Commentary on Hegel\'s Logic 8: Specific quantities

1 Third division. Measure We have seen that any qualitative structuring of situations and things can be reduced to a quantitative structuring by interpreting all apparent differences as merely aspectual: different situations or things are actually mere moments in a greater whole. The result of such an interpretation was a quantity, and because it was possible to divide this quantity, many different quanta. Now, quanta produced by the same manner could switch their places: we could arbitrarily map one position in a number system to another and thus see the quanta as mere aspects of the number system or the relational framework to which they belonged. Some of such mappings occurred within a number system, like direct ratios, but others mapped quanta of one number system to quanta of another system, like indirect ratios, or in a simpler manner, relations of different powers. The end result was then the construction of a function from one quantitative system to another: a framework of both quantitative and qualitative elements or measure. The third division of this book has a bit problematic character. On the one hand, it is like the division on quantities, in being not as essential as the first division. The issue of the division is the structure of measure: not just a static state where situations and things are classified according to both qualitative and quantitative divisions, but also a method and a process by which to get from the qualitative division to quantitative and vice versa. In effect, this division merely continues the study of mathematical structures begun in the previous division, only in a further level of complexity. On the other hand, the division also serves as the transition to next book on essence: one aim of the division is to show how from any situation one could show some essence or general structure that it exemplifies. In line with this twofold character of the division, its threefold classification falls into two parts. The first two chapters construct evermore complex measures, while the third and thus the most important chapter shows how the transition to the essence is made.

1./569. Measure is a structure uniting qualitative and quantitative structures. State of being is any situation with a consistent characteristic, but from such a situation others can be constructed, and then it is not the only possibility anymore. In a state of quantity, the determined position in the larger framework is irrelevant and arbitrary to the state of being itself: such a state is determined only by its relation to other quantitative states. Third sort of situation is a selfcontained system of variable quantities, which as a system is related to other systems of quantities: it is in some sense quantitative, in another qualitative.

As Hegel himself notes, the idea of the third moment as a combination of the two previous moments is still rather abstract and meaningless: such an explanation does not yet sufficiently describe the third moment in any Hegelian triad, whether the division is natural or artificial. Here, a measure is

2 not just something that is qualitative in one sense and quantitative in another, but a method for producing one type of structure from the other. The movement from the quality and quantity to measure is not an arbitrary combination of two structures in some manner, but is justified through seeing that we have the capacity to construct quantitative structures from qualitative and vice versa: the assumption of the possibility of the new structure type is then natural. Hegel begins the division with a summary of all that has happened thus far, in order to remind us why he has the right to assume the possible existence of a measure. He does not begin from qualitative structures – indeed, he apparently ignores them completely – and starts from an even more abstract phase: a state of being or any situation unrelated to other situations. Yet, he admits that this beginning is still determined, hence, it must already be a part of some qualitative structure: there must be other situations, either immediately given or made through construction. Such an immediately determined situation can be sublated, that is, it can be interpreted as a mere aspect of the framework. The actual situation or position we happen to take in this framework is then arbitrary: the formerly qualitative framework is then interpreted as quantitative. In a quantitative framework the situations still have determinations, but only in relation to some arbitrary reference point: two meters is large compared to two centimeters, but small compared to a kilometer. The value of the situations is then not constant, but constant is only the relational framework or quantitative system in which situations are arranged. This relational framework forms a situation of its own or a sort of superstructure: compared to the quantitative situations within it, it is independent, qualitative situation. As we saw, we could construct other such relational frameworks beyond this one: meters can be compared to square meters. Every position in a relational framework is then in one sense or in one context quantitative – two meters compared to one meter – but in another sense qualitative – two meters compared to a square meter.

2./570. In Kant quality and quantity are followed, after relation, by modality: this includes categories which add nothing objective to the cognition of thing, but describe its relation to cognition – the Kantian categories of modality will be studied later. Kant has not applied form of triad to the genera of categories.

