Commentary on Hegel\'s Logic 9: Real measures

1 Second chapter. Real measure

The previous chapter presented us a series of more and more complex measures, starting from a simple classification that could be interpreted both qualitatively and quantitatively, continuing with a specifying measure where a number system with a mere quantitative meaning was related by a function to a number system with a qualitative meaning and ended with relation of number systems both having qualitative meanings. The reason behind these constructions was something called realization of measure: we had to construct a measure that had as its aspects simpler measures or a function that related one measure to another – such a complex measure could then be interpreted as not just a relation between situations or objects, but as an independent object itself. By varying the number systems involved and their relations, we could actually construct many different measures or functions as objects. The aim of this chapter is then to investigate the relationships these measures or functions have to one another: in effect, we are looking at algebraic systems consisting of functions. Here Hegel could actually have done a lot more in this area, but the state of mathematics was not yet ripe for such discussions: Hegel is satisfied with describing structures that could be used in describing chemical issues. This task of constructing framework for combining measures etc. is actually mostly left to the first division of this chapter, while the two latter divisions concentrate on constructions familiar from the sections on quality and quantity: the second division shows how one can always construct further measures from given measures – like one could find qualities from given qualities or discover quantities larger or smaller than given quantities – while the third division points out that we could interpret these different measures as aspects of one underlying “infinity”, or as it is here called, measureless.

1./614. We are now investigating relationship of measures that characterize certain objects: previous measures could be relations of mere situations, like space and time, but these structures are exemplified e.g. by chemical relations – although space and time are aspects of the relationships, they alone do not determine them. Because the measures are here relations of simpler measures, we don’t have to assume they are any more complex than direct ratios.

The purpose of the previous chapter was to provide us with objects to work with: similar moves have been occurring since the beginning of the Logic where the state of nothingness or the empty situations served as the first object known certainly to us. The reason for this maneuver is to find some common objects for all those who are acquainted with measures: even if you and I otherwise applied measure-structures to different objects, we could still both structurize “pure measures” or abstract functions according to them. This possibility, of course, also presupposes that we can arrange measures as objects into measure-classifications, that is, classify them both qualitatively

2 and quantitatively. This task is performed by the first section of the new chapter, which provides us then with first measures that certainly will have to be classifications of objects, while previous structures, on the other hand, could have been classifications of situations or structures of situations: thus, we cannot anymore use mere relation of space and time as an example, but we will have to use material objects in our physical examples. Interesting question is why Hegel suddenly accepts direct ratios as measures, when he spent whole previous chapter in constructing something more complex as an example for measure. The main reason is the familiar movement to the most abstract form of a structure type involved in a transition to another chapter: we have found an example of a framework of a plurality of measures of functions, thus, we should now begin from the simplest type of such a framework by ignoring all the unnecessary details.

2./614. (A) A measure or function as an object can be combined in a specific manner with other functions: this combination produces new measures, and the original measure is characterized by the different measures it produces in those combinations.

The first section of the chapter is evidently a construction of a model for chemical combinations, although Hegel makes a half-hearted attempt to suggest that other phenomena of nature could also be structurized in a similar manner. Yet, we need not assume that Hegel would here have to rely on arbitrary experience, because there is a purely mathematical instance to which this section could be applied, namely, any algebra of functions. Indeed, this possibility is foreshadowed by Hegel’s own text, where we learn that we are about to investigate combinations of measures, which essentially are certain functions between different variables. The result of this construction shall be that we obtain a characterization of functions in terms of their place in this combinatory algebra: what function is produced when two functions are combined in some manner?

3./615. (B) The functions characterized by combinations can be classified qualitatively and retain some of their qualitativeness in combinations, but in another sense they can be classified in a merely quantitative fashion: we have a structure where quantitative progression is broken with qualitative leaps.

Although the issue of the algebra of functions would be interesting, Hegel develops the subject matter only as a means for further constructions. The idea of measures preferring certain combinations is evidently taken from chemical phenomena and it is difficult to translate into terms of the algebra of functions. Yet, this is no great loss, because Hegel is merely trying to classify measures both quantitatively and qualitatively: the “exclusive combinations” provide Hegel with the qualitative “leaps” in the otherwise quantitative or combinatorial classification. Such double classification can undoubtedly be given to the algebra of functions also, thus making the reliance on

3 chemical terms unnecessary. The partly quantitative and partly qualitative classification serves then as an example how a difference in quantity can result in a difference in quality or how we can change one measure to another through mere quantitative alterations: we met this phenomenon earlier, but then it wasn’t still certain whether we could have measure-structures with objects.

4./616. (C) We can interpret the different measures classified both quantitatively and qualitatively as aspects or states of a measureless or infinite.

The final section of the chapter ends with a familiar transition. We are acquainted with a classification – previously we have had first qualitative and then quantitative classifications, now we have a combination of both. Then we notice that an object in one position of the classification can be transformed into an object in another position: an object could have different qualities and quantities, or as in here, it could be characterized by a different measure. The result is that we could interpret the situation anew: the supposed classification of objects could be taken as a classification of aspects of one underlying object. This underlying object is then “infinite” in comparison with its limited or “finite” states: or as in here, it is not characterized by a single measure, hence, it is measureless.

A. Relationship of independent measures

We begin from an arbitrary framework of many different measures as objects, whether they be mathematical functions or chemical matters characterized by their specific gravity: Hegel’s purpose in this section is to describe in general terms the behaviour of all such systems. The first two subsections succeed in this task: they investigate, firstly, how measures can be in some manner combined to produce new measures, and secondly, how these combinations can then be used to characterize the individual measures within the system. The third subsection is more of a disappointment, as it concentrates on the question how certain combinations of measures seem more natural than others: a chemical phenomenon that is hard to transfer e.g. to a mathematical example.

1./617. Measures are at this stage not just simple functions, but functions between simpler measures and can thus be taken as objects.

The crucial difference between the measures of this chapter and measures of the previous chapter is that here the measures are not just measures – relations or functions between different number systems – but also objects. Hegel goes even so far as to call the “realized” or complex measures

4 material. This is a perfectly understandable analogy: objects in general are to situations in general like material objects or things are to the spatio-temporal contexts in which they exist. Furthermore, material objects can also be seen as one species of complex measures, as we are about to witness: matters are characterized by their specific gravity, which is essentially a function from the weight of the object to its volume.

2./618. (a) The relation of such measures is at first stage merely given: the measures are independent objects that are externally combined.

At the first stage of investigation we look at the framework of measures as something immediate or merely given: we have a set of measures as independent objects that at first sight have no essential connection to other measures. We are studying what results come about from combination of different measures: what new functions we get from combination of familiar functions or what new chemical stuff is produced from a mix of different chemical matters. Because of the independency of the objects, there is nothing to tell which combinations would be more essential or natural: we merely connect measures in some arbitrary manner.

3./619. (b) The independent measures are qualitatively classified by their quantitative characteristics and the results of combining them with other measures: measures are now seen as combinable with other measures and thus as having a definite place in a system of measures.

It is possible to combine a measure not just with one, but with several measures: thus, for every measure we can find a series of possible combinations. In the case of chemical matters, the combinations can occur only between certain sorts of matters – e.g. between an acid and a base, but not between two acids – and indeed, we can divide a certain group of chemical matters into two sorts (acids and bases) according to their capability of being combined with other matters. Hence, Hegel introduced the notion of affinity, referring to the relationship between measures that can be combined with one another: in the case of functions this notion can be applied only in a trivial manner, as all functions can be combined with all other functions. What is common to all combinatory system of measures is the possibility of characterizing measures by their possible combinations: e.g. a chemical substance is defined by the results of combining it with other substances.

4./620. (c) We can find some combinations that are more natural and which measure thus prefers over others.

The final phase of the section concentrates on a phenomenon that is almost exclusively instantiated

5 by the chemistry, where e.g. a base might “prefer” combining with one acid rather than another: actually, this has undoubtedly nothing to do with any preference on the part of the base, but is caused by further properties of the matters involved. In the combination of mathematical functions no similar “natural” exclusion of combinations is involved: at most, we could say that some combinations seem to us more natural than other, for instance, the combination of a function with its converse. Then again, we should not think on this account that the algebra of functions would be a bad example of what Hegel is looking for: even in the combination of tones or notes – a musical example Hegel himself mentions – the natural combinations are determined more by our likes than the nature of the combined notes.

a. Combination of two measures

The result of the previous discussions was that we have constructed an example of a measure that could be interpreted as object, because it is an underlying unity behind two aspects (namely, the qualitative number systems or constituent measures). The example Hegel presented was that of a power function, but here we abstract from this requirement: we are looking at any measure or function between number systems that is seen as an object, even one that is a mere direct ratio. The next task shall be the investigation of what further properties the functions have and what relations such functions have to one another. Particularly interesting are the combinations of functions or measures, as they differentiate measures from both qualities and quantities: qualities cannot even be combined in a proper sense, while quantities combine in a simpler manner.

