressure and thus it must have some way to appear through extensive quantities. This happens via the effects of the weight. When we set an object of certain weight into scales, we have to put to the other side a certain amount of objects of some unit weight: this amount of objects is clearly an extensive quantity.

7./397. Temperature has a degree, but this degree is measured by its effects on the extension of bodies or by the extension of warmed area.

The case of temperature, according to the natural science of Hegel’s time, was somewhere between purely a mechanical and a more concrete physical phenomenon: even Hegel was aware that motion and warmth have some connection. Nowadays we even know that warmth is nothing but motion: warmer matter means that atoms are moving more quickly and absolute coldness is an absolute lack of all atomic vibration. The modern science has thus proven that temperature truly is an extensive quantity. Yet, even Hegel succeeds in arguing that the intensive quantity of temperature – how warm we feel the object or the environment to be – is connected with some extensive quantity; this is even a prerequisite for us being capable of measuring temperatures. The change of the temperature causes some measurable effects on the extension of the bodies: increase of the temperature makes matter to take up more space, while a decrease has the opposite effect. Of course, we wouldn’t perhaps go as far as to identify temperature with the extension of the matter involved, but the connection between the two is so lawlike that we can use one as the indicator of the other.

8./398. A more intensive note makes more vibrations, louder note can be heard from farther away: more intensive colour can be used to paint a larger area in the same way, brighter colour can be seen from farther away.

Hegel’s examples move farther and farther away from the mechanical cases. Sound and colours are both sensations which can be characterized by two different quantities. Loudness and brightness are easily presented as extensive quantities, through the test of how far away they are sensed. Intensity of a sound is also a simple case, because Hegel is aware that sounds can be quantified through the vibrations they make. It is the fourth quality which is a bit of a problem. Science of Hegel’s time had not yet established that colours could be interpreted as vibrations of a sort: thus, Hegel has to measure intensity of a colour awkwardly through seeing how large an area one can paint with it

36 while producing a similar sensation. Note that Hegel passes over all biological cases and jumps right to the psychological quantification. Of course, even the latter is somewhat inappropriate, as we shall see in the next remark, and it perhaps was a conscious choice to leave the biological case out, because the life cannot be properly expressed in a quantitative form according to Hegel.

9./399. In spiritual cases, higher intensity of talent is connected with wider and deeper effects.

With every example we have come farther and farther from a strict identity of extensive and intensive quanta towards a mere lawlike connection between extensive and intensive quanta. When we step into the realm of spiritual or human, we face yet another modification. Intensity of a character or talent should be connected in a lawlike fashion with its effects. Yet, in order that we could meaningfully speak of a connection between two quanta, we would have to be able to measure at least one of the sides: obviously, the extensive side of the equation. But it seems impossible to measure e.g. the effects of a person’s artistic talent: we couldn’t give any reasonable scale for measuring the worth of artistic productions. The mere mass of products is the only thing we perhaps could measure – and one hardly has great artistic talents, if one has not produced anything – but this quantity seems to be no true correlate of the talent: surely it shows more talent to produce a number of masterpieces than a same number of mediocre works. Even in the case of some form of intelligence, where we can produce quantitatively measurable tests, external factors may contribute in determining the results of the test and thus are nor very reliable and certainly not strictly lawlike correlate of the intelligence of a person. This case exemplifies then an even more weak relation between the sides: not a relation between intensive and extensive quanta, but between what Hegel later calls inner and outer sides of a phenomenon: this distinction resembles the earlier between dynamistic and atomistic interpretations of matter. Hegel is willing to say that we can speak of what happens “inside” some phenomenon or object or what that object is “in itself” – in this case, the talents and characteristics of a person – only if this inside has some effects in the outside or in the way how we perceive the phenomenon or object – in the behaviour of the person in question: this criticism is targeted, for instance, against romantic philosophers who thought that a person could be a genius or a saint without taking any action in the common world. Furthermore, the example might be taken as an instance of negative criticism. We might say that consciousness has no intrinsic connection with quantities or its essence is not measurable. Now, if someone retorted that this might be true of extensive quantities, but might be untrue of intensive quantities – that is, consciousness could have an intensive quantity – Hegel could reply that this intensive quantity should be measurable through some measurable extensive quantity: in fact, we shall see this criticism used against Kant in the next remark.

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Remark 2.

The last paragraph of the previous remark guided us to the theme of this remark, that is, the possibility of quantifying human behaviour. Like in the previous remark, the reason for the remark comes from Kant, who had suggested that consciousness could have an intensive, although not extensive quantity. While the previous remark was connected to an earlier remark of the behaviour of matter in Kant’s system, this remark points further to the final book of Logic, where among other things the structure of cognition is discussed: thus, the main thing to learn here is that quantitative structures, whether extensive or intensive, are not adequate for describing all aspects of human behaviour.

1./400. Kant suggested that consciousness of a soul, like any object we experience, should have an intensive quantum, although soul could not be measured extensively: thus, soul could decay by losing gradually its consciousness. But a singular state of a spirit is not essential to it, and thus, we shouldn’t quantify it at all.

We can discern here a string of critical remarks dedicated to clarifying in what terms we should – and especially should not – describe consciousness. At the bottom level are the crude materialists who straightforwardly identify consciousness with some pattern of material objects. If this view would be true, then consciousness would inevitably be something that can be measured, e.g. through a brain scan or similar. Metaphysicians disagreed with this assessment by noting that consciousness is not extensively quantified like length or mass is: we cannot literally cut consciousness into smaller pieces of consciousness. They suggested, on the other hand, that consciousness should be seen as an indivisible object, perhaps connected with some pattern of material objects or body, but still at least conceptually separable from it. Because consciousness was indivisible, it couldn’t be cut to pieces and thus it was indestructible, at least if we discounted divine means of obliterating whole substances at once. Now, metaphysics supposed consciousness should still be an object of the same level of abstraction as material objects, although of a different kind. Kant noted that in order to speak meaningfully of such an object we should have some sort of experience of it: as we indeed do, Kant continued, in the form of what he called inner sensation – we sense ourselves in some way. But as an object of experience, the consciousness had to obey the rules of possible human experience: as a real entity, it should have some degree of existence or intensive quantity. Thus, it was not as indestructible as it metaphysicians thought: its intensive quantity could diminish, that is, it could become minute by minute unclearer in its thoughts and finally vanish into non-consciousness. Hegel remarks now that Kant’s suggestion fails because intensive quantity of consciousness would imply

38 that it is also as an extensive quantity in some sense, which was already prohibited by metaphysicians. Hegel’s own theory is that consciousness is not at all an object of same level as material objects: consciousness is not defined by it constitution – thus, it could well have a material basis – but by certain capabilities it has – an object is conscious if it can do something.

c. Change of quantum In the previous two sections we have learned that there seems to be two species of quanta – extensive and intensive – but that this difference is actually one of mere interpretation: any extensive quantum can be seen as intensive and vice versa. Thus, the aim of these sections has merely been a negative one: we do not have to consider extensive and intensive quanta separately, because there is no true difference between the two. This section then has a far more important task. It tries to show not just that quantum of a thing or a situation could change to another quantum, but that a quantum itself could change place in the framework of quanta – or more precisely, that the reference point of the framework (zero or unit) could change place and thus the determinations of all other quanta in relation to the reference point. Like the similar section in the chapter on Dasein, this section ends in pointing to the possibility of changing the quantum into an infinite quantum, that is, quantum that is larger than any quantum in some given context.

1./401. Extensive and intensive quanta are interpretations of the same quantum: quantum in general is in one sense a determination, but in another sense it could be interpreted as another determination. When quantum is understood as extensive, these two sides can already be discerned, but when it is interpreted as intensive, they can be seen even more clearly: an intensive quantum is in one sense independent, but in another sense its determination depends on its place in a framework of quanta.

The main result of the two previous sections was that extensive and intensive quanta are mere interpretations of the same quanta. Thus, we can concentrate our study to one of the two forms – in this case, to intensive quanta – and nothing essential will have been lost. The general definition of a quantum has been that it is in one sense a determination, that is, a definite place in a framework of quantitative situations: say, five. Yet, the place that it has in this framework is also arbitrary: five is five only in comparison to some arbitrarily selected unit. These two contexts are present already in extensive quantities: a set or a collection of five units is a set of five units, only because we have chosen a suitable unit, while in relation to a unit half the size it would be a set of ten units. The two contexts are even more apparent when we interpret a quantum in an intensive manner, that is, as an ordinal number. Fifth degree in a scale is fifth just because we have selected a certain unit degree: then it has four units before it. But this selection is arbitrary, and with halving the unit this degree

39 would change its place into the tenth place.

2./402. The characteristic of a quantum is that it could be identified with a quantum that is different in some context: there must be a context where this quantum has the same value as that quantum has here. This is an essential characteristic of a quantum: quantum is not stable, but arbitrary limit.

It is easy to read Hegel’s meaning here in a close, but not completely correct manner. We could think that Hegel is speaking only of the fact that a quantum of anything could be changed to another: for instance, if we have a rod of five meters, it could be lengthened to a rod of six meters without changing the rod essentially. This is a part of or an implication of what Hegel wants to convey, but it cannot be the whole picture. Otherwise, it would become difficult to understand why Hegel would have said that such changes must happen: although we obviously can modify the length of a rod without any essential change, there seems to be no reason why we should change it or why it should change by its own. Hegel’s meaning becomes clearer when we understand that the quantification of anything – like a rod – is arbitrary in the sense that the number we choose to describe a thing depends on what we choose as the reference or unit quantity: rod of five meters is at the same time a rod of five hundred centimeters. Thus, any quantum must change: that is, there are many equally correct ways to refer to a quantum, depending on the reference quantum we compare it with. Furthermore, the quantum in question is then not “being”, but “becoming”: that is, a quantum is a structure consisting of many situations or interpretations, depending on where we put the unit in out framework of quanta – a quantum is not a single number, but a series of possible numbers.

3./403. From a unified object could be constructed another version of this object: similarly, from a quantum could be constructed another version of this quantum. Because quantum is always determined by its relation to other quanta, the constructed version is not completely identical, but seems like a different quantum compared to the first version: it is more or less than that.

In repulsion we took some object – no matter what, just as long as it was taken as an independent unit – and then copied or multiplied it: that is, produced another independent unit. How the repulsion was done is inessential: it may have been the abstract construction of accepting the situation of an object as a new object, or a literal copying of something or also the natural movement of time where one time slice of the object moves into another. Because the result of the repulsion was in some essential sense identical to the original unit – both were independent units – we could say that the result was actually another version of the original. Thus, the one in its repulsion is infinite in the sense that its apparent differences can be interpreted as merely aspectual. Now, Hegel points out that a change of the reference quantity in a quantitative framework produces

40 constructions that are in some sense repulsions. Suppose we have a rod of five (meters) and then we change our reference quantity from meter to centimeter: then the rod has become five hundred (centimeters). Yet, during this whole “change” or reinterpretation from five to five hundred, the quantity itself has in a sense remained identical: we have merely produced another version or interpretation of the same quantity. There is still some difference between the pure case of repulsion and the reinterpretation of quantum. The units in the repulsion are supposed to be completely identical – or at least we abstract or shut our eyes from all differences: there is thus no determination that could change in a pure repulsion. On the other hand, quantum is essentially determined by its relation to other quanta: this quantum is five in this framework because it has a certain relationship to quantum called one. The changed or reinterpreted quantum has in some sense different determination than it had before, although in another sense no true change has occurred: five hundred is larger than five was. Similarly we could have also reinterpreted the quantum as 0.005, when compared to a kilometer. Quantum could be reinterpreted as larger or smaller, depending on the reference point – five meters is a large compared to a microbe, but small compared to a country – and the reinterpretation makes its role in the quantitative framework completely different.

4./404. The result of a change of a quantum is in one sense a determined quantum, but in another sense, this quantum can also be changed: we can change quantum indefinitely.

Suppose we interpret certain quantum as having a larger value than it now has, say, five as a five hundred. The resulting interpretation is also a mere interpretation. In fact, it is not even the largest interpretation: we can choose a still smaller unit and thus “enlarge” the quantum into five thousand, five milliard etc. Similarly five is not the smallest interpretation possible: we could take ever larger and larger units and thus let the quantum slowly diminish in comparison. What we get is two series of values, one of them growing and the other diminishing. In fact, these series are endless or there is no largest or smallest value to which they would end: the framework of quanta is unlimited in both ends. In the words of unrefined mathematics we usually say “the series of values approaches infinity”, when the series grows without any limit: and as we shall see, in a sense, this is the only infinity Hegel accepts for quantities.

C. Quantitative infinity

We have been studying quanta, that is, quantitative structures or situations related to other quantitative situations: for instance, this rod being five meters. We were then introduced to two

41 different species of quanta: extensive quanta or numbers – e.g. collection of hundred pieces of sand – and intensive quanta or degrees – e.g. the fourth note in the C–scale. Actually, we learned that these are not two species, but two interpretations of quanta: every quantum is extensive in some sense and intensive in another. Finally, we learned that all quanta can be reinterpreted so that it has as large or as small value as we want: quantum is more like a variable than a constant in all contexts. The subject matter of this section should then be quantitative infinity. One may wonder why this infinity has not yet been introduced, that is, why we haven’t been shown how to construct infinite quantity from a given quanta. But actually we have been: we have been shown that every quantity could have any value, no matter how large or small it should be. The following section aims, among other things, to show that this is the only sense in which we can speak of an infinity of a quantum – that is, that according to modern conceptions, there would be no infinite numbers. The division of this section copies the division of the corresponding section in the division of quality and thus reproduces all its characteristics. The first subsection is merely introductory, explaining what Hegel means by infinity of a quantum. In the second subsection we are then shown that this sort of infinity is only relative to a context: for every context with a largest or smallest number we can find another context with even larger or smaller number – we must remember that Hegelian number system can never be finished. In the final subsection we are explained what the true infinity should be in the context of quantities: it must be the rule or function which regulates the possibility of constructing new quantities from old or of reinterpreting old quantities as having new values.

a. Its concept

We begin with rather uneventful explanation of why we can call a quantum infinite in one sense: because all seemingly different quanta to which this quantum is related can be interpreted as mere versions or reinterpretations of this quantum, just like in the general introduction of infinity we reinterpreted things having apparently different qualities as mere aspects of one thing. Interesting is Hegel’s note on differences between qualitative and quantitative infinities: the reinterpretation of a quantum as infinite is more natural, because the determination we give to a quantum is already arbitrary and depends on where we put the reference point.

1./405. Quantum can be changed to another, and this change can be continued indefinitely. We can identify changed quantum with the result of the change – also a quantum: thus, they are not anymore limited by each other, but infinite, aspects of the same whole. In one sense, quantum is independent, but from this independency we can construct another quantum to which it refers as a unit: on another sense, we can always identify different quanta.

In the realm of qualitative we saw how a finite object could be interpreted as infinite: we first

42 replaced the finite object with another finite object and then noticed that in some context the starting point and the result of the change could be identified, that is, taken as not limited by each other and thus as infinite in some sense. In the realm of quantities something similar happens. Take any quantum of a certain scale or framework of quanta. This quantum can take the place of any other quantum, that is, it can assume the value that the other quantum has in some context, when we change the general reference point of the scale. “Five” is not a stable determination, but depends on what we take as “one”. Thus, all the possible values of a framework could be constructed from merely one quantum merely by reinterpreting its place in the framework: we can construct many quanta from one quantum. Then again, we can also interpret the apparently different quanta as mere versions of one quantum. The difference between the quanta is only one of reference point: this is five, but with a suitable modification that could also be five. We might thus take any quantitative framework as a framework of possible states of one variable quantum: one, two, three, four etc. are merely the same quantum looked from different contexts. This one quantum would then be interpreted as infinite in the sense that there would be no differing quanta from which it could be differentiated.

2./406. Quantum can be finite and infinite according to two contexts: it is finite, in one sense, because it has a limited value, but in another sense, because it can be interpreted as having a different value; it is infinite, in one sense, because it is not limited to one value, but in another sense, because it is an independent state of being. As finite, the quantum can be interpreted as having a different value, thus, it is not limited to any value and is infinite; as infinite, it is independent of its relation to other quanta, thus, we have abstracted from the possibility of reinterpreting it and it thus has a determinate or finite value – finity and infinity of quantum are only a matter of interpretation.

Quantum can be interpreted as finite and as infinite, but this is still not enough: it also depends on the context which criteria we use to indicate the finity or the infinity of a quantum – that is, what sort of interpretation of a quantum makes it satisfiable. Suppose we look at the fact that we can change the value of a quantum arbitrarily. One could say that because of this arbitrariness quantum is not as it should be: it has no truly determinate value, because this value depends on the choice of the reference point. Then again, another person could insist that it is just this arbitrariness which makes quanta so perfect: they are not restricted to any one value, but could be interpreted as large or as small as one wanted. Suppose then we chose some determinate value for a quantum and decided that we shouldn’t change it – say, we fixed it as the largest quantum in our system of quanta. The first person would be convinced that we have made the quantum perfect by making it independent of all other quanta through the determination of the quantitative framework. The other person, on the other hand, would insist that the quantum had become imperfect, because its value range has been limited: it cannot become even larger, although it might be the largest quantum in some

43 limited context.

3./407. The difference between qualitatively infinite and finite is essential and we can construct infinite from finite only by abstracting from the concrete differences: qualitative determination is not relative to a reference point and hence cannot be reinterpreted as another determination; quantum , on the other hand, is relative to a reference point and can be naturally reinterpreted. The transition from qualitative finity to infinity happens on an abstract level, while quantitative finity can be naturally changed to infinity: it is possible to change the value of a quantum indefinitely.

Hegel reminds us of the previous moment where we constructed infinity from finity: there we were faced with a qualitative differentiation of objects or situations and we were then asked to concentrate on the characteristics that these objects had in common – such as finity or beinglimited-by-the-other-object – and abstract from all their differences, and lo, we could imagine that the two objects were indeed mere aspects of one infinite whole. This transition required change of identity conditions: the formerly essential qualities were reduced to inessential, aspectual differences. Nothing of the sort is required here. Quantitative differences are already inessential, because they are relative or context-dependent: five is five only compared to a certain one. Here it is more natural to assume that two different objects or situations are mere versions or aspects of one totality: the whole scale of numbers is one big unity, and we merely slide its middle to different places. What has this to do with what Hegel calls infinite progression? Suppose we have a quantum of certain size: at once, we can think of, say, a slightly larger quantity. Now, we can obviously reinterpret the quantum to be of that larger size. Both of the operations can be repeated indefinitely, leading us to ever larger and larger values that this quantum could assume.

b. Quantitative infinite progress

In the previous subsection Hegel explained how we could interpret quanta as infinite: because the difference of quanta is one of reference point we can without ado regard an apparently different quantum as a mere version of this quantum. Every quantum could thus take the place of any other quantum, no matter how small or large this quantum was: every quantum could be made indefinitely large or small through a simple reinterpretation of the quantitative framework. This section then proceeds – as the respective section in the realm of qualitative – to investigate this possibility of infinite or indefinite progress. Hegel’s task is now far more simpler than then, firstly, because the main results have been gained in the qualitative section, and secondly, because the progression is easier to understand in the quantitative case. An interesting novelty is the concept of infinitely large or small: a concept that was still in use in the infinitesimal calculus of Hegel’s time and that Hegel shows to refer to no object, except when it is understood in some quite trivial

44 manner.

1./408. Indefinite progressions are born from repetition of two incompatible situations, here from situations involved in dealing with quanta. In one sense quanta are finite, in another infinite, just like in the qualitative realm: but it is natural to interpret quantum as having a different value, and even an infinite quantum [of some context] must have a determinate value.

Sometimes we have an ability to produce a situation with certain characteristic from a situation with another characteristic and also with an ability to make a similar production to the converse direction. In this case, we can repeat these two constructions indefinitely and this repetition of situations with two alternating characteristics produces what Hegel calls progression into infinity, although more appropriate would be to speak of an indefinite progression or a progression with no natural end. In quantitative indefinite progression, we are presented with the following two constructions: firstly, the possibility of finding criteria which a certain quantum doesn’t fulfil – that is, criteria for being small or large enough, or indeed, smallest or largest in some definite context – and secondly, the possibility of reinterpreting quantum as having any required value. These two constructions are more natural than those we had to use in the qualitative case. In order to make finity into infinity, we had to abstract from the qualitative differences between apparently separated objects: here, it is at once allowed to reinterpret the quantum as having any value, because all quanta have values only in relation to other quanta. This is already familiar from the previous section. Furthermore, the converse direction from infinity to finity is also easier in the quantitative case. In qualitative realm, we had to provide some new criteria by which to judge the supposed infinite as unfulfilling certain criteria: for instance, the criteria of not-being-identifiable-with-finite-objects. Here, all possible quanta are already naturally ordered in such a manner that for all positions there can be found both larger or smaller positions: thus, because the quantum as infinite in some context – as the largest or smallest quantum in relation to some quanta – is still a quantum, it is always possible to find another context in which there are still larger or smaller quanta than this.

2./409. Indefinite progression only repeats the two sides without connecting them properly: in it we merely seek the true infinite without ever finding it and without ever overcoming the realm of quanta. Quantum can, by its structure, be changed into something else: in a qualitative sense, this something else differs from quantum or is infinite compared to it; in another sense, this another is just another quantum – or version of the same quantum – and no true change has happened. Thus, in some sense, quantum can be made infinite, but in another sense, this is only a contextual infinite or limited in another context.

The problem that the indefinite progression aims to solve is the changing of a finite quantum into infinite. In one sense, it does just that – we are able to change quantum into what is considered

45 infinite in this context – but in another sense, it constantly fails to do that – we can always find another context with a new infinite. All of this is already familiar from the qualitative case, the only difference being that the constructions used are far more natural: we are already capable of producing a series of contexts with ever-growing largest quantitative state, and furthermore, we are capable of sliding the values of quanta to any position we want. Hegel even uses the terms “beyond” and “this side”, familiar from the qualitative infinite progression. In some context, we have a determinate situation of all finite quanta, that is, quanta up to some definite value, and besides or “beyond” that situation, we have a situation with another, “infinite” quantum. If this would be a case of qualitative differentiation, we would have to change the context radically in order to find some construction by which to interpret the beyond as belonging to this side. Here, we can easily just change the reference point of the qualitative framework and the formerly largest quantum falls right into the situation with finite quanta: only problem is, of course, that we have then a new and even larger largest quantum and thus a new beyond. 3./410. Quantum and infinity are seemingly combined in “infinitely large” or “infinitely small”, but when both are understood as quantities, they are not stable and thus not infinite: the variability of quantum is merely constructed explicitly in these terms and both can always be differentiated from a further infinite. No matter how large a quantum is made, it is still not infinite, or being a quantum and being infinite are qualitatively different: enlarging of a quantum makes it no more infinite – infinitely large merely should be quantum and infinite, but it cannot be them at the same time. Similarly, infinitely small is always too large to be infinite.