It seems puzzling why Hegel would want to connect measures with what Kant calls modalities, as there seems to be not much in common with the two issues. Even the category group of relations appears to be closer to the Hegelian measure: at least measure is some sort of relation, although the general notion of relation has undoubtedly been introduced earlier. Yet, there is a reason for Hegel’s statement. Modality for Kant refers to the manner how a thing is related to cognition: possibility, for example, means just possibility for us, that is, compatibility with the presuppositions of human

3 cognition. Kant’s idea of modalities is essentially connected with the transcendental nature of his philosophy, which Hegel was not very impressed of: even if Kant’s theory did not mean it, it still suggested a crude pseudo-psychological explanation of the birth of experience, which demanded a reference to something beyond all possible experience. What is then left of modalities when the connection to the cognition has been dropped is an idea of a framework of different possible situations, some of which are actualized at the moment, while others are not: as we shall see, in Hegelian context these possibilities must be such that can be constructed from one another. Now, the structure of measure consists just of such a framework of situations, the only difference being that in the measure the connections between situations or possibilities are more determinate: it is not just a question of some indefinite possibility, but of a possibility, the structure of which can be definitely calculated through the quantities within this situation – somewhat like the position of a falling stone at the next moment can be calculated from the position and speed it now has.

3./571. The modus of Spinoza is the third term after substance and attribute and refers to a mere affection of a substance or to a manner how the substance is in some context: the mode is a mere arbitrary position, instead of being independent in any sense.

It becomes evident why Hegel began his summary in the beginning of this section from the state of being, which corresponds better with the Spinozian substance: like a mere state of being or situation in itself is not related to anything else, so there can be only one substance for Spinoza. Now, Hegel proceeds to construct or discover situations to which the original situation could be related, while Spinoza merely presents the same substance from different aspects or develops different attributes of one substance. Still, as Hegel proceeds to argue that the situations in the framework of possibilities differ merely in a quantitative fashion – that is, their position depends only on the arbitrary choice of the reference point – the differences between the two accounts disappear. It is the third moment in this series where the true distinction lies. Spinoza continues merely by finding even more external standpoints in which the substance can be conceived: these are the modi of the substance. Hegel, on the other hand, has just shown how the unitary framework of quantitative situations can be differentiated in a qualitative manner from another framework. Thus Hegel succeeds in finding some (relatively) independent entities (the number systems) which still are related to other entities: while for Spinoza the individuals can be nothing more than mere modifications of one unity, Hegel has shown how in any context we can construct individuals distinct from each other. Although the individuals could then be reduced to a further unity, the possibility of producing evermore qualitative differences would still be a reality.

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4./572. This lack can be seen in all forms of pantheism: the essential point is the first moment, while the second moment introduces finity, although understood as a facet of the original unity, while the third moment is a mere downfall from the essence. Thus, in Hindu trinity, Brahma is the unity, which proceeds to concrete world in Vishnu and to mere destruction in Shiva: individuality does not emerge as truly independent, and its highest goal is submerging into Brahma.

Hegel was undoubtedly well-informed of the Indian culture according to the standards of his age, but this is still not very much: of the wealth of the different varieties and modifications that Indian religious life has Hegel knows only a few general features connected with certain abstractions Brahmins had made of the diverse facets of what is called Hinduism. Thus, Hegel’s account of pantheism is mostly dedicated to Spinoza’s philosophy and especially to the threefold division of substance, attributes, and modi: another unmentioned source is undoubtedly Neo-Platonism. What Hegel sees as a common feature in all pantheism is the disregard of individuality. The essential point is the unity or divinity in which all things share: the standpoint according to which everything is one (note that Hegel still confuses Brahma, the creator god, and Brahman, the unity behind all). After this perfect beginning, rest of the development can only be downhill. Spinoza’s attributes are one way of degrading the original substance: they are supposed to be the substance, when it is regarded from some limited standpoint, such as extension or thinking. One step lower yet is the world itself, regarded not as a unity, but as a sum total of everything: in Spinoza’s terms, this would be the infinite mode, which Hegel ignores or is not aware of, and in the Hindu trinity Vishnu the preserver might play this role. The final phase of this emanation is then the total disappearance of all essence into nonbeing: Spinoza’s infinite mode can be divided into finite modes, partial aspects of the world and God, while the Hindu trinity ends with Shiva the destroyer. The only way how an individual human being could gain some worth is to reverse this flow of decadence, either through viewing everything as an aspect of God, as in Spinoza, or more concretely, by nullifying the individual existence and so returning to the original unity, as in Neo-Platonism and Hinduism. The comparison with the Christian trinity presupposes Hegel’s unorthodox reading of Christianity, dedicated to medieval mysticism. There we may still speak of God as emanating into the world, but the God also becomes incarnate or takes a human form: in Hegelian interpretation, the human behaviour, at its best, can be seen as holy. The final moment of this newly interpreted Christian trinity is then the phase of spirit or the ideal community, which the human beings should strive to fulfil on this world: human being does not become holy through purification, but through action.