1./621. We face an object that is a function between two qualitative number systems: one of the qualities is essential and makes the object an object (e.g. intensive weight or extensive largeness), while the other represents arbitrary situations in which the object exists (e.g. space) – the relationship of these number systems forms the nature of the object (e.g. the specific gravity). The non-independent situations must be taken as units or reference points, while the intensity of the underlying object is what is counted: because the measure is taken as an object, the relationship need not be one between different powers, but a direct ratio.

We know already that the issue we are dealing with here is an abstraction from the result of the previous stage. What is not so apparent is what characteristics to include in the current structure. One clear example that should not be included is the relation of powers that was in Hegel’s eyes the important element that separated a complex measure from immediate. The measures we now deal with can be direct ratios. Hegel goes even so far that he says that they must be direct ratios: this seems somewhat of an exaggeration, although a natural conclusion if one is mainly trying to model chemical relations – like Hegel undoubtedly is.

6 Beyond this one characteristic, almost all elements of the previous stage are retained, albeit with a slightly different reading. Hegel particularly mentions the fact that one of the relata or qualitative number systems should be an essential unity, while the other is merely an external collection. Previously, in a relation of qualitative number systems, we had e.g. time as a unified, ongoing process, while the space or distance was a mere collection of independent locations: this difference in the qualities was supposedly reflected in the roles that the number systems played in the function connecting them – time was a root, space a power. Here, on other hand, the different roles of the number systems are connected more with their relations to the total measure or the object: the essential number system reflects more of the characteristic of the object itself, while the arbitrary number system is connected with the inessential situations or aspects in which the object appears: for instance, weight is a property of a material object or collection of objects, while volume is only a property of the space it occupies. Note that here the meanings of the reality and ideality are somewhat reversed, due to the presence of the object. Previously, space was characterized by reality in the sense that it was a multiplicity of truly independent situations, while time was described as ideal, because it was a framework of situations which were merely aspects of one unitary process. Now, the concepts of reality and ideality cannot mean same as they did, because in relation to the underlying object all frameworks of multiple situations seem merely aspectual: here, then, the reality refers to the object or the underlying unity. Another apparent difference concerns the relation of the conceptual pairs of reality-ideality, extensive-intensive and unit-amount. Their relation seemed clear in the previous chapter: the number system representing the ideality was naturally intensive – that is, described order relations – and also the reference point from which the values of the other number system were calculated. Here, these clear relations are confused or at least made more fluid. The number system connected with the object can be interpreted both intensively – if we view the object as a true unity – and extensively – if we see it as consisting of smaller objects: indeed, Hegel appears to be quite indifferent as to how any object – such as matter – is in this context interpreted. Now, Hegel speaks as if the external number system should always be interpreted extensively, but this seems another exaggeration: if we interpret the object itself as a collection of objects, the quantity of the “space” they occupy could be interpreted as a unit in an ordering. Similarly, Hegel’s statement that the measures studied here are then functions from the external to the essential number system depend on the viewpoint. If object is interpreted extensively as a collection of “smaller” objects, we calculate the number of these objects from the intensity of the situation they occupy; if object is interpreted as an intensive unity, we calculate the number of the situation it occupies from its intensity.

7 2./622. The direct ratio of an object belongs to the nature of the object, but its quantum is determined only by its relation to the direct ratios of other objects: still, because of its quantitativeness, it can change, and with it, also the object. Such change is occasioned by a combination with another object (for instance, the combination of two metals): in one sense, both direct ratios or functions remain independent, in another sense, they specify or affect one another.

At this particular stage, the objects we are interested in are essentially characterized by the functions relating different number systems: indeed, in the mathematical example, objects are nothing more than functions. When we restrict our attention to direct ratios, as Hegel does, we have a natural quantitative ordering for the functions. We might ask whether this natural ordering could be extended to other functions. At least in the case of algebraic functions this seems plausible, although the ordering would not be isomorphic with the usual ordering of the rational or real numbers. Now, looking at the direct ratios, their numeric value is most natural to interpret in an intensive manner: function “3x” does not consist of functions “2x” and “1x”. Thus, in order that the numeric value would have any meaning we must have the ability to construct examples of other functions from examples of one function: Hegel suggests that combining measures to produce new measures would be an example of such an ability. In the paradigmatic case of functions this is clearly possible in a very strong sense. We have at least two different sorts of combinations, the summing of functions – (f + g)(x) = f(x) + g(x) – and combination of functions – (f o g)(x) = f(g(x)). Furthermore, we need only one function to begin constructing new functions: e.g. (f + f)(x) and (f o f)(x) usually differ from the original function. In Hegel’s own example – the combination of matters – the producing of new measures or “sorts of matter” is not as easy: we need at least two matters to begin with. Furthermore, as Hegel himself admits, there are restrictions as to what sort of matters one can combine. Note that Hegel is not attempting to prove apriori that matters can be combined: he is merely pointing out that certain physical phenomena could be described in terms of models that could be constructed “apriori”. The combination of measures or functions works in two distinct manners. Firstly, as in combination of quantities, we can “separate” or “analyze” the combination, that is, present it as a combination of independent measures: e.g. we can say that f o g = h. Yet, the combined measures are not completely independent in the context of the combination: while previously the functions f and g worked as independent regulators between two number systems, now their results have been modified by the other function.

3./623. If the combination were merely quantitative, then the values of the related variables in the combination would be combinations of the respective variables in the two combined measures: but only the side connected with the object changes in such a manner, while the mutual specifying of direct ratios modifies the combination of the values of the

8 inessential or aspectual side – for instance, in a combination of different matter, the volume of the combination may differ from the sum of the volumes of the combined matters.

The paragraph appears to rely heavily on the empirical phenomenon of combining different matters, and especially on an outdated interpretation of that phenomenon, thus making it rather pointless for the modern reader. Yet, the phenomenon Hegel describes has a clear counterpart in the example of mathematical functions. Suppose we investigate a combination of two functions, f and g – it is indifferent whether we are interested of f + g or f o g. Now, let us fix one number system connected by the function, in the sense that when we are given quantity a with f and quantity b with g, we are then always given quantity a + b with f + g or f o g: it is natural to assume that this number system is the one connected with the essential nature of the “function as an object”. Then, the value of the other number system is not determined by the formula f(a) + g(b) – as should be, if the second number system would also “change only quantitatively” – but by the formula (f + g) (a + b) or (f o g) (a + b). While Hegel’s descriptions correspond with what happens in the paradigmatic example of mathematical functions, the case is different when Hegel tries to apply this structure to the chemical phenomenon of combining matters. True, when we combine e.g. alcohol with water, the weight of the combination is the combination of the weights of the alcohol and the water, while the volume of the combination is somewhat smaller than the combination of the volumes of the separate matters: it is unclear whether Hegel thinks there is some formula by which the volume of the combination could still be calculated. Yet, the applied structure contains elements that might not be present in the empirical phenomenon: we should assume that material objects would not consists of atoms – or this would not be an essential description of them – but would be like force fields, which regulated the space around them through the intensity of their weight – the combination of two such matters would make the force fields modify one another. As the explanation Hegel rejects – that the diminishing of the volume is caused by there being empty space between atoms – appears to be the correct one, it seems that Hegel has here applied a structure of Logic in a wrong place: no fault of the Logic, but only of the application.

4./624. What can be changed is not just one qualitative number system, but the whole framework of related number systems: thus, the measure as the specific quality of an object is determined by its relation to other measures or functions.

The investigation of the specific characteristic of the combinations of measures was merely a side trip: the important issue is the possibility of combining measures at all, not its particular properties. What does this possibility imply? It shows that a measure can be changed into other measures –

9 although perhaps with the help of another measure – just like quantities can be turned into other quantities. Indeed, we may abstract from the properties that the measure or function is supposed to have in itself and concentrate only on its possible combinations with other measures or functions. The abstraction results in an abstract calculatory system of measures or functions, which is the subject matter of the next subsection.

b. Measure as a series of measure-relations

We began the study of measures or functions as objects from the possibility of combining them: we noted especially that the combining of measures differed essentially from combining of mere quanta. Next task shall be to extend these investigations: a function can be combined not just with one, but with a series of other functions. The results of these combinations provide a characterization of the function and of the whole system of functions or measures in general. The subsection follows the traditional threefold division of “introduction-analysis-transition”: first Hegel shows that one must consider all the possible combinations of measures, then he describes the results of these combinations and finally he argues that we can interpret some of these combinations as more essential than others.