Infinitely large or infinitely small should be something that is both a quantum – thus, something that can be more or less – and infinite – something that cannot be more or something that cannot be less. Evidently, this is possible only if we understand the expression as indicating a Hegelian contradiction, that is, as speaking of the same thing in different contexts. Now, there are two ways how this contextuality can be understood, although Hegel speaks only of one. Firstly, it could be the case of a mere contextually infinite quantum. This quantum is largest, that is, in some context. No matter how much we increase it – no matter what is the context where it is the largest – we can still find a further context in which it is not the largest: thus, “infinitely large” in this sense is always too small to be truly infinite, similarly as “infinitely small” in the same sense is too large to be a true infinitesimal. The other possibility is that we have a quantum in some sense, which in another sense or compared to another quantum is infinite or qualitatively different. Let me explain further. A line and a plane are two quanta in some sense: both lines and planes form a quantitative series of their own. Yet, towards one another they are not related as quanta, but as qualitatively different things or situations: being a line differs qualitatively from being a plane. Thus, e.g. plane is in some sense

46 quantitative – when compared to other planes – and in another sense infinite – compared to a line. Hegel does not yet mention this possibility, because we haven’t yet constructed any examples of qualitatively different quantitative series: this is something we do in the future. Still, even such a “infinitely large” as plane wouldn’t be what is usually meant by quantitative infinity, because when plane is taken as infinite or qualitatively different – compared to a line – it is not taken as a quantity. Thus, plane is not formed of infinite number of lines, similarly as line is not an infinitesimal of plane.

4./411. The infinite that should be completely separated from all finite quanta is bad quantitative infinite: like its qualitative version, it involves a perennial movement from a limited to non-limited context and from non-limited to limited context. In quantitative progression we are not moving between merely different situations or objects, but between different quanta: still, the progress is mere repetition of positing and denying – we aim at separating finity and infinity and end up with combining them.

The reason why we always miss our goal in a quantitative infinite progression is similar as the reason for failure in qualitative infinite progressions: infinity is separated so completely from finite that it is impossible to construct one from the other. In qualitative infinite progression the supposed infinity without finity was revealed to be finite, just because it failed to permanently satisfy the criteria of finity: it was finite in some other sense or in some other context. Here, in quantitative infinite progression, we never even face an infinity that at least seems to be separated from finity: all infinities we can reach in a quantitative progression are mere contextual infinities. Hegel is, thus, in some sense against Cantorian realm of infinities: or more precisely, he would say that these infinite cardinals or ordinals are not quantities in the same sense as ordinary cardinals – they are qualitatively different cardinals, at most. Furthermore, Hegel would with pleasure point out that even in the realm of Cantor’s cardinals no final or largest cardinal – the cardinal of all cardinals – can be found.

Remark 1.

Qualitative bad infinity was expressed as a realm untainted by anything finite and was therefore proved to be finite in another sense: it was something that couldn’t become finite and was thus imperfect, not capable of existing everywhere. In quantitative bad infinity we add the notion that the realm of infinity is unreachable through any, but endless progress: in other words, that it holds a place beyond a progression of all finite realms and objects. This notion is historically important, as it was endorsed by almost all philosophers in the nearest period preceding Hegel: Kant, Fichte and even Schelling. Virtue, knowledge and absolute were supposed to lay infinite steps beyond human

47 realm. Furthermore, between this ideal and any arbitrary finite member of the progress should always be further members, nearer to the ideal than this member: thus, we could always be better in our actions, cognitions etc. What Hegel wants to argue is that such an unreachable infinity is a meaningless idea, as the qualitative bad infinity already was, and that the supposed better members beyond this member are not as important as Kant and companions had thought: quantitative progression is merely a refinement of what one already possesses or a repetition without anything qualitatively new.

1./412. Quantitative progression into infinity has been considered sublime and final truth: actually, it is only fleeing of objects and enlarging of own subjectivity.

God, absolute, virtue and truth are infinitely removed from the affairs and capabilities of human nature: we can arrive at them only through an infinite progression, that is, never at all, if we regard only finite times. With such words it is supposed to elevate e.g. God, as being not tainted by the ordinary and mundane: God is meant to be an object or thing high above any objects that we are aware of. Yet, the God, absolute etc., if they are understood in this manner, are not something we could call objects: they do not exist, or more properly, we cannot say that they exist: if we want to speak meaningfully of God etc. we have to understand the terms in a different manner. In fact, we are merely fleeing all those objects we are concretely aware of, when we indulge into an indefinite progression: we are never satisfied with what is before us, although it would be good in its own context. Instead, we are placing our imagination above things and objects: we can always fancy something better and larger and altogether more positive than those puny things we are aware of.

2./413. Kant, for example, finds it elevating when we think of the endlessness of stars and planets and their movements; such thinking, he says, ends only with fainting.

Kant had long been intrigued by the idea of an endless universe with stars beyond every star and “worlds” – or galaxies, as we might say – beyond every world. Indeed, in his precritical phase he had described a natural history of such an infinite, evermore organized and thus in some sense expanding universe with no limits, from the supposed chaotic beginning to its never-ending phases of order and chaos. The faith concerning the existence of this infinite universe had never been shaken, not even by the critical turn. Only difference was that in his critical years Kant accepted that this universe, as a spatio-temporal entity, was perhaps only something that appeared to us and not “world as it is in itself”. Furthermore, he suggested that we could only experience finite portions of it, and thus, although we could think of it – that is, speak of it as existing in its infinity – we could never truly cognize it as a whole: we can never be sure that an infinite universe existed,

48 although we have an urge to not accept any finite region of space and time as the whole universe. Our mental journey from star to star could not ever end: we could not imagine how the universe would stop, either in space or time. Thus, it makes sense that Kant would have admitted that such a hike of thought could end only in a complete exhaustion, when the person searching for an end would finally fall asleep unfulfilled. 3./414. Best part in Kant’s presentation is the description of the end: boring repetition of finding and overcoming limits leads to an exhaustion and in seeking infinity untainted by finity we can never even master finity.

It is interesting to see what Hegel takes to be the main point in Kant’s description. It is not the greatness of the universe or the number of stars or any quantitative perfectness that interests Hegel: these multitudes are mere indefinite repetitions of familiar things. It is the only possible end for this repetition – the fainting from exhaustion – that Hegel finds most intriguing. The indefinite repetition has no other possible end than giving up, that is, the supposed totality of all stars or all universe is a mere fiction for us: we cannot find satisfaction in this manner, but only in noticing in a second-order investigation that any finite context – and these are the only contexts we shall ever become aware of – are quite within our grasp: in noticing that nothing in the world of experience can be truly beyond our comprehension. 4./415. Kant has called Haller’s description of eternity fearsome and trembling. 5./416. Actually the poet himself notices that the true eternity can be found only by leaving the infinite progression.

A similar discrepancy in viewpoints of Kant and Hegel as in the previous paragraphs can be found in the opinions of both on the poem of Haller, which Hegel quotes only partially. The theme of the poem is eternity, a fitting subject matter for bringing out the true differences between Kant and Hegel. Kant, of course, emphasizes the impossibility of ever completing an eternal process, no matter how many moments of time we happen to compile together: a finite subject can never reach an infinity, although she has an urge to seek for it. Hegel, on the other hand, holds in ridicule such a notion of eternity: if we can never reach it, why should we even believe in its existence, let alone seek for it? Indeed, as the poet admits, if we let go of the futile attempt to reach eternity by counting, we can seek the eternity within a moment, that is, we can investigate generalities and recurring characteristics. And to find those, we need not repeat any progression indefinitely: indeed, it would never be a sufficient proof of true generality.

6./417. Astronomers are proud of their science, because it investigates immeasurable collections of similar objects, although its true worth lies in discerning laws of stellar movement.

49

It is quite clear that Hegel does not valuate highly the idea of an infinite, everlasting universe with an infinity of matter and stars within it. Of course, we shouldn’t say that he held it to be a false view of the universe, and especially we shouldn’t see Hegel as a precursor of the idea of so-called threedimensional elliptic world, as Hösle in at least one of his writings has stated. The infinite or indefinite universe was the astronomical model in science of Hegel’s days, and Hegel is not to be taken as an enemy of science for light reasons: and especially we shouldn’t hail him as a prophet of relativity, just because we who live after Einstein feel some connection between the modern physical theories and suggestive words Hegel uses. We shall see in the next remark more closely what Hegel actually thinks of the shape of the “world”, if we can speak of such a thing. By now it should be clear that he admits that we can find new stars after all stars and new space after all space. This is true, but it also makes space and stars an uninteresting, monotonous topic: thus, it is no great news if one new star has been discovered in the skies. Far more important discovery is Kepler’s and Newton’s finding of laws that govern the movement of planets and stars. These laws are something general that recurs throughout all space and that regulates the positions of stellar objects: they are true infinite, compared to the mere relative perfection of large distances and masses of stars.

7./418. Kant [or Fichte] sets the intuited infinity of stars against the free self-identity of I. 8./419. The pure I is in some sense an infinity present to us, in another sense it always remains unfulfilled when compared to all possible content beyond it.

It is unclear where Hegel gets the so-called quotes from Kant in this remark. There are some clearly Kantian elements and even phrases in them: the whole opposition of infinite universe outside me and the freedom of moral law within me is situated at the end of the Critique of practical reason. In this particular quote, there are some Fichtean-sounding phrases such as pure I (reines Ich). Still, at least the tone is unmistakably Kantian: human reason is free or we must assume that it is completely free and autonomous, in order to explain morality or make it possible. This autonomy is something to be praised in Kant’s opinion: it is sublime, although in a different sense than in which the unlimited universe is sublime – an autonomous subject is determined by itself and not by anything outside itself. Hegel admits also certain perfection for self-consciousness and its autonomy: while for any collection of stellar objects we can always ask what lies beyond it, in some sense a similar infinite series does not arise in case of self-consciousness. Self-consciousness is a concrete example of apparently different aspects forming one totality: within self-consciousness we can always discern between e.g. the person who is conscious of and person of whom the first is conscious, but we can also identify these two persons. Problems arise when we oppose this self-consciousness with the

50 world beyond it. Self-consciousness can never, of course, cognize all that there is to cognize, because there are potentially infinite amount of things to know: Kant would indeed admit as much. Furthermore, even as practical or acting, the self-consciousness is never completely perfect. Actions are directed towards objects, which are modified to agree with human reason. Yet, if there are potentially infinite number of objects in the world, it is impossible ever to find a final stop in the series of actions. Self-consciousness is thus perfect only within the restricted context of its own ego, while the relationship to further matters makes it in some sense imperfect.

9./420. Even Kant admits that these two sublimes regulate our research, but can never fulfil their lack.

Hegel’s standpoint is not necessarily as anti-Kantian as has been traditionally thought. Ideas of a complete or total universe and an autonomous person – the goals of cognitive and moral striving, we might say – are in some sense nothing more than ideas: something that can regulate our cognition and action, but that cannot ever be actualized. All of this is something that Hegel and Kant would agree on: the main difference being that Kant still retains some faith in the slight possibility that some other form of consciousness could fulfil these ideas, while Hegel rejects such ideas as useless fancies – at least from our perspective. Note how the two unreachable goals have almost opposite reasons for being impossible to achieve. For universe, it is the overabundance of possible content that keeps the completion from its grasp: because we can find larger and larger “universes” none of them can be the world or totality of everything spatio-temporal. For autonomous I, it is the lack of content: the overabundant world with its own laws will always remain beyond the power of human control and thus also the natural side of human itself will in some measure be uncontrollable.

10./421. Moral progress is especially conceived as an infinite progression: I tries to determine nature and thus free itself from it, but because of the infinity of the nature it can never achieve its goal completely – it can only become more or less moral. This is supposed to be a blessing, because morality wouldn’t be possible without a struggle.

Hegel has slowly shifted his gaze from the stray heavens to the mind and especially will of a person and its relation to the world outside it. The autonomous morality of Kant has been the topic in the previous paragraphs and now Hegel mentions the explicit connection to infinite progression it has. For Kant, it is especially the relationship of morality or pure will with the sensible or natural element within man that creates infinite progression: or actually, makes us postulate that such progression is possible. Morality would be futile if it wouldn’t have any effect on the actions of a person, but the problem is there is another source of action within all persons – the needs and wants. These needs and wants can never be made totally subservient to morality, thus, in order to see

51 perfect morality as a possible goal we must believe that we can at least come indefinitely nearer to it – which requires an infinite amount of time. Fichte extends this control of our sensible side to control over all nature: the world beyond us hinders our pure will and thus a similar infinite progression arises in relation with the struggle of mankind and inhuman nature. We can quantify the area of nature our will has managed to take into its control: because of the potential infinity of universe, the control over it is also potentially infinite. Indeed, as the human will by its definition is characterized by struggle – at least when it is good or it acts morally – it is actually good that the final goal can never be reached: otherwise our will would vanish and so would we as human, finite persons, or we would have completely dehumanized ourselves by becoming gods.

11./422. Moral and natural laws should in some sense be independent of each other, in another sense they should be aspects of the human progression in morality: these incompatible aspects are not properly put together in an infinite progression.

All indefinite progressions should arise because we are incapable of understanding how two seemingly incompatible and yet equally convincing types of situations could be combined into a whole and we are yet faced with the possibility of constructing one type from the other: thus, we end up alternating between the two types without ever finding an end – usually these two types are a) one of being limited in some manner and b) one of overcoming this particular limitation. In the particular case of moral progression the relationship of a person and his will contra his environment and inclinations can be of two sorts: firstly, the environment or natural inclinations may hinder the will, or secondly, they might obey will. Now, the moral progression arises, because firstly, if the environment or inclinations hinder us, we can act in such a manner that they obey us – we can put nature under our will – and secondly, if we have managed to gain control of some part of our environment or inclinations, we can still find further part of environment or a further inclination that has not yet been overtaken. The will of a person could thus gain control over everything natural, whether within or outside itself, but it cannot control all the nature at once, because of the indefinite boundaries of nature itself.

12./423. The image of quantitative progression is thought to help in solving the contradiction, but actually it just shows that nothing essential changes in the relationship of nature and will.

Although we can never control our inclinations or environment completely we may still progress towards this goal, Kant and Fichte suggest. For Hegel, this seems a shallow hope: it is indifferent how much of an infinity we have travelled, because there is always an infinity of steps to take. No matter how far we progress in morality, we can never completely cancel the natural forces hindering

52 our moral will. True solution would be to discard this goal as impossible to reach. We are never trying to control all nature, whatever that is, but only finite parts of it: we want to realize some definite goal, like building a bridge, or getting rid of some nasty habit, like smoking. Such definite goals can be realized, or at least they are much more possible to reach than ideal power over everything. 13./424. Fichte’s more abstract presentation falls for the same problem: a not-I is supposed independently of I and it is given its own quantitative share of what we are conscious of; as a result, we can only try to cancel the not-I, but never succeed in cancelling it.

Hegel’s interpretation of Kant is already coloured by Fichtean terms, and it is no wonder that Hegel then thinks Fichte’s philosophy is only a more abstract version of Kantianism: indeed, we might suggest that Hegel is correct when he says Fichteanism to be Kantianism in a consistent form. Of course, then Hegel’s criticism of Fichte goes somewhat astray: both Kant and Fichte try not to say how we should or could change our every-day-view of the world, but only how it could be justified. Thus, Fichte begins with accepting the existence of I – this must be the starting point of all transcendental philosophy which accepts that we are always limited to a view of how things are for us – but also accepts the existence of something differing from I – this is occasioned by empirical demands, because we can separate ourselves from something else. The question is then how to explain these two facts – a possible consciousness of ourselves and a possible consciousness of something else – without however falling into any contradictions. Fichte’s answer is that I and non-I must have their own limited spheres within the world of experience: they control different quantities of what there is. Of course, this basic limitation can never be broken – otherwise, consciousness as we know it would collapse – but only the quantities allotted to I and non-I can be changed. Still, Hegel’s basic question why we should then set such an impossible goal as getting rid of all non-I is valid against Fichte’s theory. 14./425. It was thought [by Schelling] that when all differences – such as the difference of subjective and objective – were interpreted as quantitative, we should see that everything was a unity; yet, quantitativeness of objects implies that they are unrelated to each other and identified only by an observer. We must show how objects can be constructed from each other, e.g. how qualities can be constructed from quantities.

After Kant and Fichte we end up with a nameless persona, who has obvious affinities with the thought of Schelling and young Hegel: nature and spirit of consciousness should not be two different sorts of objects, but mere modifications of the so-called absolute identity – spirit just happens to be more aware, but even nature is alive in some sense. The problem with this monistic

53 theory is that it tries to avoid differences by mere interpretation: these objects do seem different, but actually they are only quantitative modifications of the same basic stuff. Still, the same differences remain valid, although in a different, supposedly unessential level, and it is still problematic whether the sorts of objects have any true connection: it is like saying that nitrogen and oxygen are modifications of the same prime matter without providing any formula for transforming one to the other form. It is this formula what Schelling’s theory still needs: a method by which one could construct or discover consciousness from nature and nature from consciousness – it is this method which is truly worth of attention instead of any supposed common substrate behind nature and consciousness. Indeed, the whole Logic is full of attempts to find such methods, the latest one being Hegel’s attempt to show that on basis of any quantitative framework it is possible to build qualitative differences.

Remark 2.

We have faced a number of examples of indefinite progression into potential infinity. Firstly, in the section itself we met the primary example of ever larger or smaller quantities – either through reinterpretation of quantities by choice of suitable quantities or through actual construction of new quantities. Then in the previous remark we met at least two examples of such progression, one of which was the moral progression of control over our environment and inclinations. The other example of the previous remark seems especially interesting. Every finite region of space and time seems too small to cover whole material universe: as both Kant and Hegel agree, there could still be something beyond any finite space-time region. Yet, they both also admit that experience of an infinite universe is beyond capabilities of human understanding. Both Kant and Hegel note that there is a hidden third possibility to this antinomy: their solutions are interestingly different, although in some ways similar, as we shall see. 1./426. Kant’s antinomies make the opposition of finity and infinity more concrete: previously we dealt with the antinomy of qualitative finity and infinity, now the antinomy of quantitative finity and infinity.

Hegel begins the remark with a reference to the previous remark of Kant’s antinomies and especially the second antinomy. It seems puzzling why Hegel chooses to call this previous antinomy qualitative or antinomy between qualitative finity and infinity, although it was handled within the realm of quantities. One possible explanation is that the second antinomy does not concern relations between different quantities, but structure of one, unrelated quantity: the problem there is what sort of entity a quantity is. The three possibilities – or two possibilities and one

54 pseudopossibility, according to Hegel – form a sort of qualitative division of structures: quantity is either a finite collection of units, an infinite collection of units – this is the pseudopossibility – or it is characterized by a potentially infinite series of finite collections of units – or even better, by methods by which to create such a series.

2./427. Antinomy concerns the finity and infinity of spatio-temporal world: it could as well be applied to space and time, no matter whether they are interpreted as relations of things or as forms of intuition.

Already in the investigation of the second antinomy Hegel noted that Kant restricted his antinomies to things within space and time, although they applied also to space and time. It seemed problematic whether matter could be broken down to final atomic pieces or whether it consisted of infinite number of parts: but the same problematic would have to be faced when discussing of space also – in case of space we can always find only finite states of division, but still no final state of division. Kant’s answer is, of course, that space and time are and can be nothing real, because they are only forms or modes of how we intuit things. Yet, Hegel points out that this is no true answer: even then we had to face the problem that we could neither intuit some final finite phase of division in anything spatial nor intuit a phase of infinite division – at most, we can intuit space as divided to finite parts and then into further finite parts etc. The true answer to the problem was that parts of both space and matter could be introduced as existing or intuitable only after – real or mental – division: the idea that such a relation would be possible only for appearances is a Leibnizian prejudice that would have amazed Aristotle. 3./428. The proofs of this antinomy can be reduced to two statements “there is a limit” and “any limit can be overcome”.

The subject matter of the first antinomy of Kant is not how small we can go, but to how large matters we can go: whether the world is finite in size and has existed for a finite time or whether it is infinite in size and has existed for an infinite period of time. As Hegel noted in the previous paragraph, we could drop the world altogether and speak only of the question of finity and infinity of time and space: Kant seems to think that as forms of our intuition and thus as something not completely real they can be thought as infinite – indeed, many of his proofs in the first antinomy seem to be based on that assumption. In any case, Hegel's strategy is similar to his strategy in criticizing Kant’s proofs for second antinomy: Kant does not actually prove anything, but merely presupposes the statement he is about to prove – that is, he changes the situation presupposed and brings in some assumption that holds only in the situation about to be proved.

4./429. Thesis says that world has a spatial and temporal beginning.

55 5./430. The temporal finity of world is proved in the following manner. “If world would have never began, there would have been an infinite time before any point of time. Because infinity can never be completed, the world must have begun at some time.”

As the first antinomy concerns actually two questions – temporal and spatial limits of the world – Kant must give two proofs for both sides of the antinomy. Kant’s proof for the temporal finity of the world presupposes the for us a strange notion that infinity can never be completed. If infinity can never be completed, there can clearly be no infinite temporal series: this much seems clear. Yet, we may ask whether we can truly presuppose this non-compleatability of infinite series: indeed, it seems like an unbased presupposition of the required result – thus, Kant’s proof would actually be circular. Although this sort of criticism is natural for us, who are quite familiar with the idea of infinity, it was not yet available for Hegel who shared Kant’s thoughts of the quantitative infinity: indeed, Hegel even goes further and says that such an infinity is meaningless – we cannot meaningfully speak of an infinite world. Hence, Hegel’s criticism must be of a subtler nature.

6./431. Conceiving infinite space would require infinite amount of time, Kant says: thus, space cannot be infinite.

In this side of the antinomy Hegel concentrates to the proof of the temporal finity of the world’s history, because Kant’s proof of the spatial finity of the world is based on the respective temporal proof: space must be finite, because the past has been finite. This proof brings especially well out the common constructivist presuppositions of Kant and Hegel: it argues that we cannot speak of the world as spatially infinite, because we cannot have had time to conceive what a collection of infinite positions would be like. Because of this impossibility of ever directly conceiving infinities, no mere indirect or reductive proof for the infinity of the world would ever convince Kant or Hegel. Note that Kant’s proofs for the finity of the world speak only of such supposedly actualized matters as the past of the world and the current extension of the world: Kant has nothing against a potentially infinite future, because its finity will never be actualized during the course of the world.

7./432. The proof presupposes what it tries to prove: it assumes some given moment of time, which is already a limit of time.

Hegel’s criticism couldn’t seem to be wider off the mark: Kant is speaking of the possibility that the world would have begun at some moment of time and Hegel says the question is already decided when Kant has posited some limit within the time of world’s existence. Yet, there is a point in Hegel’s statement. The opponent of the thesis believes that world couldn’t have come to existence suddenly, that is, that there couldn’t have been any sudden break with the past – thus, if Kant would

56 have wanted to prove something in viewpoint of the opponent, he should have assumed that belief as true. Yet, Kant supposes some distinct point of time: such a point of time would divide the time into two parts, thus creating a sudden break – or interpreting the time as having a sudden break. Indeed, it wouldn’t perhaps be a break in context of the world, which is supposed to continue both before and after that point of time. Still, it would be a break in some context or according to some viewpoint, and a contextual break in time is all that we need to change the viewpoint we should have held on to.

8./433. Although the assumed moment ends the time, it is still a qualitative limit for the time: if it were only quantitative limit, time wouldn’t have come to an end. Furthermore, the assumed moment is also limit for the future, which would be qualitatively separated from the past according to the assumption.

Hegel has two points in his criticism against Kant’s proof of the thesis. The first one we noted already in the previous paragraph: a chosen moment of time is already an absolute beginning of some “world”, at least of the world of the future. Second point in Hegel’s discussion is that by choosing a moment of time that would end in some sense, a process of time is equally prohibited by the opponent of the thesis. Before we have separated some moment of time from others, we conceive – not an infinite, but – a unified, single time: we haven’t yet separated any moment of time from another. Within such a “singular time” it is impossible to say how long some temporal process has been, because there is only the one, everlasting process going on that cannot be compared with any other process, because no other process has yet been discovered.

9./434. Time should have been interpreted as pure, unrelated quantity, but Kant assumes that we can cut it to discrete moments, which already presupposes the limitedness of time.