1./625. 1. Introduction to the algebra of measures. If two objects with only a qualitative classification were combined, both would be lost in the result: the quantitative existence of an object with measure lets it survive combination, while its specific nature lets it affect the nature of the combination – the quality of the object is protected by its quantitativity. The combination is arbitrary, because an object can be combined with other measures also.

We begin with a reminder of the special nature of the combinations of measures by comparing it with the combination of both qualities and quantities. It may well be asked whether there is any combination of qualities, at least in the sense that there are combinations of quantities and measures: the combination of qualities is merely a sudden transition from one set of qualities to another, without any possibility to discern the previous qualities in the new situation – picture, for instance, the combination of yellow and blue to produce green, where green looks nothing like yellow or blue. On the other hand, on the combination of quantities, we always have the possibility of discerning the combined quantities within or at least in relation to the combination: 10 as an extensive quantity contains 4 and 6 and as an intensive quantity is essentially related to them within a number system. Yet, a quantitative sum is clearly simpler than a combination of measures: the combination of measures or functions involves also a specification of the values of the number system it relates – the nature of the combined measures affects the nature of the combined measure. Interesting part in this paragraph is the transition to the actual issue of this subsection, that is,

10 Hegel’s attempt to show that we should consider all possible combinations of a measure. The one combination we produced in the previous stage was arbitrary, as it depended on the choice of the measure or function with which the combination is made. Now, Hegel suggests, we should find other similar measures with which we can also combine this measure. Note that Hegel is here playing with the restrictions of the chemical example where we have strict limitations as to which sort of matters can be combined with one sort of matter: base can be combined with acids, but not with other bases. In the case of functions, the discovery of new combinations is far easier: we may just take the result of the previous combination and combine it with the original function.

2./626. 2. Analysis of the algebra of measures. The combination of a measure with different measures produces different measures as a result: these results provide us with the nature of the original measure. These results form a series in relation to the original measure as a “unit” or reference point: it is this series instead of its quantum that separates the measure or function from other measures or functions – other measures would produce other series.

Hegel explains in a rather simple manner the main idea behind the creation of a table of calculations for any system of calculation: ironical is that Hegel is not doing it on purpose, but is instead trying to describe the structure of chemical systems. We combine all objects – here, measures or functions – with all suitable objects: in the case of functions, all objects can be combined with all objects, while in the case of chemical matters, only some objects can be combined with one another. The series of results for one object – one row or column of the table – presents us with the nature of the object, at least within the context of that table. Hegel uses here the concept of unit in a peculiar manner, when he clearly wants to speak of a reference point: the object determined by one line of the calculatory table is the “unit” of that row or column, in the sense that all the objects in that line are determined as combinations of this “unit” with another measure.

3./627. Another similar measure forms other series with the same measures: problem is how to determine the relation of these “reference measures”. We need a common reference point for comparing both, but this can be found only in the series of combinations created by both: indifferent measures can be compared through their combinations with constant other measures. Beyond the comparison, the measures are reference points for a series of measures: when the measures are compared, the series is taken as a reference point.

When the confusing terminology of units and amounts is interpreted correctly, the chemical meaning Hegel has intended becomes evident. Suppose that we have produced all combinations of an acid with different possible bases: the series of combinations forms the nature of the acid in question. If we call the combination of acid with base their “comparison”, it becomes problematic how we could compare e.g. two acids which cannot be combined in the same manner. Hegel’s suggestion is obvious: we compare the series produced by both acids with all possible bases. This

11 description seems to have some difficulties in translating to the terms of the algebra of functions: in chemistry there are two different sorts of matters altogether such that one can combine only the different sorts to one another, while functions can all be combined with one another. Yet, we have something similar even in the case of functions, namely, the different roles a function can play in the combination: for instance, it is quite different whether f is on the left side of the o-combination [(f o g) (x) = f(g(x))] or on the right side [(g o f) (x) = g(f(x))]. Thus, the problem described here becomes the problem of comparing e.g. function as occurring on the left side with another function as occurring on the left side: functions in this manner cannot be “compared” by combining them, but only by comparing the series of combinations they produce with functions as occurring on the right side.

4./628. The measures of the other side can also be taken as reference points of their own series: all measures can be seen then (i) as reference points for a series of combinations, (ii) as belonging to measures that form a series of combinations for some reference point and (iii) as one among many measures with series of combinations.

The previous paragraphs have ignored the obvious fact that the forming and comparing of series could be made from either sort or aspect of object: we could compares bases as well as acids, and we could compare the series produced by the functions when they are in the right side of the combination as well as when they are in the left side. The results of all the calculations can then be presented in a form of tables, familiar for many from the time when they learned multiplication. Every measure in such a table has several aspects, which Hegel goes on to enumerate. Firstly, we may ask which series of combinations a given measure forms: then we are treating the measure as a reference point for certain series of combinations – for instance, we may investigate what functions are produced by left-combination of f with them. Furthermore, when we take some other measure as a reference point, this measure may occur as a constituent of one combination with it – e.g. (f o g) is one combination in a series of g as the right constituent. Finally, we may compare the measure and the series it produces with series produced by other measures.

12

Possible example of a table describing simple combinations of measures

5./629. 3. Transition to selective combinations. Here measure is determined by its place in a framework of measures, like a grade or intensive quantum was determined by its place in a number system: the difference is that here the framework does not consist of mere quanta, but similar is the manner of being essentially related to others. Measure as a mere given quantum is independent of other measures: it is the qualitative side of the measure that is determined by its relations to other measures, although this quality is determined by the “quantitative” production of new measure. In one sense or context, both measures are transformed, in another sense or context, the old measures remain intact in this combination, because they specify the nature of the new measure. If both measures are naturally combined to one another, they may form an exclusive unity, and this exclusiveness forms the specific nature of a measure. (i) A measure can be combined to many different other measures, if it is arbitrary to which it is related, but (ii) there is sometimes an exclusive choice to which it should be combined: we have then found an instance of a selective combination.

Until this moment we have regarded the combinations of measure as being equal: we have been interested only of combinations of pairs of measures or functions and thus there has been no need to decide the order of the combinations. When we start to speak of combinations of many measures, the question of the order of combinations becomes imminent: in the example of function, it is problematic whether in (f o g o h) we should first combine g and h or f and g. Whichever combination is made first, it may be said to exclude the possibility of the second combination. Hegel is interested here of such cases where the exclusive combinations happen somehow “naturally”, that is, where the exclusion is connected with the nature of the combined objects. Usually, in any system of calculation we decide that some manner of “reading” the calculations is the correct, for instance, that in the calculus of functions the combinations must be read from right to left: the exceptional cases can then be marked by use of parentheses, (f o g) o h. Neither of these possibilities is natural in the sense indicated earlier: the order of the combinations is determined by other considerations than the objects involved. On the other hand, in chemical combinations it can

13 well happen that one matter shows “preference” for combination with matter A instead of matter B. Does the transition to the next phase of Logic then require such a reliance on empirical phenomena or can we modify the calculus of functions in such a manner that the required examples of “natural” selectivity of combinations could be found? The answer is quite easy to fathom. Only thing we need to do is to decide that in the algebra of functions e.g. combinations of f and g are “preferred” or dealt before other combinations, if no parentheses indicate otherwise: in the context following this decision the combination of f and g would be “naturally” exclusive. This would be a rather interesting extension of algebra, although in a sense trivial: with the help of parentheses, all the same combinations as before could be still represented in this extension, that is, the extension would differ mainly on the pragmatics of calculating combinations. The only remaining question is whether there is any criterion for deciding which function-pairs to choose for being exclusive. Hegel’s only remark on this issue seems to indicate that if we have some other method of “manipulating” one measures or function in order to discover another measure or function, then their combinations would be exclusive. One possible example would be functions that are converse of one another: thus, if f and f –1 were converses of one another, then g o f o f –1 o h could be shortened to the form g o h. Note that this is actually possible in the ordinary algebra of functions also – this extension is thus no true extension. Yet, in the pragmatics of calculation, it is an optimal manner of calculating compared to the mere calculation from right to left: hence, it is in some sense “natural”.

c. Elective affinities Hegel’s reconstruction of chemical relations, suited also for the description of an algebra of functions, is nearing its end. We began by showing that we could combine measures or functions in order to produce new measures or functions: indeed, in case of functions, we could begin just with one function and few calculations and produce a whole system of functions from this starting point. Next, we noticed that one combination was not enough, but a whole system of possible combinations was needed to characterize one measure. Now, we have seen that when we combine more than two structures, we must specify which combinations we should do first. A decision for generally combining a certain pair of measures before other combinations gives us an example of what Hegel calls “an elective affinity” – in other words, a selective combination. We have already mentioned the somewhat futile nature of this subsection: although Hegel presents a possibility for an interesting expansion of ordinary algebra, it all comes to nothing, as the main purpose of this expansion is to exemplify a structure of objects classified both qualitatively and quantitatively.

14 1./630. We have used chemical concepts, because the structures investigated here appear in an especially clear manner in chemical combinations, where a nature of matter is determined by its possible combinations: similarly, in music, notes are determined by the harmonies they produce – these more concrete questions belong to the natural science.

The paragraph is interesting mostly because it enlightens the relationship between the Logic proper and its application to empirical sciences: it is not that the structures investigated here would be abstracted from the empirical findings, but the models constructed here can be applied to such empirical findings – whether they be chemical, musical or anything. The introduction of the musical relations is also intriguing, because up to now we might have thought that the chemical relationships were the only concrete application of the structures in this section. We can combine not just abstract functions or chemical substances, but also different notes to one another: a combination of different notes or tones produces a more complex sound – in this, the combination of notes resembles the chemical combinations. On the other hand, combinations of tones or harmonies are not “naturally” elective, but it is the human ear that prefers some harmonies over others: this characteristic resembles the manner how we must ourselves decide the functions, combinations of which should be preferred.

2./631. If the differences between distinct possibilities of combination were only quantitative [or otherwise external], the combination would be determined only by quantities [or by other external reasons]: in a properly selective combination, the quantities [or anything external] should not affect the combination. We could suppose that combinations were determined by the extensive amounts of the measures present: although we could present this basis in an intensive form, no real development would be made, because intensive and extensive magnitudes are mere aspects of the same object – it would be no true selective combination.

Hegel investigates a system of measure combinations where the preference order of the combinations is determined by the amounts of the measures involved. This is a clear reference to the chemical example, where we might imagine that only the amounts of available substances determine what combinations take place: we could imagine that an acid “prefers” to combine with a base that is available in the largest amount. Hegel’s outlook here is clearly somewhat restrictive: we should speak generally of external reasons for preferring some combinations: in the case of the algebra of functions, the arbitrary choice to calculate from right to left is one such example. Yet, we might apply Hegel’s chemical example of quantitative manner of ordering combinations in some manner also to this algebra: we might decide e.g. that the longest successions of identical functions would always be combined first – thus, f o g o g o g o f o f o f o f o g would be calculated as f o ((g o g o g) o (f o f o f o f)) o g. Of course, such manner of combination is not very natural in the manner Hegel wants, because it doesn’t depend on the nature of the measures involved: we could replace one measure with another, leaving just the amount of this measure unchanged – in the

15 previous example of functions, we could change function g in all places to a function h – and the order of combinations would still be formally same. Even if we represented the groups of different measures in a form of an intensive quantity – e.g. (f o f o f o f) would be represented as 4f – nothing essential would have changed: the order of combination would still be determined only by issues external to the measures.

3./632. Because the combinations here investigated involve measures instead of mere quantities, they may follow some more essential order, although in some cases quantitative [or external] reasons may determine the order of combinations: we have constructed a structure where both quantitative and qualitative elements are mixed.

The developments of this subsection end abruptly. Because we are combining measures instead of mere quantities, there may be criteria which should determine the order of combinations: perhaps the measures themselves “prefer” to be combined in some way, as in chemical combinations, or perhaps we can for some “aesthetic” reasons choose to prefer some order of combinations before others, as in music or calculus of functions. Nothing further is actually said on the subject of “elective affinities”, as Hegel has already established what he needs to continue his constructions. Not everything in the order of combinations is determined by the nature of the measures involved, but in some cases we must rely on externalities: e.g. in a calculation of functions involving no preferred combinations the order of the calculation must be determined in some external manner, like proceeding from right to left. We may now classify the different possible combinations involving at least two measures according to the order in which the individual combinations are made: some of them follow an order determined by the measures, others follow mere external ordering, while in most both external and essential elements are involved. We have thus constructed a structure or classification where both qualitative and quantitative elements are intertwined, that is, a measure structure, and this is where this section ends. While the purpose of the first chapter was to provide us with some objects, when any measure structure was given to us – we could just take the measures or functions involved as new objects – the purpose of the first section of this chapter has been to construct a measure structure for these objects. The end result is then a measure structure with objects: a result that might have been constructed in an easier manner also.

Remark

Although within the purely logical development of new structures, the first subsection has played only a minor role, within the more general context of Hegelian philosophy it should have a more prominent place, as an example of how to apply structures of Logic to concrete phenomena: it is clear that the chemical combinations have been on Hegel’s mind during whole of this section. Still,

16 this part of Hegel’s agenda seems somewhat dated now, as the developments of science have went beyond what Hegel in his time could know: although the analogy of the system of chemical combinations with the system of combinations of functions is interesting, it doesn’t even scratch the complexities involved in chemical relations. Unfortunately, Hegel even disfavours those suggestions of the chemistry of his times that were to have most success in explaining the chemical phenomena: his dislike of all atomistic explanations is a good example.

1./633. Chemical substances are the primary example of measures determined by their relation to one another: acids and bases are not truly independent, but try to combine with one another – e.g. an acid requires certain amount of base in order to become neutral. If the chemical substances differed only qualitatively, there would be only one acid and one base, but because of their quantitativeness, we have a whole series of acids and bases: because of their independency, e.g. acids prefer combinations with certain bases over combinations with other bases.

It is a fatal presupposition of Hegel that if one cannot see something supposedly material with one’s own eyes, then it doesn’t properly exist – fatal in the sense that it makes Hegel’s philosophy of nature seem very quaint and outdated. For instance, Hegel is willing to state that e.g. hydrogen and oxygen do not exist within water, but only when the water has been explicitly dissolved into hydrogen and oxygen, and generally, a chemical substance exists only in those phases of a chemical process where it can be literally observed – thus, Hegel speaks for the idea that chemical processes involve true creation and destruction of substances. Thus, acids and bases are according to Hegel existent only in their independent state, and because they remain in their independent state only through a forceful intervention, acids and bases are not “completely real” for Hegel: acids are only objects that tend to combine with bases and vice versa. Now, Hegel points out that other natural phenomena also involve similar “necessary connections” or relations of objects trying to “unite” with one another: the examples include relation of different poles of magnets and relation of positively and negatively electrical objects. The difference between these relations and the relation of acids and bases is that one object can actually be either positively or negatively electrical – not at one moment, but at least in different moments – and that both poles of magnet are present in one object and thus that in both cases one object may play both roles of the opposition, while an acid cannot be in any sense base nor base acid. Hegel’s explanation for this characteristic of chemical relations is that the chemical differences involve also quantitativeness. Hegel’s idea appears to be that because there are many possible sorts of bases that could be combined with many possible sorts of acids, we cannot change the roles of an acid and a base even conceptually, because e.g. an acid can always be determined also by its relations to other acids – somewhat unsatisfactory explanation. What interests Hegel most in this “chemical algebra” is the fact that the combinations

17 produced by mixing different substances are not always determined by mere quantitative issues. We might think that an acid would “prefer” to combine with a base which is present in the greatest degree, but this is not the case: acids prefer to combine with some bases and bases prefer to combine with some acids, not depending on how much of the acids and bases are present. Note that Hegel does not even try to provide an explanation for this phenomenon: he might even consider it misleading to try to explain such matters. He is satisfied with describing this habit of chemical substances, as it provides a good example of a natural structure where everything is apparently not ruled by quantities (the development of chemistry has since shown that “elective affinities” can be explained e.g. through quantum physical means).

2./634. It has been noticed that mixing of two neutral solutions always brings about only further neutral substances: thus, it is possible to calculate how much e.g. one base requires each acid in order to become neutral. This discovery has been later extended with more empirical evidences and hypotheses, but nothing essentially new has been found out.

Hegel’s interest in empirical sciences lies in describing phenomena and their regularities, not in explaining them. Thus, we note that Hegel is almost thrilled of the discovery of Fischer that the combinations of acids and bases follow a regular pattern: it is these regularities that allow us to describe the combinations of acids and bases in a form of “algebra of measures”. The table of possible combinations provide us with a generalized description of the phenomenon of chemical combinations: we do not have to anymore go and look what combination is produced from certain acid and base, but we can merely look at the table. Note how uninterested Hegel is of any extensions to Ficher’s discovery. At best, they are mere empirical data that do not add anything truly new to the general structure. At worst, they are hypotheses that try to explain the regularities of chemical combinations, but end up only supposing some unobservable and hence possibly fictional objects.

3./635. Berthollet has shown that in many cases apparent selectivity of combinations depends actually on external causes.