Let us go through Kant’s proof once more. We begin with time as a single, undivided and undetermined unity: we cannot say that time before was truly different from time now or time in future, because we cannot separate any times before, now or after. Then Kant changes the context and introduces a point in time: he assumes that there is some specific temporal stretch that can be compared with other stretches of time. In this context it makes sense to speak of times before this moment or present stretch and times after it – in this context we truly can measure temporal stretches and compare them with one another: only in a context with temporal limits we can say e.g. that it took longer to walk here than to walk there. The situation within this context undoubtedly must have lasted for some determined time that can be compared to another determined time, but only because this “world” or situation has been determined. Hegel does not speak of Kant’s proof of spatial finity, because it is based on his “proof” of

57 temporal finity, but we could undoubtedly discover a similar error in that proof also. At first, we should be speaking of space in general, as undetermined and unquantified. Then we would start to measure it and separate some specific space as the “world”. This world would undoubtedly be finite, but this would prove nothing, because this spatially finite “world” would have been introduced and assumed, not proved or argued for.

10./435. Antithesis says that world is spatially and temporally infinite. 11./436. Kant’s proof is indirect. “If world had a beginning, there would have been an empty time. Because nothing can come out of an emptiness, there could have been no beginning.”

The proof of antithesis is based on the principle of causality or sufficient reason: everything in the world must have a reason for why it exists as it does. If this presupposition is accepted, then we would have to have an explanation also for the existence of the world, or more precisely, for the first moment of its existence: furthermore, Kant has reason to believe in its existence, because time as we conceive it couldn’t function as it does without the principle of causality. Now, Kant continues, time as we conceive it is an infinite stretch: thus, there must be a preceding moment before the beginning of the world. What then could have instigated world’s coming into existence, Kant asks and answers that nothing, because there was nothing there before the world came into existence. Thus, the first moment of the world would have no cause for its existence, which appears to be contradiction: thus, Kant deduced that the world couldn’t have been created at any specific time.

12./437. Kant assumes that world should have come to existence from some previous conditions, although he should have not overstepped the presupposed limit of the world.

The fault Hegel finds in Kant’s proof is easy to understand. In the beginning of the proof we assume some temporally definite and limited collection of processes as the lifeline of the world. Now, Kant assumes that the first moment of this lifeline should be further explained: we should explain why this lifeline began. Why should we do this, Hegel asks. The lifeline is supposed to be self-contained and in need of no explanation – this is what the presupposition of the proof amounts to. We shouldn’t even assume the existence of previous time, but accept time also as limited. Of course, there might be a context in which this apparently self-contained lifeline is explained by some previous events and in which thus there are more temporal moments before this moment. But to move from the context of the beginning to a context with such previous moments is not proving: it is assuming.

58 13./438. Kant proves spatial infinity of the world by noting that a relation of world to empty space would be unconceivable.

Kant’s proof for the spatial finity of the world is apparently a variation of a proof of Leibniz against Newtonian absolute space. If we supposed that the world would be finite, then, says Kant, it would be within an infinite and empty space. Problem is we couldn’t say where exactly the world would be situated: it would make no difference if the world were few feet left from where it now stands. Indeed, we could even imagine that the world was in a constant motion from one place to another, as long as the inner relations of the world were retained. Kant notes that all this sounds truly absurd: how could one relate world to an emptiness with nothing it? Thus, the only option seems then to be that the world fills all of the infinite space and is hence also infinite.

14./439. Kant merely presupposed that there is an infinite empty space, which then must be filled with the world.

The fault in Kant’s proof for the infinity of space is similar as in the respective temporal case. Kant’s proof presupposes the Newtonian infinite space in which the finite world is then situated. But we might well ask shouldn’t the space end together with the world. The proof begins from a situation with a finite, self-contained world: why should we assume anything to exist beyond it? Indeed, Aristotle saw no need to assume any space beyond the farthest limit of the skies. Of course, we might once again presuppose a further context containing a larger amount of space in which this finite world would have been situated, but this presupposition immediately changes the context and is thus no true proof.

15./440. The thesis and antithesis merely assert without proving: in some sense there is a limit, but any limit can be overcome; in another sense there is something beyond limit, but any beyond can be determined and limited.

It is the result of Hegel’s criticism that Kant’s so-called proofs cannot be seen as real proofs, but as constructions that change the context of discussion: at the same time he seems to accept that such constructions truly are possible. We may have either a world temporally and spatially – not infinite, but – indeterminate and open or a world temporally and spatially limited and closed. We may suppose that the world is, firstly, open-ended: the world contains perhaps some definite regions of space and stretches of time, but no clear limit can be found. Still, we can then define or discover the limits of the world: here it ends. We may suppose, secondly, that the world has definite spatial and temporal limits. Yet, we can always find new regions and stretches that could still be added to the world.

59 16./441. Kant suggests that the world should be non-contradictory and only our conceptions of it are contradictory: Kant is too sweet for the world. Consciousness can understand how same thing can be interpreted in apparently contradictory ways; on the other hand, world, whether objective or subjective, is always essentially incomplete.

We may have accepted all that Hegel criticizes in Kant, but still fail to see how it all adds up. All right, we have finite spatio-temporal regions and beyond them further spatio-temporal regions, but what about at the end of that series: what size is the sum-total of space-time? But Hegel would deny the presupposition of the question: there is nothing at the end of that series or there is no sum-total of everything. Kantian antinomy arises only when we assume the existence of such final moment of series, thus we better deny there is any final moment. In some sense we could say that there is no world. Yet, this must not be understood as if Hegel would deny that there is anything real or even that he would think all to merely dependent of our minds: Hegel is clearly quite neutral here as to the subjectiveness or objectiveness of the assumed world, because a subjective sum-total of everything would face all the same problems. Indeed, we could also say that there are many possible worlds: we could call any finite space-time-region a world, although there would of course be further worlds beyond it. Any such finite world or situation probably is independent of us and thus objective. Now, any such world is contradictory in the Hegelian meaning of the word: according to one context it is the world, according to another it is just a part of the world. How does consciousness then overcome this contradiction? Simply by noting the lawful relations between the different possibilities, that is, by noting that beyond any finite world it can find larger, but still finite worlds.

c. Infinity of quantum.

We began the section by noting that quanta could be seen as infinite, because they could take any value – no matter how great or small – just through reinterpretation of the number system as a whole. Then we noticed that in another sense quantity is always finite – we can always find some greater or smaller value that the quantity has not yet reached. Indeed, the idea of a greatest or smallest context-independent number was shown to be, firstly, unnecessary, and secondly, even formally contradictory – unless we were speaking of a number in some other number system. The subject matter of this section is the infinity of a quantum. The title must be taken literally: it is indeed Hegel’s belief that any individual quantum can be seen as infinite in some sense. This takes us back to the point of the first subsection: any quantum can take any value and is in this sense infinite. Here Hegel emphasizes especially the methods by which one quantum can take another value: Hegel calls them relations, but we should perhaps speak of functions or mappings.

60 1./442. Infinite quantum – or an object that in some sense is quantum, but in another not – contains the possibility for an indefinite progression: the assumed final result of such a progress is a concept with no instance. In infinite progress these two aspects are explicitly constructed.

Hegel begins the new section by reminding us of the results reached so far. The concept of infinitely small or infinitely great quantum contains two incompatible predicates: on the one hand, it is supposed to be quantum and thus some determinate, limited quantity; on the other hand, it should be infinite, that is, larger or smaller than any other quantity. If these characteristics should hold independent of the context, they clearly could determine no object: infinity as the contextindependently largest or smallest quantity is impossible in Hegelian system of numbers. Still, it is quite possible that a quantum satisfies these characteristics in different contexts: this is the familiar form of Hegelian contradictions. It is when one tries to construct an example of the infinite quantum in the first sense, when one ultimately faces the possibility of indefinite progression of infinite quantities in the second sense: any quantity that seems to be the largest or any value of a quantum that seems to be the largest possible can always be shown to be smaller than some quantity in another context.

2./443. As a degree quantum is in one sense determined without consisting of other quanta: in another sense, it must then be determined by its relation to other quanta – it can take any value depending on the reference point. In one context, a quantum can thus be interpreted as larger than any quanta – as infinitely large: in another context, this largest quantum can be seen as another quantum – a quantum can be identified with what seems to be different from any quantum.

We are still on some fairly familiar ground: Hegel recounts how exactly we were able to find an instance of an indefinite progression. We began with some quantum that was in the shape of a degree: that is, it was a quantum that neither was nor could be divided into a collection of units. Any quantum could serve as an example, if it was interpreted properly, but we could take a degree of temperature as a concrete example. Now, the degree was still a quantum, and its quantitativeness was due to its occupying a certain place in a scale of potentially infinite number of degrees: temperature is a quantum, because it has a place in a temperature scale. It is the place of the degree in the scale that is marked by its numeric value, that is, it marks the relationship of degree to a certain unit degree. If we changed that unit degree, the numeric value of the degree in question would change: indeed, we could make the degree as large or small as we liked. In one sense, then, we could make degree larger than any known quantum just by interpreting the scale anew. Yet, it would still have numeric value and thus we could always find a further value it could take: this is the indefinite progression we required.

61

3./444. The indefinite progression constructs the general structure of quanta: quanta are defined by the arbitrariness of their value.

Hegel reminds us that this indefiniteness of value is not something arbitrary for quanta. Indeed, even the definition of quantum contained that it was something merely external even to itself, that is, it had value only relative to other quanta. The indefinite progression merely explicates this characteristic. The general concept of quanta is, as it were, an infallible recipe one can immediately exemplify: all one needs is to literally look at how one and the same quantum can assume different values and become as great or small as one pleases.

4./445. In indefinite progression we can move away from any determinate quantum, but also from any supposed stage of infinity: these are mere aspects of the progression. Thus, the progression solves the contradiction by showing that any quantitative stage is arbitrary: thus, we can see that all we will ever find in the progression are more quanta.

Indefinite progression is not just showing that one could always exceed any quantitative limit: that is, that from any quantum one could construct something greater or smaller than that quantum and thus “infinite” compared to it – or change that same quantum into one of larger or smaller and thus “infinite” value. There is always the other direction that from any situation with a supposedly infinite quantum we can construct a situation where that quantum seems once again limited by a larger or a smaller quantum. There are thus two aspects present in every quantum in the progression: one where the quantum is limited and another where it is not limited. There is no infinite quantum that we should try to reach, Hegel says, but only a potentially infinite series of finite quanta. It is this idea of unreachable infinity that makes the infinite or indefinite progression into something unpleasant: e.g. in moral progression it seems futile to try to make progress when the result can never be reached. Because there is no limit to this progression of quanta, it is actually arbitrary where on the scale we happen to reside: it is always possible to reinterpret the scaling system if we wish to say that the degree in question has a certain quantity. Thus, it is no shame if we still have a lot to do morally, because we will always have lot to do and will never be finished: not even after an infinite life time.

5./446. When we have constructed quantitative from qualitative structures, we have in one sense overcome limitations, because in quantitative structures qualitative limits are merely aspectual: yet, it is only potentially this, because quantitative structures consist of many different quanta with some of them greater than others. The search for the greatest quantum realizes the original potentiality: now we see that any quantum could be constructed from any other quantum, but this insight is yet possible only from an external standpoint. Still, we have determined the quality of different quanta and thus constructed qualitative structures from quantitative: we have seen that all quanta [in a context]

62 are aspects of one quantum that could be called infinite in some sense.

A qualitative structure consists of situations and objects separated by having different characteristics: for instance, a structure of colours or feelings is qualitative. Now, in the road from quality to quantity, Hegel presented a possible interpretation whereby any qualitative structure could be seen as consisting of mere aspects of one whole: this was the transition from finity to infinity. Indeed, a pure quantity or any quantity when unrelated was characterized not by one state, but by a series of states: a quantity is something that could be always divided further. Now, it seems then that in quantitative structures we should have passed all difference of situations and objects. Yet, we can multiply our first quantum by dividing it and thus produce a system of many – indeed, indefinitely many – quanta. Thus, although we seemed to get rid of differences, they return in a new guise, that is, in form of different quanta. Now, the indefinite progression or the search for an “infinite” quantum appears in one sense to be mere avoidance of the problem, because the infinity can never be truly or contextindependently be found. Yet, in another sense, as we have seen, the indefinite progression has been a step forward. It reveals that one quantity could take the place of any quantity, provided the reference point was changed accordingly. Because of this arbitrariness of value, we are allowed to make another construction, that is, we are allowed to interpret the apparently different members in the series or scale of quanta as modifications or aspects of one and the same quantum: e.g. different temperatures can be seen as possible states of one object “temperature”. In other words we might say that any quantum is a variable within its own number system. Why does Hegel call this result the return of quantity to quality? Consider the example of the scale of temperature seen as a one entity. This one entity is nothing but the quality of temperature that can take different values. Similarly, any scale or series of numbers, interpreted as a unity could be called “quality” in this sense. Of course, there must be a possibility for some different scales, in order to be meaningful to speak of qualities in this case: remember that Hegelian qualities are always separating some situations or objects. Indeed, the next task in Logic is to construct examples of different scales or number systems, that is, to construct qualitative from quantitative structures.

6./447. The infinity sought in an indefinite progression is qualitatively different: the arbitrary change of values can be stopped only by a qualitative construction or essential change. Quantum is quantum, because it can be as large or small as one wants, but this potential progress has been integrated into one unity.

This paragraph admits two different interpretations, both of which seem equally valid. First of all, Hegel may perhaps be merely paraphrasing what he said in the previous paragraph. What we looked

63 for in an indefinite progression was a thing that could be called quantum in one sense, but in another sense was no quantum or was larger or smaller than any given quantum. Now, this task is solved once we interpret all quanta in some number system, series or scale as modifications of one “quality”. Then we can take any quantum – or more precisely, any quantitative modification of the quality – and interpret it as larger than some other modification: the fact that there are still larger modifications is of no consequence, because this a mere difference of aspects instead of independent situations or objects: it is indifferent where we happen to be in a quantitative scale, because all places are only modifications of one place. The other possibility to interpret this paragraph is to see it as a reminder of the second sense of quantitative infinity. A quantitative construction can never bring us to anything that could be called infinity: adding more units to a set won’t produce an uncountable set. It is only with construction that could be called qualitative in some sense – a construction that would break the limits of different scales of quantities – that we could get into a quantity immeasurable according to one scale. In the future we shall see what sort of construction could be used in that effect: a mathematical one suffices, although not one of merely arithmetically adding more units to quantity.

7./448. A structure of determinate quantitative states is a move away from qualitative states of being: thus, when we move away from determinate quantitative states we must find qualitative states once again – we try to find it when we move to ever larger or smaller quantities and we construct qualities when we understand all quantities as aspects of one scale.

The language of negations Hegel uses is seductively simple to comprehend: of course, if quantity is negation of quality, then negation of a quantity is, as a negation of negation, an affirmation of quality. Yet, this interpretation puts too much weight on the confusingly familiar word “negation” and its connotation – and it makes no difference whether the negation here is understood as propositional or as conceptual. Negation of a negation in the familiar meaning lands us back to the exactly same result where we started: either to the same sentence/proposition or to the same term/concept. Yet, Hegel’s negation of negation in general and particularly in this case does not result in the exactly same thing as we faced at the beginning, but at most – as here – to something with a similar structure: the negation of Hegel must be understood a movement – construction – from one situation or context to another. When we understand Hegelian negation thus as a replacement of one context with another, Hegel’s description becomes more complex, but perhaps also more acceptable for the uninitiated. Quantum is a sublated or integrated quality: this refers to the fact that quantitative states of being have been constructed from qualitative states of being 1) as alternative interpretations 2) in which apparently essential differences have been reduced to mere aspectual or quantitative differences:

64 what was earlier seen as differences of quality has now been interpreted as differences of mere more or less – e.g. light is not different from darkness, but merely more luminous than it. Now, one can make changes between quantitative states of being or one can interpret one and the same quantitative state as having a different value (compare it to a different reference point, that is). In such a change of reference we may be said to implicitly look for something beyond all quantitative states: something that would be incommensurable with the quantities we know of. The solution is to ignore the difference between the different quantitative states found in the indefinite progression – or not to ignore them, but to interpret them as merely aspectual – and move to new “scale of quantities”, which is then qualitatively different from the first scale.

8./449. From quantum can be constructed other quanta, but all the quanta can then be interpreted as aspects of one scale with certain characteristic. Here we have constructed an instance of a quantitative relation [or function/mapping]: this can be interpreted as consisting either of different quanta or of aspects of one quality. Although the values of quanta are variable, their relations are not.

We know that the value of a quantum can be decided arbitrarily: we can take it to be as large or as small as we like. Now, such a change of reference point does not affect the relations between quanta: it is these relations then which are the true quantitative infinite we have been looking for. Supposing we first regarded ourselves quite small when compared to an elephant, but then changed our viewpoint and noticed that we are quite large when compared to a mouse: this change of a reference point does not alter the fact that we are still smaller than the elephant. What is then stable in a number system are not the values, but the relations of quanta: this resembles Leibniz’s idea that space and time are actually relational structures instead of anything absolute – I can change the reference point or origin of my coordinate system without changing the spatio-temporal relations between objects. Now, before Hegel can move on to investigate these relations he must show how to construct an example of them. This is connected with the discussion of infinity. We can produce or discover new quanta or new values of quanta – this was the starting point of the indefinite progression. Furthermore, we have just seen that we can interpret all such quanta produced as mere aspects of the same “quality”: they can be seen as mere non-independent places in the same scale or as mere different values for the same variable quantity. What Hegel is saying is that there are some operations by which we can change – or as we would nowadays say, map – one value of quantity into another: e.g. it is now 2, but we can double it and make it 4 – thus, what we now take as two, we can also take as 4, and thus, in some sense there is no essential, but only relational difference between 2 and 4. Quantitative relations must then be constructed, before we can admit them, hence, although Hegel speaks of stable relations, it is more natural for us to speak of functions and

65 mappings, and furthermore, interpret them operationalistically, that is, as quantitative constructions.

Remark 1: Conceptual determination of mathematical infinite

When we hear someone talk of mathematical infinity, we at once think of Cantor and his transfinite cardinals. Yet, Hegel couldn't understand what transfinite cardinals are – how anyone could make a set of all natural numbers instead of all numbers within this context would be completely beyond him. It is more the infinity in the geometrical matters that interests Hegel: here is a concrete case where infinity and especially infinitesimals occur e.g. in the relations between points and lines. Particularly the infinitesimal calculus interests Hegel: the results of the calculus were undeniable and yet the use of the notion of infinitely small appeared to go against everything Hegel had just presumably argued to be outside what could be constructed and thus meaningfully spoken of. Thus, Hegel sees it as his duty to stop the ongoing construction of categories for a moment in order to analyze the problematic of infinitesimals: do we truly need them in differential and integral calculus and how do they in general work?

1./450. Use of infinity in mathematics has led to great results, yet, it has not been justified in any other way nor has it been shown that we could construct examples of it.

The infinitesimal calculus seems so common thing to us – a mere high school kid could learn the formulaic differentiation and even the basic theoretical ideas need not much explanation. Thus, it is difficult to remember that at the beginning of the 19th century the art of differentiation and integration was still quite new – while there were some precursors, the systematic use of infinitesimal calculus was inaugurated by Leibniz and Newton – and the wealth of results opened up by the new methods seemed enormous: calculation of gravitational effects could have not been possible without the calculus. Despite the great results, the theoretical basis of the calculus was still in shambles: one had not yet learned to evade infinitesimals properly and even less was there any notion of a consistent calculus of infinitesimals. Hegel desires especially the development of the required infinities from concept: this does not mean mere analysis of what the calculus involves, but concrete construction of examples for such basic notions of the calculus as infinitely small etc. Such construction the ancient mathematicians required for all their concepts – Euclid begins by postulating the capability of producing straight lines and circles – but it seemed to have drop out from the heads of the modern mathematicians.

2./451. It is unscientific to use a method without justifying it: furthermore then no one knows when it can be used.

66 The result of the earlier mentioned unclarity was that infinitely small and other terms were used in a too relaxed manner without anyone clearly conceiving what could be done with them and particularly what not. For instance, we know that some infinite series tend to have finite sums, that is, when we add one more term of the series to the earlier partial sums we come nearer and nearer to some finite number. Now, suppose we had a series with a positive and negative unity alternating. It seems a meaningful question to ask what is the sum of that series, because the partial sums never grow too much. Indeed, such questions were asked – the answer must be one or zero, they quickly concluded – but the true answer is that there is no sum of such kind. Another possible source of amazement could have been continuous graphs with points that were not differentiabale: sudden angles in a curve, as it were. Indeed, it would not be long before even more puzzling graphs would be introduced: continuous, but nowhere differentiable.

3./452. Mathematical infinity is important, because it is truer than the so-called metaphysical infinity: when the latter is used to attack the first, the mathematician can defend herself by denying that she is studying metaphysics.

By metaphysical infinity Hegel means infinity as Kant understood it, that is, the totality of e.g. everything in nature that is unreachable by human cognition: as we have seen, such an infinity would be either impossible or then only contextual, that is, bad infinite. Mathematical infinite, on the other hand, should be true infinity, says Hegel. One may perhaps be tempted to think that the true infinite Hegel mentions should be then interpreted as an infinity that has been actualized, because mathematical infinities are actual: this is an obvious anachronistic paralogism, as the truly mathematical account of actual infinites had to wait Cantor. Instead, we have to interpret Hegel’s idea of mathematical infinities through his idea of the true infinity, which we have shown to mean a perfectness of method used in creating indefinite progressions: we shall see how this account fits with the structures Hegel takes to be infinite in mathematics. Hegel then would see no need for mathematics to account for its failings against the metaphysicians: instead, it is the metaphysics that have the wrong idea of infinity. Yet, the mathematicians of Hegel’s time were not completely certain as to how the infinite they talked of should be interpreted: they spoke of infinitely great and infinitely small without knowing what such expressions could actually mean. Thus, a philosopher like Berkeley could well ask what sort of ghosts the mathematicians meant by their “vanishing” or “infinitely small” quantities. The escape route of the mathematicians was then, as it also nowadays often is, a formalism. Mathematicians admitted that they didn’t know what the nature of the entities they used was: still, they said, one could operate with these entities, as long as one just followed the rules that had been set for operating with these entities.

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4./453. True, mathematics doesn’t have to show how to construct examples of the structures it investigates: yet, operating with mathematical infinities is in one sense different from operating with finite quanta, although in another sense it applies ordinary calculus for mathematical infinities.

Hegel admits that mathematical formalism is actually quite valid standpoint and that it is not the task of a mathematician to find examples for the abstract models they study. Still, the differential calculus of Hegel’s time had not yet reached even satisfying formalism. What was especially lacking was clear rules how to operate with the infinitesimals. In one part of a calculation infinitesimals were counted just like regular quanta were: thus, one spoke of dividing one infinitesimal with another etc. Yet, when such embarrassing things as fractions with infinitesimals were obliterated, one just identified infinitesimals with zeroes: and presto, the result with no infinitesimals was reached! It is this ambiguity of infinitesimals that does not satisfy Hegel in the “calculus of infinities”, and we shall see how he attempts to deal with it.

5./454. Sometimes the results of the infinitesimal calculus can be later verified through other means, but this is not always possible: furthermore, this is no proper justification. The infinitesimal calculus uses apparently inaccurate means to gain accurate results, although mathematics should be an accurate science.

The two remarks Hegel here aims against the differential calculus – its justification through its fruitfulness and the apparent inaccuracy involved in its basic operations – derive from same roots. It is empirical methods that are usually valued by their results: new scientific equipment is used, because they provide us new phenomena or because they let us test propositions that used to be untestable. Mathematical methods undoubtedly also provide us some results, but different from empirical results. Mathematical results should be completely certain, which empirical results can never be: there is an enormous difference between measuring three angles of a triangle and proving what their sum must be. Although an empirical method would give us correct results now, there is always the possibility that it fails to do so in some cases. This brings us to Hegel’s second point. It is empirical methods that give only results within some level of accuracy – although all of them are not as inaccurate as sniffing, which Hegel here playfully mentions. Now, the results of the calculus shouldn’t be inaccurate, Hegel admits. Still, when it was explained how the calculus should work and especially how we were justified in discarding the infinitesimals at the end of the calculation, the explanation was often based on the insignificancy of them: they were so small that our measures couldn’t detect their influence on the final outcome. Now, the final theoretical explanation of this “insignificancy” was yet to come, and thus Hegel was justified in asking how such an empirical notion as insignificancy of an addition could be reconciled with the accuracy of mathematics in

68 general: once again the problematic place of calculus of Hegel’s day between mathematics and empirical science raises its head.

6./455. Let us see how the mathematical infinite has been conceived and justified. 7./456. Mathematical infinite is usually defined as the greatest or the smallest quantity. On one hand, this is no true infinity, but such that creates indefinite progression: on the other hand, it implicitly says that there can be quantities incommensurable with the given quantities.

The most common idea of quantitative infinity is not that involved in infinitesimal calculus, but the familiar idea of a magnitude, greater (or smaller) of which there cannot exist. Now, such an infinity has not yet been introduced even in the modern mathematics – Cantor introduced a realm of infinities with no largest infinite – and it certainly goes against Hegel’s constructivism, as we have already seen. The greatest number is meaningless concept, because we can always find a larger number for any given number. Then again, the concept could be accepted as contextual – greatest number in this situation – but this greatest number would be different from one context to another and we could always find a context with yet larger number: this was the idea which led us to the indefinite progression. Still, as we have seen, there is yet another way to interpret this difficult concept. “Largest” number could mean something that was number in another context, but in this context or compared to the quantities of this scale it would be incommensurable – “larger” than anyone of them. Such a concept can have examples, provided that there are different scales of quantities: for instance, the scales of areas and lengths form such different scales.

8./457. Usually one forgets that infinite quantum cannot be quantum in the same sense as a finite quantum, but still is quantum in some sense.

Hegel points out two mistakes that can be commonly made when discussing “infinite” quanta, that is, quanta that are of different scale than some given quanta: one could imagine, for instance, areas compares to lengths. Firstly, we may make the mistake of interpreting the quanta of the different scale as same kind of quanta as those in first scale: area is quantity and length is quantity, thus, area must be a collection of infinite number of lengths. Clearly, Hegel would be against all such talk: it is nonsensical to say that an area consists of an uncountable number of straight lines. The second mistake is to assume that an “infinity” cannot be true quantum in any sense: instead, it should be some sort of absolute that cannot be quantified in any other manner, but as being “the greatest number”. Yet, Hegel insists, we can make sense of such infinity for some scale of quantities, only if it is a quantity in another, incommensurable scale, like areas form a scale of their own, while still being “infinite” in comparison to the quantities in the scale of lengths.

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9./458. Kant claims that maximal quantity cannot be identified with infinity: “there are no maximal quantities, but infinity means a whole that is incommensurable with some arbitrary unit”.

In one sense it seems that Kant and Hegel would actually be in partial agreement in how infinities should be understood. First of all, both Kant and Hegel deny that infinity could mean maximal quantity in some scale – or in Hegel’s case, that every such maximal quantity is maximal only in some context – just because it is always possible to find some further quantity in that scale which is still “more maximal”. Secondly, Kant does admit another possible idea of infinity, or that of incommensurability. We could choose some arbitrary unit – let us say, a point – and then notice that there are some quanta that cannot be measured with such a unit, because they contain “infinitely more” than every could be described in finite number of such units: a line is such an “infinity” in comparison with points.

10./459. Maximal [or incommensurable] quantities are truly always contextual, whereas true infinities [or quantitative functions] are not.

But the two forms Kant mentions are only shapes of bad infinity: they are manners of interpreting the apparently contradictory combination of infinity and quantitativeness. Their difference lies only in what contexts the quantitativeness and infinity appear. In contextually maximal quantity, the infinity lies in a limited context within the scale, while the quantitativeness or the possibility of still larger quantities lies in a wider context within the same scale: in incommensurable quantity, the infinity lies in the relation of two different scales, while the quantitativeness lies within the scale of the incommensurable quantity. Now, Hegel reminds us that there is yet a further and true mode of quantitative infinity and criticizes Kant for not noticing it: this is the familiar idea of a function regulating or providing possibilities for constructing further quantities in the same series or scale. Hegel calls this form of quantitative infinity here infinite difference: a reminder that he is going to interpret infinities in differential calculus as functions instead of concrete quanta. Now, Hegel’s criticism is bit harsh to say the least: the use of the term “infinity” of constructions or functions governing some potentially infinite series is peculiar to Hegel himself and he couldn’t have expected that Kant would have used the term in such an outlandish way. Furthermore, the infinities or infinitesimals in differential calculus were usually understood as true “infinitely small” quantities – or then one wanted to get rid of them: why then would Kant have interpreted infinities in some other manner? 11./450. True or transcendental infinity for Kant means that quantum can never be measured completely: this leads

70 clearly to infinite progress. In abstraction quantum is complete and thus limited, but consciousness can never find this determinate value, but must step over all limits.

The true fault in Kant’s account of infinity in Hegel’s opinion is that Kant seems to smuggle the impossible or meaningless idea of the uncontextually greatest quantum or number back to the discussion. Of course, Kant merely speaks of this greatest quantum – or that which cannot be found by counting – as transcendental in contrast to empirical concepts, that is, it is a term belonging to pure reason and thus being in no way straightforwardly intuitable. In this case, the concept is not even applicable to any possible experience, because it is larger than any possible experience: it is transcendent. Still, Kant would admit that there is at least the possibility, in some sense of the possibility, that there would be a determinately infinite object or collection of objects: in itself, that is, in the world as it is untainted by our consciousness and its restrictions, there might be anything that would exceed our cognition. An object in itself – such as the world – could then be “limitedly” or determinately infinite, although the world for us would have no determinate quantity: that is, we could see only greater and greater “worlds”, but not the world or totality of everything. Hegel is, firstly, not very convinced of such empty possibilities: if we can’t cognize it or something like it, then we cannot say it is possible. Furthermore, he could point out that the idea of uncontextually greatest quantum is actually contradictory even in the formal sense of the word, when quantum is understood as something for which one could always find greater values – although the object would be “infinite” in the sense that it would exceed all that we can understand, it would still have “larger” infinities in which it would be contained. This is the crucial point where the Cantorian infinities are revealed as not being the greatest quantum after all, because they can always be topped by even greater infinities: and the so-called Absolute would be for Hegel a mere fancy – the realm of entities, concrete or abstract, is according to Hegel inexhaustible.

12./451. In true mathematical infinity both the finite quanta and the contextual infinities have been shown as mere moments: the result is simple, but not like intensive quantity that is still related to number of other quanta in the same scale. In one sense, there are many quanta, in another sense they can mapped to one another: quanta are mere values of a variable quality. Values are determined only by their relations to other values: they are mere aspects and otherwise nothing.

Hegel summarizes once more the construction of true quantitative infinity from finite quanta. At first, we had to show that there were two other possible constructions, that of “infinite” or contextually infinite quantum from any finite quantum and that of interpreting any infinite quantum in some context as a finite quantum: that is, we had to show that we had the ability to find a largest number in any given context of numbers and to find a context with larger numbers for any given

71 context. These possible constructions suggested that there was no essential structural difference between any quanta – they were all “small” or “great” compared to some other quanta – and thus we could interpret all these quanta as mere versions or values of one underlying object. Hegel compares this construction with the construction of intensive from extensive quanta: there we abstracted from the fact that the quantum in question consists of other quanta, but the quantum still remained in relation to other quanta, because the relational structure between different quanta was retained. Here, on the other hand, all quanta have been incorporated as aspects into one scale or “quality”: e.g. different temperatures could be interpreted as possible values of quality “warmth”. Still, the different quanta are not abstracted of, but only interpreted as moments or aspects or possible values. Thus, the scale contains a relational system of possible values: some possible values of a scale are e.g. twice the size of other possible values. Hegel describes this relationality of the values by saying that outside these relations the separate quanta are only zeros: on one hand, this is only a figurative way of saying that quanta are no longer interpreted as independent, on the other hand, it suggests an analogy with the differential calculus where the socalled infinitesimals are only relational entities or placeholders for a function.

13./452. This abstract structure of infinity is the basis of all forms of mathematical infinity, from those that still can be interpreted as ordinary quanta to true infinities. 14./453. First form of such relational structure is a quotient like 2/7, which is a finite, but not ordinary quanta, because it is determined by two other quanta behaving as unity and amount. 2 and 7 are in one sense independent, but in another sense they are mere aspects of the structure 2/7: they could be replaced by any pair in a same proportion or it is their relational structure that counts.

Hegel has just presented an abstract schema of true mathematical infinities, and in the following paragraphs he notes that actually many mathematical structures can be described in terms of this schema. This is not to say that all of these would be true mathematical infinities in the best sense of the word: some of them are “true infinities” only within some context or situation, while some aspects of them are closer to the characteristics of regular quanta. Thus, we might anachronistically say that Hegel is trying to find the intended model for the structure in question, that is, the true mathematical infinity. We are thus looking for mathematical structures where there are in some sense different, independent quanta, which in another sense have values only in relation to other quanta: that is, structures where the potentially relational nature of all quanta has been actualized in some measure. The simplest case is that of two quanta having a fixed relationship: Hegel speaks of fractions, because two such quanta determine a fraction. We may think of a quantitative scale with two places, one of them which is a reference point – or unity, as Hegel says – and the other of which is

72 compared to this reference point. Such a structure is not a quantum in the extensive sense of the word – we cannot say that it would consist of a fixed number of unities – yet, it may be used as an intensive quantity in a scale of degrees: that is, it has a determinate relation to other similar quanta, and in a sense, even to “ordinary” quanta – it is commensurable or “finite” on the viewpoint of quanta like two or three. Now, when we abstract from the fact that one of the quanta in a fraction has been selected as the reference point – and indeed, either of them could be selected to that task – then we face just a structure of two variables changing in a fixed relation. True, we could abstract from the fact that one quantum, like 2 in Hegel’s example, is part of this larger structure: 2 is 2 independently of 7. Still, as we have seen, any quanta can be interpreted also as a variable, depending on what unity it is compared: 2 is 4 halves, for example. Now, in the fractional structure we may let the quanta vary through indefinite values, without the structure changing, as long as the relation remains similar: 2 could be altered to 6, if 7 is just altered to 21. The fractional structure forms then a kind of a method or rule for determining one number when the other number is known: thus, it already has some affinities with what Hegel calls true mathematical infinity.

15./454. Fraction or quotient is still not a very good specimen of mathematical infinity, because the numbers can be abstracted from their relation and even their relation is in some sense an ordinary quantum.

The reason why fractions cannot be the intended model for the mathematical infinity is that although in some context the fractions do have the characteristics of the required structure, in other, more essential contexts they fail to have these characteristics. The numbers in the fractions can be interpreted as mere aspects of the fractional structure, but it is more natural to see them as independent numbers that can be abstracted from this structure: 2 is 2, although there wouldn’t be number 7 to which relate it. Furthermore, the relation or function of 2 to 7 – “the fraction” – could be interpreted as an entity of another level: it is a “true infinity” in sense of being the rule that governs the variability of the numbers in the relation of two to seven. Yet, in another sense, we can also take it as a number among other numbers. In fact, we may embed the system of “regular” numbers within a system of fractions – by mapping them to functions of a unity of the “regular” system to the number in question – in such a way that the quantitative relations among the “old numbers” were retained. In effect, we would have merely added some new degrees in the scale of numbers, and degrees which would still have relationship of a number to other degrees, that is, which would be commensurable with all other degrees.

16./455. Numbers in a fraction can be replaced by indeterminate letters and we can thus abstract universal schema of fractions, but even the letters are supposed to refer to numbers, albeit indeterminately.

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This paragraph is interesting because it is one of the few places where Hegel speaks of what we would call algebra, but what Hegel refers to as universal arithmetic. Now, the general point in algebra is to abstract some general schemas from concrete cases of calculation: the modern algebra also contains the possibility of changing the conditions within that schema – dropping or adding axioms that the operations should follow etc. – but Hegel is not yet aware of such possibilities. Now, such a general schema is in some sense “more infinite” compared to a concrete case: it is satisfied by more situations and objects. Thus, a general schema of fractional relationship or function from one schematic quantity b to another schematic quantity b could be satisfied by many numbers and functions. Yet, such a general schema is a mere abstraction that is meaningful only in relation to some possible concrete case: for instance the formula “d + e” makes sense only because we know how the sign “+” is to be applied in concrete cases of calculation – even in modern algebra the abstract systems are meaningful only when some possible application can be found for them, although they need not be applicable to any numeric systems. Thus, an indeterminate fraction – fraction in itself, we might say – or the general schema of fractions is meaningful only in relation to some concrete fractions or relations of concrete numbers: thus, it isn’t a whit better example of a “true infinite” structure than determinate fractions are.

17./456. Quantitative relations have thus two aspects: a) they are in some sense quanta, but b) in another sense they consist of qualitatively opposed sides – furthermore, one can always return from this opposition and regard the relation as a mere independent quanta.

After the short excursion to algebra, which proved to be a dead end, Hegel returns to consider the concrete relations or functions between numbers. Hegel emphasizes now that the function itself can be considered from two aspects. Firstly, in one context – when compared with other similar functions – it can be considered as a quantum: it is characterized by a certain degree like 2/1 or 3/7 and these degrees can be ordered, related to one another etc. Furthermore, as has been mentioned, the so-called “regular” numbers could be embedded in the set of these fractions. Secondly, in another context – when compared with the system of those “regular” numbers – the fraction is not so much a quantum, but a rule governing the relations of two quanta: it states that while one quantum varies, the other must vary in such a manner that their relation remains same. Now, it is possible to move from one aspect of the fraction to another. We can analyze a number given in the series of fractions and notice that it has a more intricate structure than is apparent at first. But we can also idealize the “ordinary” numbers within the structure of fraction or interpret them as mere ideal aspects of it and thus return to the scale of fractions, where the fraction is a quantum with a determinate value.

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18./457. Fraction can also be expressed as an infinite series: then it is understood not as a relation, but as an amount. It is not important that the decimal expression seems to consist of decimals, that is, of fractions: similarly, in interpreting signs of whole numbers we abstract from the fact that numerical signs literally express sums of decads. We also ignore the fact that some fractions seem to have finite decimal expressions, because all fractions have infinite expressions in context of some number system.

Hegel notes that the two aspects mentioned in the previous paragraph can be emphasized in different manners. Until now, we have concentrated our attention on the fraction being a complex structure of a function between quantitative states of being, but we can also make the other aspect as our reference point: that is, we can view the context in which fraction acts like an ordinary quantity, that is, where we relate it to other “fractions”. Now, as a “regular” quantity, fraction can be interpreted intensively – as a degree in a scale of degrees – but also extensively, as a sum of smaller quantities: this is possible, if we can pick out a suitable unit, for instance, ½ can be expressed as sum of five decimals. It is of no concern to Hegel that this quantitativeness is restricted to a different scale apart from the “regular” scale of quantities. For Hegel, there is no true regular scale of quantities, or better, it is arbitrary which scale to choose as the reference scale to which others should be related. Indeed, as Hegel points out, even in dealing with the so-called “regular” scale we for sake of convenience use signs referring to multiple scales, related as scales of decades etc., and to sums of representatives of these scales. In practice, we abstract from this circumstance and speak of all these numbers as if they occupied one scale of quantities, and similar abstraction should work in the case of fractions too. Hegel also points out that in some contexts or in some number systems the fraction can be expressed only as an infinite sum: we can only produce better and better approximations of the fraction, but not a concrete number corresponding to it. As Hegel well understands, in case of fractions this is just a matter of choosing the right context: a fraction expressible in one number system is not expressible in another and vice versa, depending on e.g. the choice of units – ½ can be expressed as five decimals, but in a tertiary base it can be expressed only as an infinite sum 0,11111... where even smaller thirds of thirds must be added in order to asymptotically approach the required fraction.

19./458. In a form of series fraction cannot be infinite in the same sense, because it is no longer a relationship: still, it at least is infinite in the number of its members.

Changing the designated or the “essential” side of the fraction from its functional structure to its value in the scale of fractions must inevitably change its other characteristics. Particularly, it

75 becomes suspect whether we can reasonably speak of the fraction as infinite in the Hegelian sense. Hegel’s true mathematical infinity was supposed to be attached to functions that rule or govern the variability of quantities: thus, fraction could be seen as infinite when it was interpreted as a function relating certain variable quanta to one another in a fixed relation – given value of one quantum it was possible to construct the value of other quantum through a simple operation of multiplication. When we abstract from this internal structure fraction and see it as a mere quantum, the infinity in the Hegelian sense is lost. Yet, it seems that infinity could be gained in another manner: we could regard the fraction as a sum of “infinitely many” other quanta in some number system: that is, we could see the fraction as a beyond that could not be reached by any normal operations in that number system. Could this sort of infinity replace the infinity of a function?

20./459. The series is only indefinite progression, because it tries to express qualitative relationship in numbers: the series will therefore never reach the required sum, although it can approach this sum indefinitely.

A fraction can be expressed in some number system only through approximate sums: this involves indeed a form of infinity, but it is that which Hegel calls bad infinity. The fraction itself, compared to the number system in which we try to represent it, is something beyond reach: we cannot construct an example of the fraction within that particular number system. Thus, we cannot even say that the fraction would be a number according to that number system: it has no quantitative relation to other numbers in that number system. The indefinite progressions are formed out of a never-ending possibility of alternating between two sorts of states. Here, Hegel suggests, the sorts in question are qualitative and quantitative. We suggest some number as representing certain “qualitative” state of being, that is, certain structure of dependency between two variables. When we compare the suggested number with the fraction, we then notice that the representation is not perfect: the number is still too small – this is the transition to the first direction. Yet, we can also make a transition to the other direction: from any failed attempt of representing the fraction we can find another and better representative. Thus, we have the possibility of approaching the fraction with better and better approximations. Of course, the approach itself is unimportant for Hegel, because it will never truly achieve its goal: what is important is the mere ability that can be applied to different cases.

21./450. Infinite series is truly inaccurate, whereas true mathematical infinity [or operation of differentiation and integration] only appears to be so. Latter is sometimes explained through infinite series, but they shouldn’t be mixed.

This paragraph is undoubtedly a somewhat obscure reference to the way how the infinitesimal calculus was interpreted in those days. One way to understand it is to use the idea of an infinite

76 progression, but Hegel is against this procedure, not so much because it is mathematically faulty, but because it fails to catch what is essential in the calculus. Still, there is a more general point to be made. What makes ordinary quantitative series unsatisfactory is that there seems to be no general rule governing it: we merely sum an indefinite number of quantities together without ever reaching the result we are looking for. Now, although the result of an infinite sum cannot be reached through the addition, it can be characterized in another way, namely, through the method by which we can get better and better approximations for its quantitative representations. This rule governing the sums is the true mathematical infinite – and we shall see how this structure can be applied in infinitesimal calculus – and of a completely different nature than an potentially infinite series: its structure is closer to that of a fraction in its shape of a rule governing variability of quantities. 22./451. Infinite series is only a bad infinity or actually finity, because it fails to produce what it should; its “finite” expression is infinite in the sense of being perfect.

In the mathematics of Hegel’s time it was common to call the sum of an infinite series – if there happened to exist one – finite. This expression emphasizes the number of signs used to express a certain quantum: while the series should have infinite amount of members, its sum consists of a finite number of signs. Now, Hegel is not very interested of this sort of infinity: in fact he is outright skeptical of the existence of such quantitatively infinite collections. On the other hand, the “infinite” series presents an example of what Hegel calls bad infinity, as we have seen: every member of the series fails to fulfil the final criterion of being representative of a certain fraction and can thus always be replaced by a better approximation. In fact, every member of the series is “finite” in the sense of failing to satisfy some criterion. Euler’s “finite expression of the series” or its sum, as we would nowadays call it, satisfies the criterion and is thus in a sense “infinite”. Of course, it is not supposed to be a member of the same number system as the unsatisfiable terms of the series: it is of a different quality altogether, or it is a function governing the first number system.

23./452. Infinite series shows that there are further numbers outside it, while within fraction the differences exist as aspects: fraction is not a sum, but a relation, while infinite series consists of mere sums or collections. That what is missing of an infinite series is infinite, because it is beyond, while in another sense it is a regular “finite” quantity.

The previous paragraph argued that the usual denotations of the finite and infinite form of the series should be reversed: the series itself or its members are always merely finite, while the sum of the series or the rule which the series tries to represent is truly infinite. Now Hegel points out that the term “sum” is also unsuitable for the fraction investigated. The point is that is cannot be represented in a form of a sum in this particular number system. Indeed, its structure is not that of a sum or

77 simple quantity, but of a relation or function between quantities. It is “the negative of the negative”, that is, it is a common relation governing two different variables, which are thus “negative” in comparison with each other: it contains differences as aspects within itself. The members of the series, on the other hand, are nothing but sums or simple quantities: they have the form of a sum of a finite collection of quantities. Now, in one sense both the fraction and the “finite” or “imperfect” members of the series share the same number system: we may construct for them a common number system where to compare their values and even count the difference between them. Yet, even here the “true infinite” works in some sense as a rule to govern different quantities. The required fraction – or any quantity, as the deeper structure of the fraction does not count here – forms a kind of “universe” for the members of the series: the members can “take up space” in that “universe”, but they can never completely fill it. In one sense, the difference between the incomplete sum and the “total” quantity is something beyond the incomplete sum: it is what the sum lacks in order to become “perfect” representation of the total quantity. It even remains a constant beyond, because the sum can never empty the total quantity. Still, it is at most contextually perfect: we could as well reverse the roles of the sum and the difference and then the difference would look imperfect, because it too isn’t equal to the complete quantity. The total quantity works here as a rule for limiting the areas of two quantities: no matter which is tried to augment, neither of them can completely destroy the other.

24./453. Some series do not have final sum that would have the relation of a number to the members of the series: such incommensurability of different number systems entails a more complex form of infinity. Yet, the form of series involved is as futile as in the case dealt before.

At first it seems obvious that Hegel is speaking in this paragraph of series that do not have final sum, either because the series exceeds every limit or because it e.g. alternates between many values. Yet, Hegel clearly admits that in the case he is talking the series does have a limit, but it is only incommensurable with the series itself, that is, the sum and the members of the series do not have a relation of a whole number to whole number. Now, the fraction does have such a relation and is in some sense therefore, “number of a same kind”, that is, there is some number system where the fraction and the whole numbers are both quantities. Besides such sums, there are other sums that cannot be presented as part of the same number system – when number system is understood as a system with a common unit whose multiples all numbers are. Such series may have an irrational number as their sum, or then it is a case of e.g. plane being a “sum” of lines. Such relations are far more complex because they require the existence of many number systems – many dimensions, as it were. Yet, even in them the series itself is infinite only in the “bad” sense of the word that no member of it can ever adequately represent the required sum.