Although Hegel is anxious to show that some empirical phenomena follow the structure of “measure algebra” and especially the version with natural selective combinations, he is willing to test the proposition that chemical combinations follow these structures by comparing it with empirical data. Thus, Hegel must admit that Berthollet has shown the supposed selective combinations to be based on mere external and quantitative matters: issues like the strength of cohesion have an effect on the possible combinations a chemical substance can have and “prefers” to have. Still, Berthollet’s empirical discoveries have not yet shown that all cases of selective

18 chemical combinations wouldn’t be natural or based merely on the nature of substances involved. Furthermore, even if later discoveries of science would show without any doubt that no selective combinations are natural in Hegel’s sense, it would still be possible that other natural phenomena would show examples of natural selective combinations. 4./636. Berzelius explains Berthollet’s discoveries through a corpuscular metaphysics, where all matter consists of atoms and in an apparently continuous solution atoms of one matter are organized around atoms of other atoms: this metaphysics can be justified neither empirically nor logically.

Although Berthollet’s empirical discoveries Hegel will gladly accept, he is not so fond of any hypothetical explanations of them. Particularly hard Hegel finds to accept any atomistic explanation of chemical combinations: when two substances are chemically combined and even when one substance is solved into a liquid, it seems like the two separate substances were gone and only one substance left. Mere perceptions or experiments – at least those that could be made in Hegel’s time – could then not show that such a combination or solution consisted actually of two separate substances, which furthermore would consist of atoms and empty space between them: for Hegel, matter is continuous in the sense that between every two points of material object we can find still more matter. Berzelius’ theory seems to suggest that if we just could enhance our vision enough, we could see that seemingly continuous solutions actually consisted of small corpuscles or atoms: in relation to the scientific advancements of his time, Hegel is actually justified to say that such suggestions are still little more than fabrications of an imagination gone too wild – that is, if there is no other evidence for corpuscular theory. Hegel also suggests that Logic couldn’t either justify corpuscular theory – rather uninteresting statement, because it seems that Logic has little to do with so empirical questions – but adds also that Logic appears to be able to do just the opposite, that is, prove that the corpuscular theory is necessarily untrue. This is a rather bold statement and requires some justification. The one case where Logic has been able to “deduce” – or better, to demonstrate – something concrete was when the existence of actual quantitative infinities was shown to be dubious: Hegel could point out, firstly, that the mathematical constructions used in ordinary arithmetic never let us reach anything beyond finite numbers, and secondly, supposing we had found “an infinite number”, we could then always find a larger or “even more infinite” number, thus making it inappropriate to call the first number infinite: such arguments were no direct proof of the impossibility of infinities, but at least indirect arguments which demonstrated that “infinity” in one context is not always infinity in another context. Could a similar indirect “proof” be discovered against atomism? Hegel at least appears to think so. Such “proof” would show that if we were given an atomistic description of a chemical solution or combination, we could always find another, finer description where the

19 supposed simple atoms of one sort of matter actually consisted of smaller entities of both sorts of matters – no pure Logical proof, but a proof demanding certain physical presuppositions, like the infinite divisibility of matter.

5./637. The laws regulating the combinations of acids and bases tell only how much e.g. of an acid is needed for neutralizing a base, but they do not involve atomic theory: we could translate the theory as well into intensive terms of Kantian dynamism.

It is clearly the atomistic metaphysics that Hegel mostly dislikes in Berzelius’ explanations: the laws of neutralization concern the relation of a group of unities of one material to a unity of another material, but by unity Hegel does not refer to atoms, but to some arbitrary reference point – unities of first material are not independent, but all parts of one “mass”, while the unit of the second material is still divisible to further parts of the same material. If the atomic theory is ignored, Hegel is indifferent as to how the simple material objects should be described. They could be characterized as masses: thus, we would emphasize their divisibility and interpret them as having extensive quanta. On the other hand, they could be understood as force fields of resistance: then we would emphasize their unity and self-identity and interpret them as having intensive quanta. Hegel is quick to notice that these two interpretations are nothing but different interpretations of the same matter, of which, furthermore, neither can have any advantage over the other: what can be expressed in one interpretation can be expressed also in another interpretation. Thus, when Berzelius tries to distance his theory from e.g. Kant’s dynamism, he does not succeed, because if the atomist metaphysics is removed from Berzelius’ explanations, his theory coincides with Kant’s theory of matter.

6./638. When selective combinations have been explained quantitatively, they are no longer properly selective: when they are explained by external situations like cohesion, they are still qualitatively selective – similarly, stopping of the movement of the pendulum is caused by the weight of the pendulum, although this requires also friction.

Although Hegel previously appeared to admit that Berthollet’s experiments raised serious doubts of the existence of true “elective affinities”, he is now willing to state that they actually are no disproof of naturally selective combinations. On a closer look we see that Hegel’s change of mind is more a qualification of his previous statement. In so far as apparently selective combinations are explained by quantitative or other external reasons, the selective combinations have been “disproved” or shown to be no truly selective combination: thus, in so far as the preference of one chemical substance by another can be explained by mere quantitative or other external conditions, this preference has been “explained away”. Now, although an individual case of apparently selective

20 combination could be explained by external conditions – e.g. substance A prefers substance B over substance C in this context, because of the effects of stronger cohesion of B – we might still say that the capacity of an individual substance to react to different conditions would be part of its qualitative nature: while substance A would react in such a manner in these conditions, substance D might react in a completely different manner. Something similar seems to be behind Hegel’s obscure statement on the effects of mass and friction on the movement of a pendulum: the fact that pendulum has mass or weighs makes it possible that the pendulum slows down when it comes in contact with other bodies or endures friction.

7./639. Explaining elective affinity through electricity is futile: electricity can work as a cause of chemical combinations, but it doesn’t capture the essential characteristic of these combinations – electrical combinations are not as stable as chemical combinations.

Supposing hypothetical connections between magnetism, electricity and chemical combinations was fashionable in Hegel’s time: even Schellingian philosophers of nature were eager to see something great similarity in the fact that in these phenomena material objects seemed to attract and repulse one another – in this case, science has shown that the philosophers of nature were on to something important, as we now know that electricity and magnetism are aspects of the same force and that electricity has something to do also with the chemical combinations. It is thus intriguing to find out that Hegel was not very fond of this “identity theory”. The first reason is obvious: making unwarranted hypotheses is something Hegel is nor fond of. Yet, there is something more going on. Hegel is willing to admit that e.g. electricity might be cause of a chemical combination: for instance, in phenomena that Hegel calls galvanism, electricity is connected with chemical reactions. What Hegel is against to is the suggestion that electricity as a phenomenon would be completely identical with the phenomenon of chemical combination: he is against identifying their descriptions. Clearly, electrical phenomena differ in their characteristics from chemical phenomena: flow of electricity from one object to another is more temporary than chemical combinations. If we then ignore all hypothetical explanations of these phenomena, as Hegel does, it is just natural to say that electricity and chemical combinations are two quite different issues.

8./640. Empirical results show that we should speak of more or less electrically positive and negative substances, instead of electrically negative and positive substances: then the intensities of elective affinities can be reduced to the intensities of electrical polarities – which is no true reduction. 9./641. Like in Newton’s case, Berzelius’ empirical discoveries have made his followers uncritical towards the weaknesses of his metaphysics.

21 When Hegel is discussing electricity, he is describing certain electrical phenomena and not their – at the time – hypothetical cause: Hegel’s dislike of hypothesis is once again shown by his passing comparison of Berzelius’ chemical hypotheses with the physical hypotheses of Newton. Most likely it is cases like that of electrically opposite objects attracting one another or that of metal poles of different electricity producing sparkles between them that interest Hegel most. Hence, it is just understandable that he would insist that negative and positive electricity form a non-quantifiable opposition: in the context of a single electric phenomenon there is just one negatively and one positively charged object, although these charges do no form a classification of object – in this situation negatively charged object may be charged positively in another situation. Characteristics such as electronegativity and electropositivity describe not what role e.g. a metal takes in an electrical phenomenon, but its tendency to become either negatively or positively charged – its tendency to give or receive electrons, as we nowadays know – and form thus a quantitative scale of many positions instead of an opposition of two characteristics. Even in times of Hegel, chemists knew that the quantities of electronegativity and electropositivity where somehow connected with the tendencies of combinations of matters, and Hegel is not trying to deny there could be some connection. Instead, Hegel is trying to point out that from the very abstract viewpoint of ontological structure nothing much is gained by transition from a scale of intensive quantities of “elective affinities” to a scale of intensive quantities of electronegativity and electropositivity: we are still admitting that the combinations of substances can be described only with a scale of intensities.

10./642. In addition to selective combinations, the densities of specific substances form a system of quantitative relations, but the law governing this system is still unknown.