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25./454. True mathematical infinity has been called relative [by Kant?], while metaphysical infinity has been called absolute: actually metaphysical infinity is perfect only by satisfying a criterion that an independent “finite world” does not satisfy, while in viewpoint of mathematical infinity both finity and its beyond are mere aspects which it regulates.

It is perhaps Kant whom Hegel has in mind when he speaks of thinkers who presented the difference between mathematical and metaphysical infinite as a difference between relative and absolute infinity. The idea implied in the distinction is obvious. Mathematical infinites, for instance, a plane when compared to a line, are infinite only in some context: e.g. plane is still finite when compared with other planes. Now, a metaphysical infinity or the unconditioned as Kant called it should be something perfect according to all contexts: and we could at least think of such absolute perfection, although we couldn’t even understand or imagine what the absolute perfection would be like. Hegel, on the other hand, is not convinced even of a possibility of such absolute perfection, because for a perfection according to some standard it is always possible to construct a context where it isn’t perfect. The so-called “absolute” infinity or perfection is indeed, for Hegel, only relative, and furthermore, very low form of infinity because it cannot be exemplified in any concrete, fully described situation: it exists at most in some abstraction. Mathematical infinity, that is, the rule governing the variability of quantities or quantitative system – a function of some sort – is in comparison much more perfect, because it is embodied in the relations of some concrete number systems: it is absolute at least according to those systems.

26./455. Spinoza has made a similar distinction. 27./456. He defines infinity as absolute affirmation or independence and finity as negativity or determination, that is, relatedness to something else. True infinity should also be active construction of independency out of dissimilarity, but Spinoza’s infinite substance is still too rigid and lifeless.

The relation between the thought of Spinoza and Hegel is a fascinating subject, but it is often misunderstood just how different the ideas of these philosophers actually is. Hegel here compares his own account of types of infinities with Spinoza’s distinction between finity and infinity. A common characteristic of both is that finity is defined through relation to something else: all determination is negation, as Hegel has emphasized earlier also. Now, the interesting point is the difference in the accounts of infinity. Spinoza deduced that if finity means relatedness to something outside oneself, then infinity must mean unrelatedness or mere self-relatedness. When infinity is understood in such a manner, then an infinite or perfect creature can only be such that has nothing outside it: in fact, only the whole world satisfies such a definition. Hegel, on the other hand, maintains that such an unrelatedness can be true only within a

79 limited context, where an object just hasn’t yet been related to objects that exist in other possible contexts: no absolute totality is possible. Yet, he admits of another sort of infinity, where the state of independence has been or could be actively constructed from a state of being-related-to-otherobjects. Even an ordinary independence of physical objects satisfies this criterion in some sense: a rock can be taken away from tree. Even more adequately the definition is satisfied, for instance, by a conscious subject who can e.g. regard the different content within itself as a mere aspect of its own cognitive processes. In fact, any structure equipped with such method of comprehending differences as mere aspects satisfies this definition of infinity, and it need not be restricted to any supposed absolute totality of everything.

28./457. Spinoza exemplifies true infinity with the image of a circle drawn within another in a non-concentric fashion: although the space within two circles is finite, there is a potentially infinite variety of lines between the two circles, because no final collection of all possible lines within that space can be determined: that is, the variety of lines cannot be presented by mentioning all the possible lines within that space. Spinoza calls the infinity of series imagined infinity, but true infinity actual infinity of thought, because in it we have infinity present in some context: it is not important that the true infinity is finite in some context, because the infinity in question is characterized by structure instead of size.

Spinoza separated three – or four – forms of cognition, the first one being that reliant on senses – and having as a subspecies the cognition through learning from others – the second one relying on deduction and the third and the highest being what could be called intellectual intuition, that is, an immediate certainty of some truths that we grasp with our thought. Now, when he opposes the infinity of imagination to the infinity of thought, Spinoza opposes infinity cognizable by the first with the help of second form to the infinity cognizable by the third with the help of second form. By senses we see merely finities or e.g. areas limited by further areas and can note that combining these areas larger, but still finite areas are produced. Thus, the only infinity we can conclude on that basis – discursively, as Kant would say – is some sort of imagined final phase in the series of combination: a sum of all possible areas. Through thought – or in Kantian terms, intuitive understanding – we can at once understand or at least deduce from immediate adequate cognitions that within the area between two non-concentric circles we could discern potentially infinite amount of straight lines of different size: thus, for Spinoza, the cognition of the true infinity between those circles requires a sort of intellectual intuition immediately or mediately through inferences. This simile of Spinoza had fascinated Hegel for some time: at least from the time of the Glauben und Wissen, where Hegel emphasized the fact that here a true infinity was given in an intuitive, immediately discernible form. In Logic, this reference to the intuitiveness of the simile has been dropped out: clear sign of Hegel’s wish not to base science on immediate information that varies from one situation and observer to another. Still, the simile can be used as an example of true

80 infinity: in fact, it is a special case of the relationship between what Hegel here calls continuous and discrete quantities. This distinction shouldn’t be confused with the distinction of continuous and discrete earlier, which was a distinction between aspects of quantities: one quantity – like an area – could be seen either as a unity and thus continuous or as a collection or combination of smaller quantities and thus discrete – e.g. area is a combination of smaller areas. Importantly, the area does not consist of infinite number of smaller areas, but it could be divided infinitely, that is, it could be always interpreted anew as a collection of still smaller areas: an area contains a possibility of finding potentially infinite number of smaller areas. Now, the difference Hegel here mentions is such that holds, for instance, between the area between the circles and the lines within that area. Here we can never say that the area would consist of lines, yet, we could still say that the area contains a possibility of finding potentially infinite number of lines. Area is like a rule or method which regulates the possible existence of lines: within these limits we can find lines. Thus, there is a certain analogy between the relationship of lines and areas and the relationship of series and its sum: area does not consist of lines like sum does not consist of the members of the series – they are of different number systems – but both area and sum regulate the possible existence of lines or members of the series – lines within these limits cannot exceed certain length, members of the series cannot exceed certain size. 29./458. The incommensurability in the Spinoza’s example is an instance of the incommensurability in the functions of curves and in the functions of variables in general. 30./459. The variability in question is not similar variability as in the sides of a determinate or an indeterminate fraction.

At first sight it seem may seem to require a leap of imagination to begin from Spinoza’s example – essentially, the relation between planes and lines – and end with an account of functions. Yet, there is an important connection between the two subject matters. Both relationships between lines and planes and functional relationships can be understood as relations between different number systems: planes and lines, on the one hand, and the sides of a functional relationship exemplify difference of dimensions, in a sense. Thus, it becomes more understandable why Hegel would want to differentiate the functional relationships from all fractional relations: that is, relations where the relata can be interpreted as quantities of the same number system. Thus, Hegel is justified in saying that the term “variable” describes the sides of a functional relation very poorly: it is not the main issue that the sides can have different values, but that their values change in a different rate.

31./460. Sides of a (determinate or indeterminate) fractional relationship have value also outside the relation: furthermore, the relation can be represented by a number. In a relation of a number and a square it is two variables [or number systems] that have been related: furthermore, the relation is qualitative. The relationship of different powers is

81 the essential element in the infinitesimal calculus, although simpler relations are dealt in it for the sake of completing formalism.

Hegel’s emphasis of the relation of powers or potencies as the main issue of infinitesimal calculus is made clearer when we understand that it is the difference of different number systems or dimensions that is important for Hegel in the functional relations. Indeed, the relationship of numbers and their squares is perhaps the easiest example of a relation of two variable series where the growth rate of the both series is different. Of course, Hegel seems at times to make the false assumption that potencies are not just the primary example of different number systems, but also the only one: witness, for instance, his recurring complaint that “formal” quantitative relations – that is, functions without potencies – should not be included in the subject matter of infinitesimal calculus. Hegel’s point is that by itself such relationships or functions do not implicate that the related series would be anything else, but mere aspects of one and the same number series. Hegel relates two differences that the mere fractional functions and functions between different number systems have. Firstly, there is the already mentioned fact that fractional functions can be interpreted as relating only aspects of the same number system – they merely change the same quantity into a different form – while true functions should relate one quantitative system into another and are thus more qualitative functions. The second difference is closely related to the first: the sides of a fractional relation can be interpreted as independent numbers – 2 and 7 in the relation of 2 to 7 are also independent numbers, and even a and b in some indeterminate relation are meant to symbolize some determinate numbers – but sides of a truly functional relationship should represent not just fixed numbers, but whole number systems, thus being variables in a more complex manner than a and b in the fractional case. 32./461. Even number and its square can be seen as quanta, but the so-called infinite differences – sides of the relation dx/dy – are mere aspects of the whole relation. 33./462. In such a structure the remaining independency of quanta has vanished, and yet the structure has quantitative applications.

Hegel mentions a still further move away from the quantitative structures, namely, that of differentiation: this suggestion needs some explanation. Although in a functional relationship we are not concerned with the relationship of two quanta of same number system, we are at least thinking of a relation between different quantitative systems, which can be investigated also in abstraction from the relation and the other system. Now, we can picture both number systems through series of numbers – or actually, through more and more finely grained series of numbers. We may consider the difference between members of one series and the relationship this difference

82 has to the similar difference in the other series. Through finer and finer divisions we gain relations between smaller and smaller differences. None of these “finite relations” can represent the relation between the rates of change in the number systems, that is, how fast the other is growing or diminishing in comparison to the other: calculating such a relation by usual means would require the use of infinitely small differences, which is impossible. Thus, this relation of “infinite differences” is in a sense “infinite” or unreachable compared to the “finite relation”. Now, we can in some cases – and Hegel knew only such cases – determine the general method for determining the relation of the rates of change for values of number systems having a functional relation: this general method is what is usually called derivative of the function. The derivative is not then a relation between concrete quantitative differences anymore – there are no infinitesimal stretches between numbers. Thus, the sides of the derivative relation are in a sense no true quantities in comparison with the sides of a “finite relation”. Yet, the derivative has at least quantitative applications, supposing that the other required values for those applications are known: an example is the common problem of solving the equation of a tangent to a certain point in a curve. 34./463. The structure of infinite difference has been criticized, because mathematicians haven’t been able to justify the use of such structures: it exceeds the capabilities of ordinary mathematics, because it enables to interpret curves as lines and in general different number systems as similar.

The question of the infinitesimal calculus was still somewhat muddled in the days of Hegel and it wasn’t still sure whether any sense could be made of the apparent contradictions it involved. Hegel’s own attempt to work out the problems may seem a complete muddle in itself, but it is actually quite interesting, even if wasn’t the way infinitesimal calculus was eventually integrated into mathematics. We should remember that the derivative represented for Hegel a functional method connecting the known values of two number systems with e.g. number systems of the growth rate of one number system in relation to other. In a sense, it served as a connection between “curves” and “lines”: every curve could be interpreted as a “collection of very small lines”, that is, the derivative gave a method for telling the tangent of the curve in a given position. Now, Hegel admitted that infinitesimal calculus and its operations still required some confirmation of its possibility, and the only method for the confirmation was to give an example or instance of the concepts involved in the calculus. Yet, this confirmation could not take the form of a mere search of one model for all the concepts involved. Particularly the derivative could not be instantiated within any one model, because it was essentially a method or functional relation between different situations: it was a method for changing situation. Thus, the derivative could be exemplified only through use: we must show that derivative can be used meaningfully in finding e.g. tangents for curves.

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35./464. It seemed wondrous how there could be a state between being and non-being; there isn’t, but there is a method for making states of being into states of non-being.

We are now in a position to evaluate in a more detail Hegel’s remarks in the paragraph 145 where he already revealed something of his attitude concerning the so-called infinitesimals. Hegel refrains from using terminology that would suggest that there would actually exist some infinitely small entities: there are no states between states of being or states non-being. Instead he prefers to speak of vanishing quantities. By becoming or change from one situation to another Hegel refers to the connection or method of transferring from a situation to next. Thus, by vanishing quantity Hegel means to suggest the possibility of a process of making a quantity into something that equals zero in this context: of course, it might still differ from zero in another context. In effect, this equals the socalled epsilon-delta –definition of limits if we just abstract from the difference of Platonistic and constructivist interpretation, as we have shown earlier. In this context the number systems have been divided in some manner and within this division we can find quantities that lie in the same segment as the investigated “point of contact”, although we can then perhaps find divisions where the quantity and the reference point fall into different segments.

36./465. Orders of infinitesimals have been criticized because infinities could not be compared: but infinitesimals are not quantities and thus not independent of their relations to other objects, therefore they can be compared.

One peculiarity that occurs in some arguments for certain derivatives are the differentials or infinitesimals of higher order (“dx squared”). On a common way of understanding infinity as amount (in this case, the smallest amount possible) such ordering of infinitesimals seems contradictory: if the first infinitesimal is already the smallest quantity there could be, how could there still be quantities even smaller. In Hegel’s interpretation the problem vanishes completely, because the infinitesimals are not supposed to be understood as quantities, but as mere phases in a possible process of finding ever better approximations for the derivative. An infinitesimal of higher degree is then nothing more, but a method which works in a faster rate compared to a method of lower degree: the difference becomes closer to zero with the “infinitesimal” of higher sort.

37./466. The mathematicians have implicitly accepted this account of infinitesimals, although they have had difficulties in explicating it 38./467. Newton’s explanations of the issue are among the best, if we just abstract from the connection to movement he suggests. For Newton, infinitesimals are not indivisible objects, but vanishing divisions, not sums or relations, but limits of sums and relations: they are not entities existing after the vanishing of regular quantities, but symbols for the vanishing process.

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It is not so much criticism of mathematicians, but a sort of defense of them Hegel proceeds to in the rest of this remark. Hegel maintain that in the practical dealing with the differentials mathematicians have usually been on the right track, but that their theoretical explanations of it have been rather less successful. The first example of the mathematicians is Newton, to whom Hegel here gives very uncharacteristic praise: as a good patriot, Hegel was always ready to prefer German scientists instead of Newton. Hegel admits that the basic ideas of Newton’s fluxion calculus are, on the whole, correct: provided we forget the references to movement, which hide the basic abstract structure of differentiation from our view. We find in Newton the already familiar terminology of vanishing quantities and also the idea of differentials as limits, which was taken up by the formalizers of differential calculus in 19th century. Furthermore, Newton clearly sees that the differentials cannot be entities of the same sort as regular quantities are: they are not something after “the vanishing”, but something with “the vanishing”: that is, they are mere indicators of the possible method of epsilon-delta-movement. 39./468. Newton merely explained the structure of differentiation, but he didn’t show that it had any possible instantiation: still, his explanation was good. Derivative is a mere limit, thus not a true, but only virtual relationship of mere aspects: infinitesimals are no indivisibles that could be seen as independent of the relation.

It is no wonder then that Hegel is willing to praise Newton in this issue. Especially Newton’s denial of any indivisibles Hegel accepts willingly. If there would exist some infinitesimal differences, smaller than any regular differences, these would have to be indivisibles, because otherwise they would consist of still smaller differences: thus, the name of indivisibles. Newton states correctly, in Hegel’s opinion, that differentials are not such indivisibles: they are not independent quanta, but only aspects of the differentiating process, as Hegel thinks. Despite the correct analysis of Newton, Hegel still does not think that Newton has done all that he should have in order to truly justify the conceptual background of the differential calculus. Hegel’s point is essentially Kantian. We can analyze concepts as much as we like without ever in this way showing the possibility of objects corresponding with these concepts. Thus, we need still something else, namely, a model for that concept: such a model Hegel deems to have shown how to construct in his discussion of true infinities.

40./469. Newton also notes that the final relations are not relations of any real quanta, but limits for any relations of such quanta. If he would have understood that the infinitesimals are mere aspects of the process, he wouldn’t have had to reintroduce the idea of an indefinite progress.

85 Hegel makes another critical remark on Newton’s ideas of infinitesimals, this time of the notion of infinite series Newton is forced to use: Newton describes what he calls the final relations – the derivative – as the limit which an infinite series of relations of differences can never reach. Hegel notes that the limit is not to be understood as an end point of such an infinite series. Instead, it is a method that governs that series: what is important in the differential is not the possibility of continuing indefinitely towards the derivative, but the epsilon-delta-rule that states that we can make this indefinite progression in a systematic way, i.e. by choosing the next member of the series as being indistinguishable from the derivative according to a finer division of the numbers system involved.

41./470. Other mathematicians (like Carnot) have suggested that the relations remain after the disappearance of the quanta because of a law of continuity: this is correct, if continuity does not mean continuity from one quantum to another, but continuity from concrete case to its abstract structure.

The law of continuity to which mathematicians grounded the differential calculus was an obvious precursor of the idea of continuity of functions and number series in the modern mathematics, with the obvious differences that the continuity in those days hadn’t been defined accurately through the epsilon-delta –definition, and furthermore, that it wasn’t yet known that there are functions which aren’t continuous anywhere. The “law” of continuity would thus obviously justify differentiation: if the series of relations of differences is continuous it must have a limit, that is, the derivative. Yet, although Hegel’s account of differentials is close to the idea of the epsilon-delta –definition of continuity, it is not this what he is referring to with continuity. He clarifies that he is not speaking of continuity in the sense that it has in an indefinite progression, where we can always produce something new with quantitative operations, but then at once notice that this new entity is also a quantum and thus could be identified with the original quantum: such a continuity means just that quantitative operations do not produce anything outside the number system to which it is applied. Hegel explains that the continuity he is referring here is one of between Dasein and concept. The concrete quanta and their differences and relations of difference within different number systems form the side of the “concrete existence” here. Now, Hegel says that we can produce from this concrete example some general model of similarly related number systems: we can produce a rule that limits the possible values that the concrete differences and their relations could have – this is the derivative. The continuity here is one of characteristics remaining similar during this process of model constructing: the “concept” or the abstract model shares some characteristics with the concrete case.

42./471. Newton also calls infinitesimals principles of quanta or generating quanta compared to the generated, finite

86 quanta: he has correctly differentiated between a concrete quantum and a method or general model for creating them, but the use of terms like addition return us back to the idea of infinitesimals as independent quanta.

The mixture of praise and criticism of Newton’s ideas continues. The critique here is mostly terminological: it concerns the concepts of increment, growth, addition etc. which holds on to the idea of an indefinite progress towards some true infinitesimals within some number system. On the other hand, Hegel greatly favours Newton’s notion of these increments as beginnings or principles of quanta. Hegel interprets Newton’s concepts through his own terminology. Beginning or becoming of a quantum refers here to a general method that can be used to generate the quantum: this is also what Hegel shall later refer to as concept, that is, a general structure equipped with a method of modelling it. Thus, what is called differential is just a method which can then be applied to some concrete quantitative cases in order to produce some definite quantitative relations: this we have already seen to be the essence of Hegel’s thoughts on differential calculus.

43./472. Notion of infinitely small quanta and the possibility of ignoring them in comparison with finite quanta is far worse: Leibniz endorsed it and Wolff popularized it through analogies.

The legend often told that Hegel would have based his appraises and criticisms of scientists to some sort of patriotic need for defending Germans and attacking their foreign opponents is not verified in this remark. We just saw Hegel praising Newton and now he is about to criticize Leibniz, the opponent or competitor of Newton as it comes to inventing differential calculus. Indeed, the whole idea of some strange quantities called infinitesimals that obey some equally peculiar operations is quite foreign to Hegel’s way of thinking, as we have seen. It has sometimes been said that the socalled non-standard analysis would have shown Hegel to be essentially wrong in his attacks. Yet, we must remember that the first systematization of differential calculus took a completely different route which we have seen to share some characteristics with Hegel’s own viewpoint. Furthermore, the non-standard analysis depends on the peculiarities of modern mathematics that Hegel would have found completely alien, such as the idea of infinite cardinals. Many of the ideas current in mathematics today, such as the non-standard analysis, require either a formalistic setting – we are just writing some signs according to some operations, no matter what they are supposed to mean – or then they presume a Platonist assumption of strange abstract entities with no apparent connection with reality: a muddle of which Hegel with his constructivist tendencies would feel the need to ask what it is all about. Hegel insists that e.g. differential calculus and the tools and terms it uses must have some connection with reality that we intend to describe by it: and Wolff’s reliance on comparisons is not enough to provide such a connection, Hegel thinks.

87 44./473. Mathematics cannot accept such inaccuracy, yet the method seems depend on it.

The reason why Hegel cannot accept Wolff’s analogy – even as a mere analogy – is that it is essentially faulty. Concrete measuring of physical objects with other physical objects always involves a level of inaccuracy: one cannot determine by mere eye whether e.g. a house is one millimeter longer or smaller. A mathematical “measuring”, on the other hand – whether it is made through traditional geometrical means or through differential calculus – is always completely accurate: there is no room for error in saying, for instance, that the area of a circle is pi times the square of radius. Thus, Wolff’s analogy fails to do anything else but muddle issues. Yet, there is something in Wolff’s account: the differential calculus does depend on the idea that certain parts of an equation can be discarded as insignificant. This insignificancy can be accurately defined in relative terms: it is sufficiently insignificant compared to any required level of accuracy. The mathematicians of Hegel’s time still struggled to understand this apparent inaccuracy that produced accurate results: the solutions of Weierstrass were yet to come. 45./474. Euler suggests that differentials can be regarded as zero – that is, as quantitatively insignificant, although they have a qualitative significance – but he does not emphasize that differentials have a role as aspects of the derivative. Lagrange opposes that it makes no sense to speak of quantitative relations of quantitative insignificancies: Euler’s justification of this idea is not convincing.

Euler tried to make the Leibnizian use of infinitesimals more systematic: he tried to apply infinitesimals in places where they perhaps shouldn’t be. Nowadays Euler’s inventions may not seem as fantastic as they did in Hegel’s time, which is due mostly to the discovery of the nonstandard analysis. Yet, one cannot but feel a lack of strictness in the way how Euler almost plays with infinitesimals. Hegel notes, following Lagrange, especially Euler’s idea that infinitesimals are sort of zeros and can thus be discarded: Euler tries to justify this assumption by arguing that there can be many sorts of zeros – the proof he gives is obviously faulty. Yet, even Euler deserves some praise in Hegel’s opinion. The idea of infinitesimal as zero implies the correct insight that in differentiation we abstract from the particular quanta involved. Still, Euler misses the fact that differentiation brings about something positive with it: although the independent quanta are abstracted away, the relation of the rates of change in the number systems – the derivative – remains. 46./475. Mathematicians have thus implicitly understood the issue of differentiation, but haven’t been able to explicate its basis structure sufficiently. This has led to the apparently inconsistent way of first operating with infinitesimals like with regular quanta and then discarding them.

88 The investigation of Newton’s, Leibniz’ and Euler’s ideas has shown for Hegel that mathematicians on the whole are aware of what the differentiation is all about, although they have had difficulties in trying to explain their insight and although the correct viewpoint is often shadowed by use of mysterious infinitesimals, in Hegel’s opinion. In differentiating, we start from the actual changes within some number systems that have a functional relation and proceed to abstract the general method for determining the rate of change that one number system has in comparison with the other. This differentiating involves what Hegel calls true infinity, because it results in a rule that regulates different number systems. Yet, the work of the mathematician is not yet over. The actual derivation of the derivative has not yet been justified: the mathematicians do know in a hazy way what they are looking for, but they haven’t been able to provide conclusive evidence that the methods they use are more than mere humbug. Particularly, the operation involves a discarding of all terms with the increment i after the formula [f(x + i) – f(i)]/i has been reduced into a form where the i does not occur as a denominator. This is somewhat problematic as the mathematicians seem to be using double standards here: at first they suppose that i is nothing but a regular quantum, but then toss it aside as quantitatively meaningless.

47./476. Let us see how mathematicians have tried to solve the problem. 48./477. Older mathematicians merely used calculus, while later, like Lagrange, have wanted to make it as evident as Euclidean geometry: this is a futile hope, because it deals with structures that cannot be immediately intuited.

Thus begins a new section of the remark, dedicated to investigating how the operation of differentiating has been justified by mathematicians. Hegel quickly mentions the earliest developers of differential calculus, but just in order to explain that they didn’t yet feel the need of justifying the calculus, as they had enough problems in perfecting it. The later mathematicians, on the other hand, required some evidence for the differential calculus. This evidence Hegel clearly seems to interpret to relate to intuition. There are two obvious motivators. Firstly, the Greek geometry was based in a significant manner on the possibility of actually witnessing the geometrical constructions through one’s own eyes. Not only were the Euclid’s proofs combined with diagrams explaining them, but also firstly the definition and secondly some of the basic propositions, such as the fourth proposition in the first book, required intuition in order to be justified. Second initiator is, of course, Kant’s idea of mathematics based on a priori intuition: an idea itself based on the way how Euclid’s geometry was constructed. Now, Hegel cannot accept the idea that the differential calculus could be made evident just by looking at some diagram. Differentiation involved, in Hegel’s opinion, what he calls true infinity, that is, a method or rule by which different number systems were regulated. Such a rule cannot be just intuited: we cannot merely look at a graph of a function and state that its rate of change follows

89 a certain formula. Instead, we have to study the way how the function itself is formed and how it regulates the relationship of two number systems: only after this preliminary investigation, which already involves something besides mere intuiting, can we deduce anything of the derivative. Similarly, a proper science cannot be done by merely looking: one must also interpret one’s experiences in order to find some laws governing the empiria – this is the point of the analogy Hegel makes between differentiating and science. 49./478. Some mathematicians have tried to avoid infinitesimals: Lagrange mentions Landen’s method where variables are first introduced as different and then later supposed identical, but also says that it is not easy to operate as differential calculus. Landen’s method resembles Descartes’ tangential method and has a different interest than differential calculus, which essentially concentrates on the relation of a function and its derivative.

Hegel apparently sympathizes with mathematicians who try to avoid the problems in differential calculus by discarding the idea of infinitesimals. Lagrange is especially esteemed by Hegel and here Hegel mentions John Landen whose residual analysis was a sort of precursor of Lagrange’s method. The intricacies of Landen’s method need no long explanation. Landen tries to avoid the use of small increments by speaking merely of the relation of variables x and y in [f(x) – f(y)]/[x – y]: at first x and y are supposed to be unequal and when the denominator has been eliminated they can be supposed to be identical. The method is perhaps a little more difficult than ordinary differentiation, yet Hegel sees an even more great lack in it. Landen’s method, according to Hegel, misses the main issue in differentiation, that is, Landen makes differentiation into a completely mechanical operation, but fails to note the important difference between the derivative and the original function. He fails to see that the essence of differentiation lies in the relation of a function to its rate of change or rule that regulates its values.

50./479. Other mathematicians just assumed infinitesimals were insignificant in comparison with regular quantities and thus proved the fundamental relation between a function and its derivative: similarly, in a characteristic triangle, the curve was supposed to differ from tangent only insignificantly. These assumptions made differentiation into a muddle, because infinitesimals were still operated like regular quanta.

Although some mathematicians had tried to avoid infinitesimals and problems involved in using them, especially the earlier mathematicians, but also many later ones were content with the unclear idea of infinitesimals being so unnoticeable that they could be discarded at the end of the operation. This should be a familiar muddle by now, but here Hegel mentions also another problematic change of meaning. In relation with differentiation it was sometimes spoken of a characteristic triangle – the expression was used first by Leibniz. The idea is to think of a figure resembling triangle one

90 side of which is parallel to the x-axis and another side of which is parallel to the y –axis: these sides are connected by graph of some function. Now, when the triangle is made smaller, the part of the graph should resemble more and more a straight triangle. At the point in which the lines become infinitesimal, the graph should become the tangent – this final result is the characteristic triangle. Hegel notes that a similar criticism could be held against characteristic triangle as against whole operation of differentiation: one thing is first used like a regular quantum and then it is discarded like a zero. Here, we have a curve that should differ from all straight lines, yet, when it is made sufficiently small it can then be interpreted as straight.

51./480. By the way, Newton tried to deduce the derivative of xy by calculating the difference of (x + dx/2)(y + dy/2) and (x – dx/2)(y – dy/2): this cannot be correct.

Although the remark seems to come out of blue, it is closely connected with the previous account of characteristic triangles. In his Principles of natural philosophy Newton introduced differentiation through – not triangles, but – rectangles: we were supposed to think of a product of two variables in their “fluctuating states”, that is, the rate of change of two functionally related variables. Newton begins with an easy case where both the variables merely are not raised to any power: if we suppose the variables to be identical, we can then deduce the differential for different powers etc. Now, Newton’s method is to investigate the difference between two such “rectangles”: one rectangle is produced from variables with an added increment, and the other rectangle is produced from variables from which the same increment is supposed to be taken away. The difference between these rectangles is then ydx + xdy. Newton’s procedure is meant to avoid the difficult to interpret term dxdy, but as a justification of differentiating it still faces the familiar problem of how to explain the ambiguous way of operating with the infinitesimals. Furthermore, it has a problem of its own. Of course, we know that when the increments are made sufficiently small, the difference just calculated and the difference (x + dx)(y + dy) – xy can be assumed to be almost identical. Yet, Hegel is quick to note that at the phase of mere algebraic operating with infinitesimals this identification cannot be made. Thus, Newton has to rely on the infinitesimal nature of increments quite early on, which can be seen as a setback. 52./481. Newton’s other derivations are connected with concrete motion. He uses form of series to show that the result can be made as accurate as one wants.

The basic principle of Newton’s view on differential calculus was correct, but his justifications of the operations used in this calculus are less satisfactory. In the previous paragraph we saw an attempt of algebraic derivation of some basic results of differential calculus, which failed because

91 Newton used operations that already presupposed the principles of differential calculus. Now Hegel once more goes through the rest of Newton’s attempts at grounding differential calculus. The empirical justification on basis of nature of motion is undoubtedly not a truly rigorous mathematical justification. Thus, Newton is left with the use of infinite series, and here Hegel’s reception is mixed, as always towards this idea. What is true in Newton’s method is that there is a rule or method that regulates such an indefinite series: the stricter standards we set for the result, the better approximations of the derivative we can get. Yet, the idea of the series seems too closely associated with the idea of the series having a determinate result after infinite steps: a result that is a similar entity in comparison with other quanta, although happens to be “infinite” compared to them. 53./482. Lagrange has shown that Newton’s mistakes were based on discarding terms of equation because of their supposed smallness, when they should have been discarded because of their qualitative insignificance.

Hegel’s hero of differential analysis has already been mentioned occasionally, but here Lagrange for the first time makes a positive contribution to the story of differential calculus. Based on Lagrange’s ideas, Hegel points out that different terms in what he calls developed function – essentially the Taylor series – are not just different quantities, but different quantitative systems produced by different functions from the basic quantitative variable: the first term describes the original function itself, the second term or the first differential describes the rate of change of the original function, the next term or the second differential describes the rate of change of the first differential etc. Thus, the different terms form different aspects of the whole structure: the developed function is not just a sum of arbitrary quantities, but tells exactly what sort of qualitative factors form the phenomenon of quantitative change in question. Hegel presents an example from mechanics: there the first term can be identified with velocity, second with acceleration, third with resistance etc. Thus, the decision to discard some of the terms in the series is not just a matter of the level of accuracy, but one of interest: what qualities or aspects of the phenomenon are we about to investigate?

54./483. If this qualitative significance of the terms was emphasized, then differentiation would be naturally seen to be fulfilled in the discovery of the first differential: the further differentials merely repeat the relation between a function and its rate of change that has been discovered in the first differential.

Hegel’s suggestion in the previous paragraph was that the terms of the developed function correspond to different number series that together determine different moments contributing to the actual phenomenon that could be called the “change” of a function. Now Hegel makes a further suggestion that it is actually the first differential which is the essential term in the whole

92 development. This coincides well with Hegel’s view of what good cognition is all about: we are looking for general rules or methods which regulate the potentially infinite number of particular instances. The first differential presents a paradigm of a relation between a function and its “rate of change”. What the further differentials do is only to apply this relation recursively to further differentials: second differential gives us the rate of change of the first differential etc. Thus, once we know how to construct the first differential from the original function, the whole problem of differentiating is over: it then becomes only a question of application how far we need to develop further differentials, that is, what qualitative moments are we studying.

55./484. Carnot explained the differential calculus in a clear manner, but his actual operations are muddled by the use of infinite smallness and by the justification through results.

In paragraph 470 Hegel praised Carnot among other mathematicians for explaining differentiating through what was called a law of continuity. Of course, Hegel’s interpretation of this law was rather peculiar and probably not intended by the mathematicians: when Hegel spoke of the continuity between an original function and its derivative, he meant that certain structural elements were common to both, i.e. derivative was an abstraction of some general element – rate of change – from the original function. It is now peculiar that Hegel finds then Carnot’s actual operating with differential calculus a disappointment. In the previous paragraph Hegel mentioned briefly Carnot’s idea that in differential calculus one used mistakes to patch previous mistakes – a similar idea, by the way, was suggested later by Vaihinger in his als ob –philosophy. For Carnot, the differential calculus assumes somewhat fictitious nature: the phases of operation do not necessarily describe anything real. Thus, it is just natural that Carnot would justify differential calculus more by its use: differential calculus is an efficient tool for deriving correct answers for certain problems – a viewpoint which Hegel cannot view neither philosophically nor mathematically tolerable. 56./485. Lagrange has returned to Newton’s method of series and has thus made differential calculus more precise and general: yet, he has also fallen back to the idea of discarding differentials because of their smallness, although in applications he speaks of qualitative discarding.

Even Lagrange does not survive Hegel’s criticism unscathed. Hegel is willing to admit that Lagrange has done much in making differential calculus more rigorous. Furthermore, Hegel is especially thrilled of the way how Lagrange understands the application of the differential calculus, as we shall see: here comes to the fore the idea of differential calculus finding different number systems, some of which govern the change of other number systems. Yet, when Lagrange in the phase of operation still clings to the idea of differentials being discarded because of their smallness,

93 Hegel can’t follow him. We might say that Lagrange concentrates on the continuity of the series involved in differentiating: on the fact that one can derive results that are as near to the supposed place of derivative as required. This is one important part of differentiation, but it is not a sufficient description of it: continuity is not yet differentiability. What Hegel misses is the fact that this supposed limit or derivative is supposed to be another “quality”, that is, another scale, which has the necessary connection with the original scale that it regulates the relations between different parts of that scale: just the ingredient is lacking which Hegel thinks he has found in the application of differential calculus.

57./486. The idea of limit, investigated by the theory of series, implies the correct qualitative notion of differentiation where the sides of differential are mere aspects and no independent quanta: yet, this structure does not yet sufficiently determine the relation of a function and its derivative. Still, limit should be a limit for something: in this case it should be the limit of “finite” relations. The idea of a limit instantly brings out the problem that the differential should normally vanish when the increment is turned to zero. The simple answer is to proclaim that in differentiating we are mostly interested with the coefficient of the first differential: then it is just problematic how to justify the application of this derivative.

The notion of the limit of an indefinitely long series is not a complete failure. In fact, Hegel admits that it points out some interesting characteristics of differentiation. Derivative is supposed to be a limit to a series of values of a function: thus, it is not a value of a function in the same sense as those of the series are, but in fact forms a completely different qualitative series. The derivative is indeed more of a schema for a function or relation compared to the members of the series: a member of the series is a quantity of a concrete relation between the difference of the values of the function and the increment of the variable, while in the derivative these concrete sides have been interpreted as mere aspects. What is still lacking from this account is the essential relation that the derivative should have to the original function, that is, the fact that it is supposed to be regulating the original function. Still, as this side is connected more to the application of the derivative – to how derivative can be used in order to calculate functions related to the original function – we might say that the theory of limits does express the essential of the purely mathematical side of the differential calculus. How does one then proceed to find this so-called limit of the series? Here the mathematician faces the by now quite familiar problem that he must proclaim at some point that the so-called increment of the variable can be assumed as zero, but if the assumption is made, the derivative seems to become a relation of a function to zero. Hegel applies the simple solution of Lagrange: we merely define that the derivative is the coefficient of the first term following the original function in its development. In the viewpoint of mere operations this is actually sufficient: this mere definition

94 completely avoids all embarrassing question usually connected with differentiation. Of course, it doesn’t tell the whole story of the differentiation, namely, it ignores the question why this coefficient is so important, but this is more of a question related to the application of derivative.

58./487. In differentiating , a concrete relation of quantities is changed to a general structure of such a relation with mere aspectual relata. The idea of approaching the derivative speaks only of a variable turning into another variable of the same number system; the essential ingredient in the differentiation, on the other hand, is the generalized relating of variables of different number systems. 50./488. The essential point of differential calculus is muddled by interpreting elements of variables as increments and the limit of the series of relations as a mere similar relation: actually the dx cannot have any relation to x, because it is not a quantum in the same sense as x is.

A further problem with the concentration on the form of the indefinite series is that it almost remains locked within one series: we concentrate of how one from an approximate value for a derivative can find even better value for it – that is, how one can always find [f(x +i(n+1)) – f(n)]/i that is closer to f’(x) than [f(x +i(n)) – f(x)]/i was. Now, in the differentiation, Hegel says, such a difference between different values of one number system becomes uninteresting, because the sides of the derivative dy/dx are no longer independent quanta, but mere placeholders for two different number systems. Yet, now the relation between the number systems is even more emphasized than earlier. The derivative expresses the difference in the rates of change of the two systems. For instance, if we suppose the x-system changes in a regular fashion, the rate of change in the y-system can be expressed through the function f’(x): if y is e.g. square of x, it at first changes in an opposite direction to x – that is, it becomes smaller, while x grows – but finally it starts to change faster and faster in the same direction as x. This new function determines then another function number system or another “quality”, for instance, velocity compared to time and space. Thus, its values are not quanta in the same sense as the values of either of the original variables: velocity does not consist of time and space, although it contains them as aspects.

51./489. When the infinitesimals are interpreted quantitatively, one can too easily identify qualitatively different matters like the different axes etc. On the other hand, it is permissible to identify straight lines and curves in an “infinitesimal situation”: their difference is only quantitative and the relations of lines or angles remain same.

Hegel points out quickly another mistake in concentrating on the form of series in differential calculus: as the difference of number systems is not respected in the relation of a variable and a derivative, it becomes too easy to assume that all sorts of number systems, such as that of a variable and its function, can be just identified – at least when these quantities have been made “infinitely small”. Then again, Hegel defends the idea of identifying curve with a straight line. Such

95 identification happens in the case of the so-called characteristic triangle formed of a curve and two straight lines parallel to both axes: when the triangle is assumed to become smaller and smaller, the curve resembles more and more its tangent. Hegel points out that the difference of straight line and tangent is essentially only quantitative, if the straight line is defined as the shortest line between two points. In a normal quantitative situation, a curve and a line are undoubtedly different also qualitatively, because they are in different relations to other lines: similarly, different straight lines, like axes, differ qualitatively by having different directions. In case of different axes, the quantitative approaching of the quantities is not enough to warrant the identification, because the directions of the axes remain always different, no matter how small they were made: thus, two axes should always be interpreted as different number systems. The relation of a curve to the axes in the characteristic triangle, on the other hand, becomes more and more similar to the relation of a tangent to the axes: thus, in this case even the small qualitative difference between a curve and a tangent can be ignored, because it can made as small as one wants.

52./490. It is also often assumed with no justification that all infinitesimals are equal: thus, an object with a variable velocity would always move with a constant velocity during every infinitesimal moment. Earlier mathematicians tried always to verify the differentiation by connecting their variables to reality, but later differentiation itself has been taken as a proof for existence.

If infinitesimal is understood as some sort of entity, it must be like a point compared to longer stretches: it would be a relation of infinitely small quantities. Now, within a stretch, the relation between two variables could well change, that is, it could be different at the beginning and the end of the stretch. At one point, on the other hand, the relation can have only one value. Thus, if we take the idea of infinitesimal entities seriously, we must accept that a stretch of uneven or variable relation can consist of “infinitesimal stretches” with constant values: e.g. variable velocity during a period of time is just an infinite sum of constant velocities in infinitesimal stretches of time. Hegel, of course, cannot take such talk literally. At most it means that a certain aspect of a concrete function – its “momentanous” rate of change – can be abstracted from it and expressed as a derivative. Of course, when one has to plot the course of the actual function or its concrete values, one must take into account all of its possible aspects – or then one must admit one is only giving an approximation of it. These aspects are only secondary or aspectual objects compared to the concrete plot of function: velocity etc. are mere aspects of the concrete motion. Hegel warns against reifying these aspects. In mechanics of Hegel’s time it was common to think that forces into which a concrete movement had been solved could then be assumed as real entities. Hegel, on the other hand, favours strict empiricism in these matters and declares that only the concrete existence we can experience can tell us what exists primarily, while the forces etc. are mere abstractions.

96

53./491. Lot of charlatanry has been accepted as a proof of differential calculus, just because even a fake proof should be better than mere empirical justification: similarly Newton has been praised instead of Kepler. 54./492. Mathematics cannot prove empirical laws without empirical help, because it cannot construct models for qualities like philosophy does. Newton’s proofs are as bad his optics.

Differential calculus operates with differently qualified number systems, whereas ordinary arithmetic and algebra are confined within some number system. Thus, an attempt to algebraically justify the differential operations will always remain insufficient, Hegel concludes. Algebra cannot prove differential calculus possible. There are thus only two other possibilities. Firstly, a mathematician can accept the different number systems from experience: he may, for instance, learn from physics that there are such thing as space, time, velocity, acceleration etc. and note some functional dependencies between these different systems. The other possibility is what Hegel calls deriving the differential calculus from concept. In effect, Hegel suggests that we shouldn’t use algebra for proving differential operations, but for constructing structures in which differential operations make sense: Hegel’s favourite examples in this case are relations of different powers. Although Hegel’s criticism in this long remark has seemed quite level headed and reasonable, in these final paragraphs he seems to speak more with a voice of a patriot, preferring Kepler and Goethe over Newton with his physics and optics. Yet, we may actually discern a deeper pattern here: more is at stake than mere father land of the scientist. Kepler and Goethe have both described certain phenomenon: Kepler the movement of planets and Goethe the appearance of colours. Such phenomena are essential wholes, Hegel suggests: one cannot separate any aspects from them, except in some peculiar circumstances. What Newton has done is to make these aspects independent entities: Newton has suggested that it is singular forces revealed by differential calculus which create the movement of planets and that white light consists of colours discovered by prism. As these so-called independent entities appear only through manipulation of the concrete event, Hegel suggests that the concrete whole does not so much consist of them as it has the potentiality of producing them in suitable conditions.

Remark 2. The purpose of differential calculus derived from its application. Hegel’s stand on mathematical infinities should be clear by now. Mathematical does not mean for Hegel infinity in a quantitative sense – that is, characteristic of groups with an immense number of objects – but a function, or relation as Hegel would call it, that regulates either quantities within some quantitative system, or even better, different quantitative systems. The most complex sort of function Hegel discovers in mathematics is the one that regulates between quantitative system that

97 is function of another system and a system expressing the rate of change of the first system: this is the operation of differentiation. Now, Hegel has provided us with the true essence of differentiation and Lagrange, so Hegel believes, has found the most suitable and problem-free method of differentiation. Yet, the tale of differentiation cannot end here. What is important in the relation of derivative and an original function is that the first regulates the second: the derivative can be used in determining values of and connected with the original function. Hegel is in a sense pragmatist: mathematics must have a reason for its existence and its reason lies in how we can use it in practical calculations. It is this question of applications of differentiation that Hegel turns to now.

1./493. We now turn to the applications of differentiation. Differential operations one can learn easily, the difficulties arise when one must explain why one is interested only in the first derivative.

Hegel begins the remark with a short reminder how easy the differential calculus actually is, at least when it comes to the operation of finding the differential for a given function. Indeed, differentiating is rather mechanical task, at least in the level in which Hegel’s temporaries were confined: the rules for differentiating can just be learned by rote, like any youngster learning mathematics could tell. Even the derivation of these rules is quite easy, once one just accepts what must be done. It may undoubtedly seem problematic why some terms are being discarded along the way, but once we explain we are interested only in the particular coefficient of the development of the function, this final opposition will fade. It is only when we start to apply the results of differentiation to some concrete problems that we face some serious problems. Then we must truly justify why we have been interested only of this coefficient: we have to show that the derivative is not just a meaningless bundle of signs, but refers to something actual in the geometric or physical aspects of a concrete situation.

2./494. It is clear that differential calculus has not been developed for itself, but for some further purpose: this is common in all sciences. The differential calculus is historically a generalization of methods for finding tangents.

It is Hegel’s belief that what is usually called ground in sciences is often only what is grounded, while the concrete cases deduced from those grounds are the true ground. It may seem that Hegel has merely confused ontological and epistemic grounds: the concrete phenomena are undoubtedly the epistemic ground for assuming the explanations of those phenomena, but it may well be that the explanans is the ontological ground of the explanandum. Yet, Hegel’s statement is not a mere confusion, but reflects a certain tendency in his philosophy as a whole. The concrete phenomena are more informative than the comparatively abstract explanatory entities: in fact, it is possible to suppose that these entities are mere aspects of the whole phenomena. When a physician finds a

98 combination of forces to explain e.g. the movement of a planet, he is hence merely abstracting some facets of the whole experience. It is then against the true nature of the matter, when we are first taught of these forces and only from these is the movement of planets deduced: what we should first learn are the concrete movement of planets and only then we should be shown how to abstract the forces from it. Similarly, the operation of differential calculus is first learned in an abstract fashion and only later one is shown how to apply it. But here the concrete whole are the different quantitative systems related to one another, for instance, the curves and their tangents, from which the differential calculus first began. It should be only afterwards that we abstract the general operation of differentiation from this whole: a method that can then be used also in other topics like mechanics.

3./495. The basic structure behind the idea of infinitesimals has been the relation of differently qualified number systems: we justified this also with an empirical study of the views of mathematicians. Now we must investigate this structure further, in order to see how it could be applied.

After the short excursion to the history of differential calculus, we return once more to the determination of the task of this remark. We know the operations which the differential calculus uses, but more importantly, we also know the general structure differential calculus investigates. This is the so-called infinitesimal and relation of different infinitesimals: the infinitesimal must not be understood as some infinitely small entity, but merely as a symbol for relations of differently qualified numbers systems and their relations. We saw in the previous remark that this structure was at least implicitly understood as the main issue of the calculus by all distinguished mathematicians. Now, it is this structure that makes the operations of differential calculus interesting: by itself they would be rather senseless manipulation of equations. Hegel’s task is now twofold. The operations with a proper understanding of the structure involved are already useful: thus, we firstly should look at what mathematical advantages the differential calculus offers. Secondly, we must determine to what concrete use in e.g. physics the calculus can be put: these are the applications in the proper sense of the word.

4./496. We have already anticipated that the relation of differently qualified systems can be seen in relation of different exponents: thus, the primary issue in differential calculus should be functions of different powers.