Hegel’s interest in the elective affinities of chemical substances lies in the hope that these combinations would form a system that is based on quantitative relations, but in a qualitative or non-external fashion: the preference of one possible combination over another should at least partially depend on the nature of the substances involved. A similar “quantitative-qualitative” system Hegel hopes to find in the system of “specific gravities” or densities of substances. In this case Hegel’s hope for describing the regularities of the system has been satisfied better than in the case of planetary distances: in fact, there seems to be no necessary regularities behind those distances. Yet, Hegel’s crude descriptions are still far from the table of the atomic weights. The greatest hindrance is that Hegel is not yet properly aware of the differences between mixed and pure substances, on the one hand, and between compounds and elements, on the other hand. Indeed, Hegel’s dislike of atomism actually would prevent him of accepting the basic notions behind the table.

22

11./643. The combination of substances with different densities reveals their qualitative nature: the density of the combination is incommensurable in relation to the densities of the combined substances.

The main difference between the system of densities Hegel envisions and the table of atomic weights is that Hegel intends the system to include also all the combinations of substances: indeed, he has not yet a proper understanding of the difference between an element and a compound. In this context Hegel introduces the idea that the system of densities should be qualitative: the qualitativeness of the system follows from the fact that the density of the combination of two substances is not always a mere sum of the densities of the combined substances. Hegel goes even so far as to suggest that the densities of combinations are incommensurable with the densities of the substances from which they are combined. Hegel’s suggestion becomes more comprehensible if we think of the different densities of comparable substances as positions in some number system, where regular laws of addition would be used: then a density of a combination of two substances would belong into a new number system, because it wouldn’t be the actual sum of their densities – the “algebra” of densities is not an algebra of numbers.

B. Nodal line of measure relations

We return to a familiar phenomenon involved in measure structures, namely, a change in quantitative position related to a change in a qualitative position: indeed, at first sight it seems that nothing new has been added to the discussion in the previous chapter. Yet, something has altered. We began this chapter by taking measures or functions as objects and the previous section was used in constructing a number system or “algebra” on basis of these measures: firstly, we defined or determined the results for combinations of pairs of measures, and then we defined or determined further rules by which to select the order of the calculations in multiple combinations on basis of the measures involved – defined in algebra of functions, determined in empirical examples, like Hegel’s “chemical algebra”. Now, it belongs to the quality of the measures which combinations it “prefers”. Yet, Hegel suggests this preference or selectivity is also conditioned by quantitative – or generally, external – issues, that is, a quantitative or external change of some sort may change the preferences: the chemical preference is conditioned by warmth, cohesion etc., while the preferences of the functional algebra may be changed by our arbitrary choice. What we have done is that we have now constructed an example of measure structure where there certainly are some objects – the measures, that is – while previously we couldn’t be sure whether the situations involved weren’t empty: here is the basic difference between this and the previous chapter.

23

1./644. A combination of measures was selective or exclusive: there is yet no other rule for determining when a combination should be preferred. An exclusive combination can be changed to another combination by breaking the combination into its elements and forming a new combination: it is in some sense arbitrary which combinations are selective.

At the end of the previous section we had chosen some particular combinations of measures or functions as such that should be always combined first in our algebra of functions: for instance, we could say that a combination of a function and its converse should be preferred. This decision did not undoubtedly remove all arbitrariness of order from our functional algebra: for instance, we could always use some method like parenthesis to indicate when we wanted to form a combination in an order differing from the usual. Furthermore, there is still no non-external manner of deciding which order to prefer in cases where there is no preferred combination, e.g. in case of f o g o h, where g is not a converse of either f or h. Finally, the preference itself is only a result of our arbitrary choice: it works only in a context of that choice, but not necessarily in other contexts. Suppose we would restrict the functions in our algebra to those of the form f(x) = ax, where a is some rational number: this resembles Hegel’s own example of the “algebra of chemical substances characterized by their densities”. Then we could choose another sort of preference rule, where combinations of sort f o 2f -1 would be preferred over other combinations [here 2g would be the function defined by 2g(x) = 2(g(x))]: then our preference rule would clearly favour combination resulting in function “2 times”. Indeed, we could construct a potentially infinite series of such “preference rules” characterized by rational numbers: a quantitative classification of qualitative number systems.

2./645. These different selective combinations can be interpreted as states of one, underlying substrate, or the sudden change of one configuration of measures to another can be seen as a continuous change of this substrate.

Next step in Hegel’s construction follows a familiar pattern: we note that because two different situations or objects are in some sense similar – in this case, they are both systems of measures or functions – and because we have the ability to change one into another – in this case, through alteration of the rules for selective combinations – we may interpret these situations or objects as mere aspects or states of one, underlying unity – here Hegel calls it substrate. Indeed, nothing further should need to be said: this step takes us already to the next phase of the chapter. Yet, Hegel still has something to say on the relation of this interpretation with the earlier interpretation, where the different algebras seemed independent contexts: Hegel wants to say that neither of these interpretations is here inessential.

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3./646. We have thus a method of constructing new “qualitative algebras”, which can then be interpreted as quantitative states of one algebra: this is an example of a nodal line.

Our method of creating or discovering new function algebras is an instance of a more extensive manner of creating new “measure algebras”. We may begin with any “exclusive measure”, that is, with any context where some combination of measures is preferred over others: for instance, we could begin from some situation where certain chemical substances prefer to combine with one another. Now, we may have means to “push this context from itself” – a literal rendering of “repulsing”, which in Hegel refers to the possibility of multiplying or copying something, either through construction or through discovering. What is then required is a method of creating new “preference contexts”: for instance, a method of changing the preferred combinations of chemical substances. Furthermore, this method should be of a quantitative nature, e.g. in the chemical case, a change of some quantity involved in the combination, like cohesion or temperature. Let us then focus on one “substrate”, that is, a particular set of measures to be combined: a particular set of functions in the case of function algebra, and a particular set of chemical substances in the chemical example. Then, the change of preference conditions can always be interpreted as a quantitative or generally external change, and indeed, in some cases the change of those conditions does not affect the outcome of the combination: result of calculating f o g o h stays same if the preference conditions of f, g and h do not change. On the other hand, sometimes the change of these apparently external conditions leads to the essential or qualitative change of the result: in the chemical example it may even happen that the old bonds between substances break down and new combinations emerge. We have then an example of a structure where qualitative and quantitative classifications are intertwined: on the one hand, we have the ongoing quantitative classification of the preference conditions, on the other hand, we have the qualitative classification of the different results for the same group of measures – this is an example of what Hegel calls the nodal line of measures.

4./647. Suppose we have one combination of measures: because of its relation to quantitativity, we may change some quantitative conditions without changing the combination, but in some point the change of these conditions specify a new combination or a new object. This object is a combination of same measures and is also quantitatively alterable to the first: yet, it can also be thought as a new object, which has arisen with no help. This change can repeated indefinitely.

Let us then take any “nodal line” – a structure where the same subject matter is classified both quantitatively and qualitatively – and choose any position in that line: certain combination of chemical substances or certain state of matter, for instance. In order to make the example work, we also need some way of manipulating this original object and its position in the quantitative

25 classification: at first, we may think that this change does not alter anything essential, because it is only quantitative. Then becomes the place where the changes step over the qualitative limits, and the object is changed. Although any combination of qualitative and quantitative classifications of same subject matter would do, Hegel continues to use his paradigmatic example of nodal lines, as the reference to affinities belies: e.g. because the components making the chemical combinations are still same, the new combination is not completely haphazard. Despite the quantitative continuity between the original and the new object, we may still interpret them as different: thus, we may say that the new object has arisen, while the old object has vanished. Hegel reminds that this whole process can be repeated. This doesn’t imply that there should be potentially infinite number of possible states within the qualitative side of this nodal line: we could as well change the direction and start to alter the conditions to those that used to hold.

5./648. Positions near the place where quality changes differ only quantitatively, or the change is only gradual near this point: yet, this gradual change does not explain the change of quality or sudden leap from one sort of object or situation to another. We cannot conceive leaps through gradual change, because the interpretation of nodal line as merely quantitative destroys its qualitative nature.

It is sometimes suggested that Hegel’s interest in the paradox of sudden qualitative leaps would indicate that he supported the idea of conceptual classifications being fuzzy: the appearance of sudden leap would then result from our belief in the strictness of the classifications. Actually, Hegel prefers indubitably the opposite viewpoint: there are strict classifications, at least when these classifications are qualitative. It is not the leap itself that should be explained, at least if explaining means explaining away: if we supposed all change to be gradual, no leap indeed would happen, but then the essential characteristic of nodal line would be lost. The leap is natural, when qualitative and quantitative classifications are connected – e.g. when the series of states of water are connected with the series of its possible temperatures. It is only our amazement at the leap that should be explained away – or it should be made explicit what the amazed person has failed to notice: he has concentrated on the quantitative scale and changes in it and thus has ignored the possible qualitative changes connected with the external quantitative alterations.