We return once more to Hegel’s perplexing idea that the so-called true mathematical infinite and therefore also the subject matter of differential calculus should be relations of variables with different powers. It is undoubtedly true that e.g. the relation of a variable and its square is an example of a relation between differently qualified number systems, but surely all such relations

99 need not be of the same sort. Hegel’s idea becomes perhaps clearer, if we remember that he is stating a result of a construction instead of a result of a deduction: he is not saying that he would have demonstrated that all true relations between different number systems are relations of powers. Hegel has, instead, constructed an example of such a relation. This example or model is not just some arbitrary construction, but it is supposed to be the relation that is simplest to construct. We could have shown an example of a mere proportion between different number systems, but to show that a proportion is a relation between different number systems would need more assumptions. Furthermore, we could have produced a more complex model, but it would have been futile, when a simpler one was to be found: these models would have shown nothing new and would thus have been mere applications of the general method. Thus, it is strategic reasons that make Hegel call the relation of powers the true subject matter of differential calculus: it is not the only, but the primary example of relations between different number systems.

5./497. It is still unclear how to separate differential calculus from other parts of mathematics dealing with powers: this separation can be discerned only from applications which are the true ground of differential calculus.

Difference of powers is only the exemplary subject matter of differential calculus, not the only possible. Similarly, all aspects of raising to a power are not part of the proper subject matter of the differential calculus. For instance, the mere operation of raising a number to an arbitrary power is still an operation that happens within some number system and is thus not a proper part of the differential calculus. There are indeed many different operations, problems and questions involved in exponentiation, and it is problematic which of them should be handled by differential calculus. This is a further reason for studying applications of differential calculus. By looking at what we are trying to do in differential calculus, we could learn that in differential calculus we are not e.g. trying to find values for unknown variables in equations. 6./498. α) The subject matter of differential calculus. Differential calculus studies relations of at least two different variables where the value of the variables is undetermined: furthermore, at least one of the variables should be raised to a higher power. 7./499. The variables in differential calculus should be indeterminate, yet, the nature of the differential calculus cannot be completely deduced from operation with indeterminate quanta. The constants of the equation are in some sense arbitrary, in another sense they can be seen as functions of the variables: this is done e.g. in integration.

We may perhaps be once again skeptical of whether the task Hegel here suggests as being the primary issue in differential calculus could be the only task of differential calculus: yet, we may perhaps accept it as the primary task in the sense of being an exemplary case of what differential

100 calculus is all about. Firstly, Hegel begins, differential calculus is about equations. It is not so much about determining the variables within some equation: indeed, as Hegel says, the equations dealt in differential calculus are such that have no determinate values. The equation determines thus a number system that consists of possible solutions for the unknown variable: or actually more than one number systems, as there should be more than one variable involved. The main issue is to ensure that the variables indeed are of different number systems: an equation like y = 2x may be said to determine not two independent number systems, but two different aspects of one number system. This is ensured by the assumption that one of the variables at least has been raised to a higher power: squares form a different number system than their bases. Simpler equations, on the other hand, belong to differential calculus only in a very external manner: one can widen the differential calculus to cover also these cases, but its essential problematic is not yet adequately expressed in them. Hegel notes also that it is indifferent which terms in the investigated equation are to be taken as constants and which as unknown variables. Usually the differential calculus is about the determination of the derivative, when the form of a function between the two variables and the constants involved are known: with the derivative, further problems, like the function of a tangent to a certain point of a curve, can be solved. In these cases the constant coefficients of the variables are nothing but given magnitudes. Yet, nothing speaks against taking something else as given and e.g. the coefficient of x as the unknown to be solved. Indeed, in integral calculus and applications involved with it, the problem is somewhat converse than in differential calculus in itself: we know the derivative and we want to know the original function and the constants involved in it as coefficients.

8./500. The equations of differential calculus differ from other indeterminate equations by one of the variable being of a higher power: thus, the change of variables has been determined qualitatively and the continuity of the change is of a higher kind than mere identification of different parts of one number system. Hence, simple equations need not be differentiated for their own sake.

The result of the investigation of the subject matter of differential calculus should be clear by now: it is the relation of different number systems which can then be seen as functions of one another. The term “function” receives completely new connotations in this case: it is not just a function within some number system, but a function from one number system to another. The change of variables is qualitatively determined, that is, the variables change at a different rate. Thus, the word “continuity” is also further qualified. Within one number system, the continuity means only the fact that any point in a number system could be changed with any other point in the same system or that all numbers are functions of one another. Here, on the other hand, this continuity is connected with

101 a difference from other continuities: the points in this number system can be seen as simple functions of one another, but the points of other number system cannot be constructed by similar functions, but only by more complex, qualitative functions. After this characterization of differential calculus Hegel gets another chance to explain why the functions without higher powers are not yet proper issue of differential calculus: the differential of the equation ax = y is merely the constant coefficient a or y/x, which is not so much a function between different number systems, but a mere indication of the relation between two points of the same number system. 9./501. β) The differentiating and its applications. A variable that has been raised to a power can be interpreted as a quantum or a sum of smaller quanta: the essential can be seen when it is represented as a sum of two quanta. The theory of series studies the form of the quanta that together form the power of that variable: the differential calculus studies the relationship of the original variable with these partial powers.

After we have learned the basic structure differential calculus studies – the relation of differently qualified number systems or number systems with different rate of change – it is time to see how the actual operation of differentiation can be used in the investigation of that structure. We start with a group of number systems, one of which at least is of a higher power than one other: a minimal requirement is two number systems, one of which is, say, a square of the other. Now, the number systems or the variables that represent them can be understood also as sums: this is their nature as quantities. The base variable or representative of the base number system – e.g. x – can be interpreted as a sum of many quanta of the same number system: here the number of the quanta is minimally two, and indeed, as Hegel says, it is unnecessary to have any more quanta in the sum – x can be seen as x1 + x2. Now, the variable that is the power of the first variable can then also be represented as a sum, but not as a sum of different numbers in the same system, but of different aspects that constitute this number system: they are different qualitative facets of the function that have their own number system – if y is a function of x, the y can be represented as af'(x) + bf’’(x) + ... The differential calculus is not so much interested of the form of the series involved: indeed, it is a potentially indefinite series that can be made more and more accurate. What is interesting is the relation that these aspectual number systems have to the base number system: what sort of function of x the f’(x) is?

10./502. The sum that is derived in differentiation is less important than the relation of the variables: the question of representing the variables as sums rises naturally from the idea of the variables as functions of each other, although the form of a sum can be used also merely for the sake of applications.

The differentiation uses the idea of a variable being expressible as a series of other variable quanta,

102 but it doesn’t study such series: differential calculus is no theory of series. Yet, Hegel is still willing to explain how the form of series arises, although we could think of it as a mere means for differentiation. The important issue is not the representation of the base variable as a sum of smaller quanta, but the representation of the power as a sum of the aspectual number systems. This sum is not a mere sum of quanta, but a function of those number systems: it is indeed a similar relation between differently qualified number systems as is investigated in the differential calculus generally. The variable that is power of the other variable hence contains “plus” or the form of a sum within itself: that is, it contains the possibility of being divided into the qualitative aspects that come up in differentiation.

11./503. The increment should not be understood as a true quantum, but as mere means: we might even represent it by number 1, as we would then have no trouble of removing it. The form of series of differentials is arbitrary, because all differentials are related to the previous differential as to an original function: this relation is the important element in the differentiation.

Hegel makes somewhat radical sounding suggestion that the so-called increment in the differentiation could be replaced by number 1. Firstly, the suggestion would make the results of the differentiation somewhat easier to discern, as the 1 would essentially vanish from all the members of the series, when (x+1) would be raised to a power, and only the required coefficients or differentials would be left. Secondly, as Hegelian mathematics would allow the reinterpretation of the unit as smaller than now, we could imagine that the 1 would represent arbitrarily small quantum. If one wanted to remind oneself that e.g. the first differential is supposed to be a coefficient of a first power of the increment, one could use some indices to indicate which differential the coefficient is supposed to be. Even this is futile, according to Hegel, as the form of the series is not very important: the essence of the differentiation has already been discovered, when the relation of the original function to the first differential has been found out, as all the rest of the differentials can be discovered by applying the same method as in finding the first differential.

12./504. The main interest in differential calculus is the possibility of modelling concrete problems by the structures it investigates. 13./505. The differential calculus involves the change of a dimension: thus, it can be used in problems which involve different dimensions, such as occurs in geometry or in mechanics.

After the subject matter of differential calculus and operations used to investigate it have been determined, it is time to turn our attention to the next task of finding suitable applications for differential calculus. Even here Hegel does not base his study on arbitrary experiences, but on

103 investigating the structure studied in differential calculus in order to see where it might be most profitably used. Now, the differential calculus is, according to Hegel, all about the relations between differently qualified number systems, the main example being the different powers: in other words, it is all about relations of different dimensions. Hence, the most natural applications of differential calculus can be found in problems where the values of one dimension have been determined and the values of another dimension are to be found out. Such examples can be found, firstly, in the spatial relations between e.g. lines and planes: problems in this area involve, for instance, determining a size of a plane that is formed by turning a curve around an axis. Second realm of application is opened by the study of movement and its component elements, such as velocity and acceleration. Here the problems include e.g. determining the acceleration of an object as a function of time, when the velocity can be expressed as a similar function. It is these two applications that we shall take under observation now.

14./506. The applications of differential calculus have some apparently arbitrary characteristics which derive from the nature of the object to which the calculus is applied: furthermore, it must be understood where the calculus can be applied. Only the movement from higher to lower powers is the task of differential calculus, while the other direction is investigated by the integral calculus.

The theoretical framework of differential calculus can be easily described without any reference to experience – it is only a matter of different number systems – but when the framework is being applied, the empirical element brings with itself certain modifications which make the differentiation more difficult. The primary problem is, of course, to determine how to interpret the abstract structures of differential calculus, that is, what should be their references in a given area of application. Beside this, every subject matter has its own quirks that further determine the manner how differentiation can be applied: these quirks are result of the basic nature of the subject matter in question, but in view of the mathematician using calculus they might seem quite arbitrary. The application of differentiation causes also the diversification of the calculus into two components. In theoretical framework, there is no difference in whether we look at the matter from the viewpoint of the original function or from the viewpoint of the derivative: the framework merely speaks of a relation between two number systems. In applications, it is vital to know whether we have been given a function with a lower power – derivative – and a task of finding a function of a higher power – the original function – or conversely. Thus, it is because of the application that the need for separate differential and integral calculus has arisen.

15./507. Simplest example of the differential calculus is that of the relation between a curve and straight lines attached to it: given an equation of a higher power describing a curve, the task is to find a simple relation between certain

104 straight lines like tangent.

The differentiation is an operation for finding out from a given a function the aspects – the rate of change etc. – that together form the original function. Thus, it is a movement from something concrete to something more abstract: from equation to mere relation, as Hegel puts it. The simplest example of a geometrical application for this operation is given in the form of an equation of, say, the second power. This equation determines a relationship between two variables, one of which is of higher power than the other, and the equation is represented by a graph known as parabola. Now, at every point of the parabola, we can draw a line such that merely touches the parabola at that exact point, but does not cut it: this is the tangent of parabola at that point. At the point of contact, the tangent and the curve move, as it were, into the exactly same direction. A tangent is a mere straight line, representing an equation of a first power, and in Hegel’s terminology, not being a true equation at all, but a mere relation determined by the coefficient describing the direction to which the tangent is going. Now, one of the aspects that can be abstracted from the original function by differentiation is just this direction of the curve at a given moment: we can present the direction as a function of the base variable in the original function. Thus, we can actually determine the relation of tangents to a curve with the aid of differentiations. Further relation provided by the differentiation is the relation of the original function and the derivative, which gives us the length of the so-called subtangent, which is the projection of the tangent to the x-axis.

16./508. The first discoverers of the rule for finding the tangents, such as Barrow, merely used it without pondering how it works: it was only Leibniz and Newton who felt the need of justifying it.

The mathematics in the early years of the modern age was still in a stage of fermentation, like all sciences: it was a time when alchemical, numerological and generally mystical issues were mixed with solid results. Science in general was considered an art that required more skill and genius than hard work. Mathematics, in particular, was conceived as a skill for finding answers to some numeric and geometric problems. One important example was the problem for finding solutions to equations with higher powers. Different people had different techniques and contests were arranged where the mathematicians could match for the title of the quickest and surest discoverers of the solutions for the equations. Another problematic was the discovery of tangents for different curves, and similar battles of different methods were arranged for finding tangents. A good mathematician was apt to hide his method from publicity instead of announcing it: after all, he would lose his prestige, if the method would be out for all to know. Indeed, it is quite remarkable that men like Barrow would publish their methods. In such circumstances, there was understandably little need for justifying one’s

105 method: it sufficed if it led to correct solutions efficiently. Only when similar methods were seen to work in other areas of application also, it became apparent that they would need some further justification: this was a task that Leibniz and Newton were the first to tackle with seriously.

17./509. The differentiation produces a relation of straight lines from a function of a curve: application of the relation needs still knowledge of what straight lines are in such a relation, which the ancient mathematicians knew already, while the modern mathematicians have merely found a quick method for discovering the relation.

Hegel notes that mere differential calculus is not a very informative theory. The differentiation itself would be quite meaningless operation if no reference were connected with it: the relation found by differentiation is useless if we do not know what it relates. Thus, in the example of a curve, we have to know which straight lines the derivative relates: otherwise, the differentiation has revealed us nothing. In this matter, the modern mathematics has made no step forward from the ancient mathematics, Hegel thinks, for even in antiquity it was known that in a triangle drawn with tangent as one side the direction of the tangent expressed the relation of the two remaining side: the ordinate and the subtangent. What the moderns have added has been the method, by which this relation can be found almost mechanically, when the equation of the curve in question is known: a practical instead of a theoretical discovery. 18./510. Lagrange has differentiated these two tasks: first one finds the required differential through a general operation, then one applies it to the curve and shows geometrically that the tangent and the line determined by the relation coincide. Although the latter proof speaks of increments, they are not used as infinitesimals, but as a method for showing that no line can be drawn between the tangent and the curve.

While earlier mathematicians confused the two steps in finding the tangent for a curve, it is once again Lagrange who has earned Hegel’s praise by separating them. The confusion of the two steps brings with it the awkward question of infinitesimals, but the separation of them manages to avoid that problem. The first part of Lagrange’s method is the familiar differentiation, but the second step is then a geometric proof. This proof has a form similar to many proofs of ancient mathematicians which showed that some quantity cannot be larger than another quantity, but also not smaller and must thus be exactly the same size: e.g. Euclid proved in this manner that two circles must have same relations to each other as the squares on their radii. Lagrange’s proof argues that the area left between a tangent and the corresponding curve must be smaller than any area left between the curve and an arbitrary line that does not cut the curve, and furthermore, that this straight line does not cut the curve: the obvious result being that it is the tangent of the curve. Such a proof does use the increment, but merely to show that any straight line with the increment differs from the line

106 determined merely by the derivative: no problematic diminishing of the increment is required. 19./511. Even Descartes’ tangential method discovers the values of normal and subnormal of a curve geometrically.

Hegel is anxious to show that the equation of a tangent and lengths of related lines can be found out even without the help of differential calculus, in a pure geometric fashion: thus, the entire muddle with infinitesimals could be easily avoided. In addition to Lagrange’s method Hegel mentions Descartes’ method of finding tangents, described in latter’s book on geometry. Descartes sets out for himself a task for finding the slope of the normal for a tangent of a curve: with that information, the slope and the equation of the tangent can be easily determined. As the normal may be thought as the radii of a circle, the problem may be thought in the terms of finding a circle that touches the curve. I shall explain how the Descartes’ method works in the simplest case, where the curve is parabola. Descartes notes, first, that the triangle formed by the normal, ordinate and the projection of the normal or subnormal is straight-angled, thus the square of the length of the normal is representable as the sum of the squares of the ordinate and the subnormal: this is the standard form for an equation of a circle. As the subnormal is then representable as a part of the abscissa (x – v) and the ordinate is known as a function of the abscissa, we are left with a quadratic equation of x with parameters v and s, where v occurs in a coefficient of x in first power. Now, Descartes notes that as the curve and circle only touch without cutting each other, the equation of a second degree can have only one solution or root. As the coefficient of x in the first power is always the sum of the roots of the quadratic equation – and in this case, as the double of the only root – the coefficient and thus also v can be represented in terms of x. Because the variables v, x and y determine the slope of the normal, the slope can now be expressed in terms of x alone. 20./512. Descartes’ method finds the same relation as differential calculus, but differential calculus has not yet proven that the relation obtains between certain lines: this must be done by a study of the concrete figures.

The importance of the geometric proofs of Descartes and Lagrange lies in that they provide a connection with the concrete problems which mere differentiation does not. It is a common error even nowadays among students learning differential calculus that by learning the mechanic operation of differentiating one has mastered the differential calculus completely: the result being that the students differentiate everything, even when the differentiation would not be needed or is even inapplicable to the problem. A good introduction to differential calculus would then teach not just the mastery of the mechanical operations, but also understanding of why and where differential calculus is useful – which undoubtedly is true generally of teaching all mathematics.

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21./513. In Descartes’ method the constant term need not be investigated, because it is known that the result can be determined from the coefficient, while in the common differential calculus the removal of the constant seems arbitrary.

Another reason for praising the geometric proofs is that they give a proper explanation for the apparently unjustified ignoring of the further differentials: the differentials present aspects that make up the whole of the original function, and it thus seems like an unwarranted abstraction when only the first differential is sought for. Mere assertion that the first differential is what we are looking for is not sufficient, because we still have to explain why the first differential suffices for determining the equation of the tangent. Descartes’ proof, on the other hand, makes it clear that the first coefficient in the quadratic equation is, firstly, all that is needed to determine the slope of the normal and the tangent, and secondly, that it depends only on the value of x by being its double. Similarly, if we were required to know the constant term of this equation for some purpose – it contains the length of the normal, for instance – it could be solved merely from the fact that it is the square of the x. 22./514. The words “differentiating” and “integrating” are inappropriate, because differentiating abstracts from all differences, while integrating returns them.

Names and words are always a bit arbitrary, and reasons for naming an object in some manner are usually more historical than rational. The word “differentiating” tells us something: it points to the “infinite” differences that the first theoretical forms of differentiation used. Similarly, “integrating” refers to the idea that integration involves adding up an infinite sum of infinitely small rectangles. As both words refer to such suspicious ideas as infinite sums and infinitesimals, Hegel is consequent in his criticism. Yet, he has also another point, namely, that it would be more appropriate if the words would change their references in this occasion. Differentiation begins from a function consisting of many different aspects. Then it abstracts one of these aspects – the first differential – and treats it like those other “differences” would not exist: it is a movement from diversity to abstract unity. The so-called integration, on the other hand, tries to solve the concrete function when the first derivative is given: in order to do that it must find all those different aspects that were abstracted away in differentiation. Hence, integration is a movement towards greater diversity.

23./515. In the study of movement or mechanics, a mere constant movement does not offer anything to differentiate, but from the equation of fall, one can differentiate the relation 2at: it is futile to interpret this term as part of the actual movement or as the movement without the effect of the gravity.

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After his take on geometry, Hegel turns to the other example of the applications of differential calculus, that is, mechanics. Here it actually seems problematic whether the calculus can do any good. As in geometry, a mere equation with no variables raised into a higher power requires no differentiation: in a constant movement, the result of the differentiation – the velocity – is known already as the simple relation of time and space. More complex movements, such as the constantly accelerating movement of an object falling to ground, have more complex equations, and thus, they seem a proper subject for differentiation: the equation of falling can be divided into aspectual equations constituting it. The problems arise when one should interpret e.g. the first differential of the fall or the double of the acceleration and time combined. Hegel does not accept the common explanation that the differentials describe forces affecting the movement or at least he notes that it makes metaphysical assumptions which are not based on experience: from the fact that a movement is combined from certain aspects we cannot deduce that it is caused by entities corresponding to those aspects. A more plausible explanation would be to describe the term 2at in dispositional terms: if the fall would continue after certain moment as a constant movement, it would travel twice the distance it has travelled thus far in the same time. Yet, when one should explain how such an abrupt change is possible, problems occur once again. The suggestion that a force accelerating the fall would have discontinued its effect makes also the unwarranted supposition of force affecting the movement: in fact, it would be more truthful to suggest that a forceful intervention is required in order to change the fall into a constant movement.

24./516. The simplest movement after the fall has not been found in nature; still, there is the more complex movement of planets, which should be investigated with differential calculus.

The empirical limitations for applying differential calculus are felt more clearly in mechanics than in geometry. There are curves corresponding to equations of all degree, and thus we can try to find tangents for all of them with differential calculus. In the mechanics, on the other hand, there are not so many sorts of motion known: indeed, there is no known motion that would follow the equation where the variable of distance is a cube of the variable of time. Yet, Hegel cannot resist advertising one of his favourite potential research programs here. There is a form of motion more complex than mere fall, namely, the movement of planets around the sun, where according to Kepler’s third law the square of the time travelled has a constant ratio with the cube of the planet’s distance from the sun. What Hegel envisions is not so much an explanation of this movement through the use of forces – Newton attempted it already, and Hegel wasn’t quite convinced of the worth of that explanation – but a description and analysis of the main aspects of that motion: differential calculus

109 would be used to discover those aspects which together form the motion of the planet round the sun.

25./517. More substantial results are gained from the differential [or integral] calculus, when the curved trajectories of the motion are to be determined.

The analysis of motion through differential calculus is an interesting, but ultimately merely technical problem for Hegel: it should particularly be not understood as any sort of justification for making ontological assumptions about entities like forces. More interesting questions arise when geometry is attached to mechanics, that is, when beside the variability of motion, the trajectories it produces are taken into account. The task is to determine what sort of curve the motion of an object constructs, when we know certain aspects affecting the movement of an object. This problem is in fact quite the reverse of the problem of analyzing motion into its differentials: it is more of a reconstructing of the motion from the differentials – it is what Kepler did when compared to Newton. Indeed, this is not anymore a task of differential calculus proper, but its converse, the integral calculus.

26./518. The theoretical part of integration is not about indefinite series, although it has to occasionally use them, but of finding the original function for a derivative: thus, the theoretical facet of integration is already given with the theoretical part of differentiation.

There are two ways how to introduce a person to integral calculus and it is the value of these ways Hegel is here considering. Firstly, the historically earlier manner is the idea of integration as making an infinite sum of infinitely small rectangles. As this idea of integration is stained with the muddle of infinite sums and infinitesimals, it is no wonder that Hegel does not endorse it: thus, his repugnance to the name “integration” which obviously hails from the idea of summing small areas. The other way is to understand integration just as the converse operation for differentiation: the final characterization of integration depends then on the way how differentiation is understood. As we should already know, for Hegel, differentiation means essentially the method of finding for a function describing the relation of one number system to another a function which would be an aspect of the first function in that it would describe the rate of change of the first function – the derivative or first differential. The integration is, then, essentially meant to be a converse operation: given the function describing the rate of change of some function, find the original function of which it is the derivative.

27./519. In some sense the application of integration is already determined by the differential calculus: with the relation of function to its derivative the converse relation is also given.

110 28./520. Yet, there is still the further problem of deciding in a concrete case which relatum is already known.

The differential calculus in general is about the study of the whole relation between number systems determined by the original function and its derivative: or there are two relations, but one is determined as the converse of the other. Now, it seems then that the division of differential calculus into two different parts – differential calculus proper and integral calculus – is somewhat unnecessary, as both are merely different sides of the same coin. The division is, as Hegel points out, more for the sake of application than for any difference in the theoretical framework. In concrete problems we often don’t know whether we have been given an original function or a derivative, and furthermore, it is a completely different thing to search for the derivative than to search for the original function. Thus, it is convenient to divide the differential calculus into, as it were, two independent disciplines: differentiation is needed in some problems and integration in others.

29./521. In differential calculus, the problem is to find which moments of the situation at hand have the relation indicated by the derivative; in integral calculus, the problem is to show that the result that is to be searched corresponds with that determined by the original function.

The actual operations of differentiation and integration are then nothing but reversals of one another: with knowhow for the differentiation we have knowhow for the integration. It is the application of the results of operation that truly separates the two disciplines. In differential calculus, the problem was to determine which aspects of the concrete case were regulated by the relation or function that the derivative was: what number systems – e.g. lines attached to a curve – were related to each other in this manner. In the integration the problematic is somewhat different. Here we already know what number systems are regulated by the relation of the derivative – for instance, we have a concrete graph of the derivative – and we have to find out what function it is differentiated from. Hegel notes that usually problem is not actually put in such words: we are not asked e.g. to determine the aspect of the case which is related to a given aspect as derivative to original. Instead, we are presented with a task of finding certain aspect, and in order to do that, we have to determine whether to use integration or differentiation and then to prove that this was the right choice: e.g. we might be given a problem of determining the volume of a body described by the movement of a given curve around an axis, and no hint is given whether this is a problem for differentiation or for integration.

30./522. The ordinary integration uses the concepts of infinitesimally small areas and of their infinite sum in order to calculate areas and lengths.

111 Hegel returns to describe more fully the historically primary way to understand integral calculus, when it is applied to some definite problems, for instance, counting the area falling between a curve, an axis and two lines parallel to the other axis. Here, the area can be approximated through sums of small rectangles that fall on the same axis and between the same parallels as the area in question. The sum of those rectangles is usually either too small or too large depending on the height of the rectangle, but by making the width of the rectangles smaller and smaller, the sum can be made as similar to the actual area as wanted. We may then imagine that there is some final phase of infinitely small rectangles, the infinite sum of which equals the area required. The other example goes by the curious name of rectification of curve: it is nothing more complicated than a calculation of a length of a piece of a curve. Here the length of curve can be approximated by drawing straight lines between pairs of points of a curve. The sum of such straight lines is an approximation of the curve, which can be made more accurate by choosing more points in the curve. The final curve can then be thought as an infinite sum of infinitely small straight lines.

31./523. This method merely presupposes the proposition that a curve and a tangent of a curve relate respectively to the area determined by the curve and the original curve as a derivative to its original function.

The method exposited in the previous paragraph is somewhat problematic, not just because it uses the confused concepts of infinitesimals and infinite collections, but because it does not properly explain how it is connected to the true issue of differential and integral calculus, that is, the relation of the original function and its derivative. One cannot literally count infinite sum of infinitely small rectangles or of infinitely small straight lines. Luckily, we have the convenient result that a variable representing the area between a curve and an axis – the “square” of curve – is related to the function of the curve like an original function to its derivative; similar relation holds between functions of a tangent of a curve and of the curve itself. One can use these results to count the supposed infinite sum, but the problem of justifying that these relations hold has still been left unanswered.

32./524. The proposition is usually justified by noting e.g. that in an infinitesimal case the curve is like a straight line and related to some other straight line like an original function to its derivative: the result is achieved by taking an infinite sum. 33./525. The correct point in this method is that applying integration is not just mechanical reverse of applying differentiation, but contains a transition from infinity to finity or from abstraction to definite quantum.

The problem of justifying the results would be non-existent if the curve in question was no proper curve, but a straight line: then the results could be deduced with simple geometric considerations. Now, the division of area or curve into infinitely small pieces is meant to make such justification

112 possible, as in an infinitesimal case curve should coincide with a certain straight line, namely, its tangent. This cutting of a curve into infinitesimal pieces is, of course, highly suspicious move, and equally suspicious is the countermove of summing the infinitesimals infinitely many times, by which we should get to the definite quantum of the area or the curve. Despite its unbelievability, the method has some inkling of the proper issue, Hegel suggests. Like differential calculus does not consist of mere differentiation, but also of applying differentiation, similarly integral calculus is not just about a converse of differentiation called integration, but also of applying integration. In the application, it is the question of moving from “infinitesimal” to the finite, although in somewhat different sense than is meant in the usual method: the integral calculus makes a transition from the functions regulating some aspect of a quantum to the concrete value of the quantum.

34./526. Lagrange justifies the use of integration in the counting of the area with the method of exhaustion. 35./527. Lagrange uses the same method in determining the length of a curve: this was done already by Archimedes.

It is once again up to Lagrange to base the integral calculus in a more mathematically pleasing manner. Just like in the case of differential calculus, it is the good old method of exhaustion that will do the trick: that is, one just has to show clear qualitative limits to the quantum to be determined and then to argue that the quantum would overstep those limits if its value would be greater or smaller than it is. In case of counting the area between the curve and the axis, we can define two figures formed of straight lines, one smaller and other larger than the required area would be: then, if the area would be smaller or larger it would coincide with some such area, which it obviously doesn’t for the reason that one side of it is a curve instead of a straight line. Lagrange’s method of determining the length of the curve is of this sort, and Archimedes used a similar method in order to determine the value of pi. The method is based on the simple fact that a curve is always greater than the sum of its arcs, but smaller than the sum of its tangents. Now, if the curve would be smaller or larger than the integration would tell, then it could be represented as sum of arcs or tangents, which is impossible. 36./528. Archimedes’ and Kepler’s use of infinitesimals has been taken as a justification of their use, but the modern mathematicians have concentrated merely on the negative property of discarding infinitesimals as non-quanta and ignored the affirmative property of their being a relation between different number systems: it took Lagrange to discover this affirmative once again. 37./529. The proper subject matter of differential calculus is the relation of the derivative and the original function: all other calculations can be used in differentiation and integration, but they do not form its proper object. Even the theory of series is no proper part of differential calculus, although it shares some characteristics with it.

As the end of the remark is getting nearer, it is time to gather all the results together and determine

113 what is important in the differential calculus as a whole. Hegel begins with a short historical passage of the development of the calculus. Archimedes was the first to use the idea of infinitesimals, but he admitted that they amounted to no proper proof: one could use them in easing the determination of the areas and lengths, but one should still give a proper justification for the quanta thus found. The modern mathematicians applied the method without much thought for whether its use was justified. Furthermore, the method was used merely for the sake of convenience, without any understanding of its proper area of application: the infinitesimals were just a cheap trick by which one could get rid of certain embarrassing terms in an equation by declaring them small and insignificant. It was the honour of Lagrange, according to Hegel, to discover –or rediscover – the proper use to which the infinitesimals should be put: the study of the functional relations between different number systems, especially those constituting aspects or derivatives of more concrete systems. In the final paragraph Hegel once again demarcates the proper subject matter of differential calculus from all other calculatory operations. True, we can calculate differentials of logarithms, cosines etc., but these are merely application of the paradigmatic and exemplary case of differentiation occurring in the differentiation of simple polynomials. Results and methods from other mathematical fields of investigation can be applied in differential calculus – often they are needed to simplify the equations – but so are addition and similar elementary operations used and no one suggests that differential calculus is about addition. Especially one must separate the theory of series from differential calculus. The differential calculus can be exemplified through the use of Taylorian series, but the interest of the theory of series and differential calculus in such series is different. Theory of series is interested in finding limits that the sums of finite sets of terms of such series tend to – the so-called sum of the infinite series – while differential calculus is interested merely of the relation of the original function and the first term of the Taylorian series – the first derivative.

Remark 3. Some more forms connected with qualitative determinations of quanta.

The subject matter of differential calculus has thus been discovered through what could be called empirical investigations of mathematical problems: the main issue is the relation between derivative and its original function or an aspectual function regulating the rate of change and the respective complex whole, and thus, differential calculus can be applied in all questions where elements have such relations. The subject matter of differential calculus is one example of different qualitative number systems – this particular example is instantiated e.g. by differences of time, velocity and acceleration – but there are still other examples of such qualitative differences between

114 number systems. The examples presented in this remark are all of what could be called dimensional differences between e.g. lines and planes and particularly of the problematic possibility of expressing quanta of one dimension as functions of quanta of another dimension: as we shall see, this cannot be understood as expressing quanta of one dimension as sums of quanta on another dimension, but through a more complex dependence relation.

1./530. Relations of qualitatively different number systems can appear in other forms beside the relation of the derivative and the original function. 2./531. The relations of different powers reveal their qualitative difference completely only when they are applied to geometric objects. This application brings us to the difference between [relatively] discrete and continuous quanta and the possibility of expressing latter as the sum of former: thus, Kepler suggested proving Archimedes’ proposition on the area of a circle by dividing the circle into infinitesimally small triangles.

Hegel approaches the issue of this remark through some notes on the differential calculus and its topic or difference of derivative and original function and especially the exemplary instance of the difference between different powers. He notes that in some sense exponentiation does not still produce different number systems: we may interpret raising number into a power as a function occurring within some number system, although it is also possible to interpret it as mapping differently qualified number systems to one another. The latter application is more natural in the case of geometric applications, where squares and cubes have a natural interpretation as intuitably different dimensions compared to lines. There are thus functions between different geometric dimensions: areas are squares of lengths. As the latter function can be interpreted in a more arithmetic manner in which they are reducible to sums of numbers in the same number system, it becomes natural to ask whether we could say that the areas are also sums of lengths. Here Hegel speaks of continuous and discrete quanta in a sense differing from how he used it earlier. Officially, continuity and discreteness are mere aspects of all quantities: a quantity can be seen either as one unity or as sum of smaller quantities of the same dimension or number system. Here the discreteness and continuity have been interpreted anew, as relative properties of different number systems: a number system or dimension that is more natural to be seen as a unity can be called continuous, while the number systems, quanta of which can be separated within quanta of the continuous number system, can be called discrete – obviously, what is continuous in one sense, like areas compared to lengths, can be discrete in another sense, like compared to solids. It is clear that the two senses of the difference of discrete and continuous cannot be mixed: on one reading, they are characteristics within a number system, in another reading they are characteristics of number systems. Hegel gives finally an example of a mathematician trying to interpret function between

115 dimensions as an infinite sum. The circle can be represented as a function of its radius: it is pi times the square of the radius, or in Archimedes’ term, as large as the straight-angled triangle formed by its circumference – which is pi times the diameter of the circle – and its radii as cathetus. Kepler notes that this function of a circle can be understood as an infinite sum of infinitesimally small triangles, and thus the circle can in principle be seen as a sum of lines. As we should know by now, Hegel cannot accept the idea of an infinite sum of infinitesimals literally. Instead, as we shall see, he suggests interpreting Kepler’s notion in a properly functional manner: linear terms can “make transitions” or be mapped into planar terms through some function regulating these terms. 3./532. What is understood as summation in Kepler’s example is actually already multiplication: furthermore, geometric multiplication necessarily changes the dimension unlike arithmetic dimension – it is like the transition of points to lines. Geometric multiplication, unlike arithmetic, is restricted by the properties of space.

Hegel’s task is then clear: he must show that apparent summation of infinite numbers of infinitesimals, if it is to have any sense, cannot actually be a mere summation of quantities to produce quantities of the same dimension, but must be more like a functional determination of quantities within one dimension through quantities of another dimension. Hegel calls this functional determination multiplication for obvious reasons: the planar quantum of e.g. rectangle is truly determined by two linear quantities of height and base and in this particular case the mere value of the planar quantum corresponds to the product of the values of the two linear quanta. Yet, Hegel quickly reminds that such a geometric multiplication differs from mere arithmetic multiplication, which can be understood as an operation within some number system, because the geometric operation essentially connects different number systems: in “multiplying” height with base, we move from linear quanta to planar quanta. Hegel compares this geometric multiplication with the ancient idea of line being movement of point, which Hegel himself on many occasion toys with: he also uses the curious expression that lines are the point’s becoming out of itself. Despite misleading terminology, Hegel is clear that it is not a question of point literally moving and thus producing lines: it would be completely arbitrary if the point moved, and thus, the existence of lines would be based on mere chance. Hegel is actually saying that by determining different points one can determine also lines: given two different points one already has a line. The possibility of constructing or discovering a point, when one point is already given, depends on the nature of space: thus, the possible existence of lines, planes and solids is grounded on the nature of space. Hence, the so-called the geometric multiplication is regulated by the characteristics of the space we live in: Hegel points out as an example that the meaningful geometric multiplication ends to three terms which determine a solid, although an arithmetic

116 multiplication can have as many terms as one wants. Hegel’s belief is that this restricting role of space plays a considerable role in mechanics. For instance, he suggests often that Kepler’s third law concerning the relation of the time taken by a planet in its orbit and its distance from the sun would be somehow regulated by the nature of the space: the spatial term of the relation would naturally limit itself to the third power, while the temporal term could be of a second power, although time itself is one-dimensional.

4./533. The qualitative relation between different number system is supposed not to be determined by exponentiation, but by a summation of discrete quanta into a continuity. 5./534. The summation is actually multiplication. For instance, the trapeze can be seen as a sum of an infinite series of infinitesimal planes between two parallels: the result of this sum should be the product of the sum of the parallels with the half of the number of the infinitesimals – but this sum is already a continuous quantum or height and the summation is actually multiplication.

Hegel is anxious to characterize the topic of this remark in a sufficient manner, by separating it both from the issue of differential calculus and from mere summation: it is geometric multiplication instead of exponentiation or summation. It is the summation that Hegel feels the most need to distinct the geometric multiplication from. Hegel presents a bit more complex example than mere rectangle, where the area is truly determined simply by two linear quanta of height and base. In the case of trapeze, we have two differing, but parallel bases: if we use the simile of plane being a moving line, the length of the line going through the trapeze does not stay equal, but varies when moving from one end to another. Now, suppose one thinks of the trapeze as a sum of infinite parallel lines or infinitesimally small planes. In case of finite sums, where we add a number n of distinct quanta situated regularly between two quanta a and b, the whole sum is, as is easy to see, (a+b)n/2. Thus, if we replace in the infinite sum of the trapeze the number n with the height h of the trapeze, we get the proper formula for the area: (a+b)h/2. The problem is, as Hegel points out, that n and h are two different sorts of quanta: n refers to a number of independent objects, while h refers to an internal measure of one particular quantum. In other words, h is not the number of the lines, but the height of the area: we are not adding lines up, but determining an area.

6./535. The summation is used also in cases where it is not the question of strict determination of areas by multiplication, but merely of finding relations of different areas: for instance, the relation of a circle and an ellipse is justified by presenting both as sums of infinitesimal trapezes. This is unnecessary, because the relation is determined by the regularity of the relation of lines within them.

Hegel perceives a possible counterargument against his position that the so-called summation of infinitesimal areas would be actually determination of areas through lines functionally related to the

117 area. What about the cases where we are not determining area of a figure, but its relation to an area of another figure, the imagined opponent asks. Hegel considers the case of the relation between an ellipse and a circle with radii identical with the major axis of the ellipse. We know that taking lines parallel to the minor axis, their cross-section with the circle is always related to the respective crosssection with the ellipse as the major axis to the minor axis: hence, if we take infinite sums over these lines, we supposedly should show that the circle as a whole is related to the ellipse like the major axis to the minor axis. Hegel points out that although there is no determination of the quantum of the area here, this is still no counterexample to his general suggestion: although the lines are not used to functionally determine the area of the circle or the ellipse, their relation between the lines determines functionally the relation between the areas. 7./536. This functional determination is on the basis of Cavalleri’s method of indivisibles where basic “indivisibles” or lines regulate the quantity of some area: they do not form the continuity of the area, but determine it arithmetically.

Hegel finds time once again to investigate more closely the work of mathematicians he appreciated: as before, his interpretations change heavily the original ideas of the mathematicians. Cavalleri’s theory of so-called indivisibles is a prime example of a theory using infinitesimals: Cavalleri e.g. shows first that lines in a figure have always certain relation to respective lines of other figures and then deduces the relation of the figures or even their determinate value from this known relation. It is probable that Cavalleri himself would have admitted that the principle of deduction he uses is based on the idea of an infinite summation and his indivisibles are meant to be infinitesimals. Still, Hegel is willing to interpret Cavalleri in good light: Cavalleri understood, according to Hegel, that his method was only expressing functional relation between quanta of different number system. As Hegel says, indivisibles or quanta of lower dimension do not constitute continuities or quanta of higher dimension, but merely determine their relation to other quanta of the same dimension.

8./537. It is a difference in method, whether figures are determined merely through their limits or also through some internal characteristics like height: this is the difference between Euclid’s and Cavalleri’s proofs. Cavalleri explains that his method does not involve use of infinite number of lines, but depends merely on the relation that the potential lines have to the limiting lines of the figure.

When Euclid proved that parallelograms with same height – or actually, situated between same parallels – and equal base are equal, he made his proof in two steps. First, he assumed that the parallelograms had the exact same base. Then he could show that both parallelograms could be represented as a sum of areas A + B – C. Area A was a triangle formed by the common base line and sides of the parallelograms that crossed one another: this triangle was clearly part of the both

118 parallelograms. Area B was a triangle formed by the line parallel to base and the two sides that belonged to the two parallelograms and started from one end of the base: it could be shown through the rules of equalities of triangles that the starting point of the two sides did not affect the area of the triangle B. The triangle C was the common part of the two versions of the B: in effect, it was the area between the two parallelograms. Secondly, this result could be extended to all parallelograms between the same parallels and having equal bases: the trick of Euclid’s proof was to draw a mediating parallelogram that shared one base with both parallelograms. All in all, Euclid’s proof is like a jigsaw puzzle: there are different pieces externally connected to form a figure of parallelogram and we just have to reshuffle the pieces in order to see that differently shaped, but equally large parallelograms can be made from the same pieces. While Euclid’s proof is based on an external viewpoint where figures are regarded as mere aggregates of smaller figures, Cavalieri’s method starts from internal proportions of the figure: parallelograms of same height are related as their bases, because the lines drawn at same distance from the respective bases are related in this manner. Now, if we supposed that this proof is based on the parallelograms being sums of infinitesimal planes, we would have fallen back to the idea of the figure as a mere aggregate. Cavalieri escapes from this conclusion by noting that he does not need to think of the plane as an infinite sum: all that is required is to say that the area of the plane is functionally dependent on the length of the lines in it. 9./538. Cavalieri’s description of his method as based on the fact that areas consist of lines is undoubtedly misleading: even Cavalieri himself is in pains to show that he can justify his propositions through Euclidean means without assuming infinities.

Although Cavalieri has an implicit understanding of the issue he is studying, Hegel thinks that he, like so many mathematicians, has problems in making this understanding explicit. Cavalieri says e.g. that the continuous magnitude is nothing but the indivisibles, that is, that the quantum of one dimension can truly be a sum of quanta of another dimension. Hegel is, of course, quick to point out that it is more of a question of certain characteristics and relations of quanta that can be transported from one dimension to another: relations of line can functionally determine relations of areas. Cavalieri himself has some trouble avoiding all the pitfalls of the idea of a figure as an infinite sum, and in order to convince his readers he quickly points out that his propositions concerning areas of figures could be justified in the Euclidean manner also. For Hegel, this is actually fallback from the justification based on inner characteristics and relations of a figure to a justification based on viewing the figures as mere external aggregates of smaller figures.

10./539. Euclidean manner of proving through congruencies serves only intuition, while qualitatively e.g. a triangle is

119 determined when three of its six factors – the sides and the angles – have been determined.

In elementary geometry, there are the well-known propositions of the congruence of triangles that have certain elements equal in size: e.g. if all respective sides are equal in two triangles, the angles of the triangles are also equal. The most elementary proposition of them all – or at least the one that Euclid proves first – is the one where we know the equality of two respective sides and an angle between them and can then deduce the equality of the remaining side and the angles. As Euclid has no better way of proving the proposition, he asks the reader to move one triangle on top of another and note that the two triangles coincide completely. Now, this is no proper mathematical proof, Hegel says, and we may concur with him: it smells too much of empiricism to be reliable. Much better would be to take the two sides and the angle between them as already determining the triangle: given these elements we know all the rest of the elements of the triangle – a proposition becomes an axiom or even a definition.

11./540. It is important to choose right lines as regulating the area of figures, or else the results might be wrong.

In the previous paragraph, a triangle was completely determined merely by two of its sides and an angle between them: with them, all the rest of the sides and angles were determined and thus also the area of the triangle. Now, when it is a question of determining the area of a more complex figure, it is vital to know which elements of the figure are essential for determining the area. Supposing two parallelograms with equal bases, the comparison of the other parallel sides might suggest that one of them is larger than the other: yet, the only important issue is the height of the parallelogram. The example of a cone is similar: Tacquet tries to ridicule the idea of lines regulating the area of the surface of the cone by choosing lines that do not have any connection with the area, while Barrow shows that it is the side of the so-called generating triangle of the cone that can be used to determine the area. Hegel leaves it unexplained how one is then to know which lines do function as regulators: mere reference to propositions known by other means seems not to be the correct answer.

12./541. It has been the purpose of these remarks to criticize the muddled idea of infinite collections: the true idea of infinity is nothing but incommensurability of different number systems – such incommensurability does not exclude a functional dependency between them.

The final paragraph closes not just this remark, but the whole series of remarks on mathematical infinity with a final explanation. As we should know by now, Hegel has attempted, firstly, to criticize the confused idea of an infinite quantity or collection, and secondly, to show that it is not needed in mathematics, not even in differential calculus or in relation of quanta of different

120 dimensions. It makes no sense to say that e.g. a plane would consist of an infinite number of lines, Hegel says. What can be said is that planes and lines are incommensurable or belong to differently qualified number systems: in this sense, we may say that plane is infinite in comparison to line, that is, planes cannot be measured as multiples of lines nor lines as fractions of planes. Hegel notes that a similar relation obtains sometimes also between different sort of lines, that is, curves and straight line. This possibility is based on the difference between what are nowadays called algebraic and transcendental numbers: transcendental numbers like pi cannot even be expressed as solutions of some finite algebraic equations with rational coefficients. Thus, curves like circle, circumferences of which are related to their radii with some multiple of pi, are unavoidably of a different number system than those radii in the Hegelian mathematics which recognizes only relations of one number to another inside one number system. Now, even if pi cannot be expressed through any rational number, its value can still be approached through better and better approximations: we can know the limits within which a length of curve is situated. We can even know the relations of different circles in basis of the relation of their radii: the system of radii functionally regulate the system of circles. Generally, we may say that different number systems have such functional relationships: for instance, an area of a figure is determined by lengths of certain lines, like parallelogram is determined by its base and height. It is these functional relationships that differential calculus and the “multiplication of lines by lines” investigate, as we have seen.

Glossary: Quantum = quantum; a limited state of quantified-being or state of quantity; quantity that is related to other quantities or to some unit or reference quantity Zahl = number; an arbitrarily large collection of copies of some quantity, which has been arbitrarily chosen as a unit or reference quantity Einheit (in case of quantities) =unit; a quantity that has been multiplied arbitrarily many times to form some unified number Anzahl = amount; the arbitrary number of the units collected together to form some unified number Extensives Quantum = extensive quantum; a quantity that can be interpreted as a number or a collection of some units; corresponds to a cardinal number Intensives Quantum / Grad = intensive quantum or grade; an undivided or even indivisible unit that has a certain numerically expressible place in an ordering of units; corresponds to an ordinal number Quantitative Endlichkeit = the fact that the value of quantum is determined by its relation to other quanta

121 Quantitative Unendlichkeit = the fact that the value of a quantum can be made arbitrarily large or small Schlechte quantitative Unendlichkeit or Unendlichgroße/kleine = the supposed ”largest” or ”smallest” quantity, which can exist only in some limited context (either as largest/smallest quantity of some limited series or as a quantity compared to another quantity of different species, such as plane compared to a line) Wahrhafte quantitative Unendlichkeit = a ”relation” or function for constructing one quantity out of another (in the same or different system of quantities)