Remark

The idea of quantity turning into quality is one of the great inventions of Hegel ruined by Marxian simplification. Hegel is simply giving a description of certain structure that could be applied to certain, mostly natural phenomena: in some cases quantitative and qualitative classifications are connected and therefore a change of position in quantitative classification may result in a change of

26 position in qualitative classification. Note that Hegel does not say this would happen necessarily in all cases: what has been meant as a description of a certain class of structures has been taken by Marxists as a general law for all situations.

1./649. Arithmetical and musical relations show similar qualitative leaps: gradual quantitative alteration can cause sudden qualitative changes. 2./650. Chemical and physical changes present examples of sudden leaps: certain sorts of oxides can be generated only in certain relations, water moves suddenly from one physical state to another and generally all generation and corruption happens suddenly.

The paragraph is important evidence against the viewpoint that the section on measure would be dependent on the empirical sciences. Mostly this viewpoint appears to have been caused by mistaken conceptions of the nature of the whole Logic: one thinks that Hegel is trying to prove e.g. that all things without qualification have measures or at least that there necessarily exists something with measure, and when Hegel gives no proper proof for this viewpoint one concludes that its justification must lie somewhere else, namely, in the empirical sciences. Of course, if we understand Hegel merely as presenting a method for creating or discovering measure structures, no such need for empirical science is needed, provided we truly can construct such structures with any or no empirically given data. Now, we have seen that mathematics – more precisely, arithmetic – must be for Hegel constructible from no given data: one merely needs any situation and methods for creating new objects and identifying several objects in order to construct the first quantitative situation. Thus, if arithmetical relations already show properties of measure structures, no empirical given is needed: for instance, any surprising fact of number theory gives us an instance of a qualitative leap caused by mere quantitative progression. The possibility of constructing measures from no given information does not, of course, preclude the possibility of modelling some empirical phenomena through these constructions: for instance, the sudden leap of new arithmetical relations could resemble sudden leaps in more empirical fields. The example Hegel presents in this remark are of the familiar sort, that is, mainly from the musical and chemical areas: although we can present musical relationships through quantitative relations of different notes, the harmonies do not follow precisely any simple formula, but arise unexpectedly, and although chemical substances can be identified quantitatively and thus presented as a continuous series – e.g. through their densities – some of their combinations can occur only in certain specific conditions. Hegel’s example of water changing instantly from solid ice to liquid and finally from liquid to vaporous form is undoubtedly a classic, although it is not perhaps literally true: an ice cube does not change all at once to liquid water, but melts gradually. Despite the possibility that some or even all of these empirical examples would not

27 actually be instances of measure structures, this would still not make the possibility of such structures doubtful: this is justified already by our capability of constructing such structures.

3./651. It is futile to try to explain away all sudden generations and corruptions through gradual change, because in some contexts such sudden changes of state happen: for instance, water may change into ice instantly. 4./652. Gradual nature of change is often said to be unnoticeable: then the potentiality of one sort of object to change into another is replaced by the pre-existence of this another sort in a smaller quantity.

For Hegel, there is no final context which could fix the truth of all statements once and for all: there are only more detailed and informative contexts, but no final “world” containing everything. Furthermore, the contexts of everyday life appear in his view often more certain than hypothetical contexts presupposed only to explain the imagined peculiarities of everyday life. Thus, the possibility that all qualitative leaps in empirical realms would be explained away seems rather distant in Hegel’s view. We do experience some qualitative leaps: although a bucket of water does not turn into ice or vapour instantly, a smaller amount of it perhaps does. The experience of an uncontinuous change becomes indeed inevitable when we move into contexts where we couldn’t properly visualize any continuity in the change. It is undoubtedly possible to suppose that the change would be continuous, although we wouldn’t experience it as such. Yet, this does not cancel the fact that at least in the context of our experiences sudden leaps occur. Furthermore, there is no reason to suppose that all changes should happen continuously, although it would be easier to formulate such continuities mathematically, because they share analogical structure with the ordinary number series – indeed, modern cosmology would perhaps concur with Hegel here.

5./653. When we describe moral phenomena as mere states of being, they present example of sudden qualitative leaps after gradual change: innocence changes suddenly to crime, and states crumble by gaining too much area and population.

For Marx, the most interesting examples of “quantity turning into quality” were social: a slow development of human society could at once erupt into a devastating revolution that swept the old class divisions away. Hegel, on the other hand, is less enthusiastic of “historical or social leaps”. True, he mentions the possibility that a quantitative change of e.g. the population of a state could result in drastic upheavals in the constitution of the state: this is actually a modification of an old idea that certain constitutions work best in states of certain size. The other example Hegel mentions here comes from the realm of law and morality: a minor incident becomes suddenly a real crime when e.g. the amount of damage inflicted has increased enough. The true reason for Hegel’s disinterest is his conviction that phenomena investigated in human sciences are only partially or

28 inadequately describable in similar terms as phenomena of natural sciences. The affairs of a person or whole society are only in some aspects or in some peculiar circumstances affected merely by quantitative, external changes: particularly the ability of human beings to affect their own conditions makes the “physical” describing of human affairs difficult – for instance, not all gradual social changes need end in revolutions, because states can also make gradual reformations on their constitutions, thus making the revolution unnecessary.

C. Measureless

The beginning of the chapter had the task of constructing an example of a non-empty measure structure where quantitative and qualitative classifications of same objects were combined: the method was to take some algebra of measures and change its rules for the order of calculations in a regular manner – the result was that in the course of this apparently external alteration some combinations of measures could suddenly have different results. Now that we have at least one, although quite abstract example of “nodal lines” or classification of “finite measures” – finite, because every measure characterizes only certain position in the whole “nodal line” – it is time to find the corresponding “infinity”: note how this move reflects similar moves in the sections on quality and quantity. Hegel calls the infinity of measures usually measureless, and like in other cases, this phrase may refer to two things. Firstly, like any situation differing qualitatively or quantitatively from this is in some sense infinite – because it fulfils some condition that this situation does not – similarly every foreign position in a nodal line can be interpreted as being measureless. Secondly, in previous cases Hegel has also spoken of a “positive” or non-contextual meaning of infinity: this has meant the process or method by which one could have changed from quality or quantity to another. Here, the measureless in the proper sense refers to a unifying substrate behind different nodes of the line – we may interpret these nodes as states of one “matter”, Hegel says – while the idea that the method of constructing different measures is their common link is left for the next chapter to develop.

1./654. Because any measure structure has an aspect of quantitativeness, it can be changed and thus its relations may alter drastically: an object or situation characterized by such relations is destroyed and replaced with something that has no measure in the viewpoint of the original object or situation – quantitativeness is the weak point of all objects and situations.

The measure structures we have studied – the algebras of functions and the “chemical algebras” – have actually two sorts of quantitative aspects. Firstly, there are the different quantitative relations within the algebras, for instance, the relation that a group of chemical objects or functions have to a

29 result of their combination. Such relations are regulated by the algebras and thus cannot really have any effect on them: e.g. if we change the “measures” that we are about to combine, the result of the combination likely changes, but the relation between the combined objects and their combination still follows the rules of the system. Secondly, there is the quantitativeness by which different algebras themselves are classified – e.g. the quantitative classification of the rules of different function algebras and the quantitative classification of different conditions affecting the preference of chemical combinations. It is when we change these conditions that the possibility of sudden qualitative changes arises: a result of a certain combination of “measures” may change. Hegel describes this change now as a move into “measureless”. Here it is the first, improper sense of the measureless that is meant: the new conditions do not follow the rules of the original system, thus, on the viewpoint of the original rules, the new system appears to be disorganized and against all proper rules.

2./655. In an abstract context, we can call all external or indifferent quantitative changes measureless: in nodal lines, we have constructed an example where such “measureless” change leads to a new situation, which is measureless or unruly in viewpoint of the original situation, but in another sense qualitatively specified – the viewpoints of quantitative and qualitative can be indefinitely altered, or one can move away from all viewpoints with a definite quality, but also from viewpoints with mere quantitative change. Qualitative states of infinity could be reached only with a sudden change of interpretation, while quantitative states were already implicitly quantitative states of infinity: in qualitative classifications finity and infinity exclude one another, while in quantitative classifications their difference is one of a degree. Here we can move from states of quality to states of quantity and back by interpreting classifications differently, and thus we merely construct an example of a measure: change of one quality into another can be interpreted as mere quantitative or external change, while mere quantitative change brings about in different context qualitative changes – we may interpret the whole process as going through aspects of one matter.

More general meaning of the measureless is that any mere quantitative change is measureless: if measures are structures where both quantitative classifications are regulated by qualitative classifications, then mere quantitative classifications without any regulation are in comparison measureless. Thus, when we interpret any nodal line as a mere quantitative classification, we are essentially making what is measured into what is “measureless”: e.g. the line of possible temperatures of water is measureless, if no mention of the physical states of water is made. Now, this sort of measureless is a mere abstraction when compared to the nodal line: changing the temperature of water involves more than this quantitative alteration. In comparison, the relatively measureless states mentioned in the previous paragraph construct this abstraction of measureless within the concrete nodal line: when we move to another “node” of the line, we notice that at least the previous rules regulating quantitative changes do not apply – the temperature is no longer one of liquid, or the original rules of an algebra do not apply. Yet, this new “measureless” is measureless

30 only in one sense, as it is also a new qualitative system with its own rules: e.g. the temperature is now attached to a gaseous object or vapour. We may thus always move from one “qualitative state of quantity” to a mere state of quantity – we may change the conditions of a quantitative structure so that it does not anymore follow the familiar rules that used to govern it, e.g. we can change the rules of an algebra of function – but this new state can then be interpreted as qualitative – we can see e.g. that a new algebra has its own rules regulating it. Because we can at least make a similar alteration into the reverse direction, returning to the original system, we may continue moving back and forth between the qualitative and quantitative interpretations for indefinitely long. If we replace the word “measureless” with the word “infinite”, we find that we have twice already faced a similar pattern of alternating states. Firstly, we have faced a qualitative classification of objects, where one object was called finite or imperfect, as it differed from the other object because of the difference of their qualities and thus failed to exist in the context or situation where only objects of the other quality existed. We could then find infinity by reinterpreting the different objects as mere aspects of an underlying unity: a further move where we compared the newfound infinity with some finite objects let us reinterpret the infinity as a mere finite object in another context. The peculiarity of this pair of possible alterations was the abruptness of the changes: nothing in the first place suggested that we could or should identify apparently different objects. Secondly, we faced a quantitative classification of objects as being greater and smaller. Here, the positions of the quantitative classification could always be mapped to one another by changing the reference point or unit of the whole classification: two meters was two hundred centimeters. Thus, we could make any quantity as small or large – as “infinite” – as we liked: on the other hand, for any fixed quantity, we could always find smaller or larger quantity. The difference to the previous discovery of infinity was that here the discovery of infinity was more natural and implicit in the arbitrariness of the positions of quantitative classifications. The structure here is a combination of earlier structures – we have a qualitative and quantitative classification intertwined to one another – and the indefinite repetition does not happen between stages of these classification, but between these different classifications or interpretations. We may start from a strict qualitative classification: this algebra is defined by these rules, that by other rules. Because the change from one quality to another can be described in quantitative terms, we may interpret the whole classification as being actually quantitative – we may return to the abstract sense of measureless. Finally, we may then once again reintroduce qualitative classifications, because the changes do appear sudden. Because of this possibility of moving from one sort of classification to another, we may interpret both as mere interpretations of the same, underlying system: and because this possibility is shown by moving from one position in the classification to another, we may interpret these positions as mere states of one, underlying object

31 or “matter”.

3./656. (a) We have discovered a substrate that is common for a number of situations or structures: this substrate is already present in quantitative structures, where a position of a situation is arbitrary or determined only relatively, and in measure structures, where one and the same situation has both qualitative and quantitative meaning. (b) The existence of substrate is constructed, when we see that one measure system creates apparently different measure systems that in another sense have only a quantitative difference. (c) When we repeat the process of constructing new measures indefinitely, we construct an example of qualitative differences emerging from quantitative repetition, but also an example of interpreting qualitative differences as merely quantitative: quantitative and qualitative interpretation can be replaced by one another. The meaning of the process is the revealing of the substrate.

Hegel begins by pointing out that the structure of one underlying matter behind many different aspects is actually one that we have met before: indeed, when we first met infinity, we saw how apparently different objects were idealized or interpreted as aspects of one unity – like now, the justification for this interpretation was that we could transform the objects to one another and that there was something similar in them. The first example Hegel mentions is a position in a quantitative classification: in quantitative classification the interpretation of the positions as aspects of a unity is more natural, because these positions or quanta are determined or differentiated only in reference to an arbitrary unit and thus their difference is only relative. Beyond mere quantitative structures, all measures are already examples of idealized differences: a position in a measure structure is in one sense part of a qualitative classification, in another sense part of a quantitative classification. Any measure is thus an example of “being infinity”: it is merely a given structure that shares the properties with any unity behind differences. What is still required is that we should posit these properties, that is, we should first create a more informative situation which seemingly doesn’t have these characteristics – which apparently has different situations or objects with no unity behind them – and then to interpret these differences once again as aspects of one unity. These two tasks have been fulfilled. We have begun from some arbitrary measure – or actually, from a system or algebra of measures (point a) – and we have multiplied this algebra by creating algebras with different rules (point b). Because these algebras share some similarities – they are all algebras – and because we have an ability to change them into one another – here this ability or method can be even stated in a quantitative manner – we may interpret these different algebras or measures as aspects of one unity (point c). Indeed, the whole classification of measures we have constructed is itself a measure structure: it can be interpreted both as a qualitative and as a quantitative classification.

32 4./657. In a system of measures we have independent objects determined by functions and rules for their combinations: now all of them can be interpreted as aspects or states of one substrate – apparent change from one function to another is then only a change of a state of something remaining same.

All the different measure algebras with different preference rules, but same group of measures can be interpreted as modifications of one algebra, because we have a quantitative method of changing one algebra into another by altering the preference rules of the algebras. Similarly, we have, within any one algebra, quantitatively describable methods for changing different measures – functions or chemical substances – into one another by different combinations. Thus, we could also interpret these different “substances” or “independent objects” as mere aspects or states of one “substrate”. The result of all these reinterpretations is a worldview where there is only one substance or object appearing in different conditions with different characteristics. The difference to the first state of infinity is that now we can express the change from one modification of the primary object to another in quantitative or algebraic terms: the methods of changing the state of the object are more definite.

5./658. At first measures were situations which had both quantitative and qualitative meaning: they were functions between different classifications. We could construct more complex measures as functions between different measures: the result was algebra of independent measures where every measure was connected with a series of possible combinations. Although such series were an external manner of classification, we could further specify the measures by giving them preference for certain combinations: the preference rules were not yet interpreted as created by some method of construction, but only as possible quantitative modifications of some “substrate”. In comparison with the substrate, the measures and measure algebras seemed like mere states: measure has become a mere aspect of larger structure.

Hegel ends the chapter with recounting the story of measures thus far: note that in this manner he separates the final chapter of the division of measures as an independent study, as it truly seems to be. We began from the most abstract sort of measure, where we had a classification that could be interpreted both as qualitative and as quantitative: generally, we began from any function combining number systems with different qualitative meanings. Hegel leaps over few steps where new sorts of measures were constructed and continues from a point where a whole system of measures has been discovered where the measures have been taken as objects: in such a system we could e.g. combine measures in order to gain new measures: e.g. we could combine functions or chemical substances. These “measure algebras” could then be modified by changing the rules by which certain combinations were preferred. These different algebras could then be ordered quantitatively and finally interpreted as mere aspects of one unity: while we have made measures complex, we have also relegated their status into mere moments of a larger complex. Hegel makes

33 an interesting remark that at this stage the substrate is not “a free concept” or a self-determining process that would specify its states. The substrate is just something common to all its aspects – thus, we know that it is possible to get from these aspects to the unity behind them – but there is still no reason to suppose that there would be a method of constructing the different aspects if the mere substrate was given to us: aspects or states of the substrate appear to be indifferent for the substrate. Thus, that the substrate exists in this particular state is in no way explained by the existence of the substrate: the existence of the substrate does not even justify the possibility of the substrate existing in this state – this already points to the transition that Hegel attempts at the next chapter.

If measure algebras can be transformed to one another, they can be interpreted as mere states or aspects of a unity.

Glossary: Reaale Maaß = measure is realized, when it has simpler measures as its constituents Verbindung = (in case of measures) producing a new measure out of several measures through some procedure Neutralisirung = ”neutralization”; in a measure combined out of several measures the characteristic qualities of these constituents have been modified by one another

34 Verwandtschaft = ”affinity”; relation of two measures that can be combined with one another Wahlverwandtschaft = ”elective affinity”; when a measure is more likely (in itself or according to the observer) to be combined to one than to another measure Knotenlinie = ”nodal line”; a structure in which the same objects or situations have been classified both quantitatively and qualitatively and thus quantitative change can cause also qualitative changes Maaßlose = ”measureless”; 1) when a certain object, situation or more complex structure characterized by some type of measure (e.g. an algebra of functions) is changed to another structure of the same sort, but characterized by a different measure, the result seems ”measureless” or ”unruly” from the viewpoint of the original structure; 2) such structures can be interpreted as mere aspects or states of one underlying substrate, which is thus characterized by no mere one measure and might thus be called ”measureless”