5 5./573. Although mode or manner of a situation or thing seems unessential, it may be crucial or essential in some cases: arbitrariness depends merely on context.

The pantheistic degradation of individuality is based on the presumption that individuality is inessential in some absolute sense: whatever the context, being of an individual is insignificant. Hegel points out that what is significant depends on the choice of the context: compared to the vastness of the universe a human being is a mere speck of sand, but in her own environment she is a valuable member of society, good friend etc. A pantheist chooses to look everything from the viewpoint of the whole world – whatever that is – and in that context all else loses its significance. Hegel, on the other hand, wants to make a Christian choice and concentrates on the human life and thus raises the individuality far above mere matter – every single soul is worth more than the whole night sky full of stars.

6./574. At this stage, the individual situations are part of a structure of measure: while pantheistic unity has no measure, even Greeks knew that everything has a measure or is organized by necessity.

It is not at first obvious what this talk of pantheism has to do with the structure of measure. The answer is revealed in this paragraph, where the pantheistical worldview is explicitly compared with measures. For pantheism the greatest truth is the proclamation of the unity of everything: there is only one structure and all else is mere aspect of it. Now, as Hegel points out, this is not very informative: what we would like to know is what further structure this unity has. Pantheistic substance has no essential structure, because all determinate structures are mere inessential modifications of the original unity. Hegel admits that such an indeterminate unity is a possibility, but notes also that we now have the capability of constructing other situations and things and relating the original indeterminate situation to them through some partially quantitative, partially qualitative relations: in effect, we have the ability to construct examples of what Hegel calls measures. The paradigmatic example Hegel’s method lets us construct is still quite abstract structure of ways how different possible number systems might be combined. Still, the construction of this paradigmatic example makes it meaningful to ask whether similar complex structures could be discovered in concrete world, that is, whether the concrete situations and things are related in a similar manner: even in experience we cannot stop with mere accepting of what has been given, but we must try to find in it some general relations of quantitative dependency.

7./575. When a structure of measure is developed further, it starts to resemble a state of necessity between situations or things: if a quantitative change of a thing exceeds the limits of measure, the value of the thing is suddenly diminished. It

6 is still pantheistic to designate the structure of measure as the ultimate subject matter of all discussion, but at least it is a far more informative subject matter than mere state of being: a measured state can in one sense be arbitrarily changed, but in another sense it is a characteristic of something in every situation – it is the most complex and informative structure if we stay in the realm of mere immediate situations.

Although the paragraph is apparently a series of unconnected statements, we can discern a certain stream of thought in it. The previous paragraph suggested the idea of the measure as equivalent to the Greek goddess Nemesis, or less poetically, with the structure of necessity. What Hegel is pointing out is the characteristic of the structure of measure to be a sort of function or construction from one situation to another: as we shall see, Hegelian necessity takes the form of a function “if in this situation matters are like this, then there is or there can be constructed another situation with certain other characteristics”. Similarly here, a certain change in a situation or thing according to its quantitative aspect may imply a change in its qualitative aspect also: this property of the structure of measure will become more evident later. Now, if the structure of measure would be the ultimate structure in the sense that there could be no structure with further properties – if it were the “predicate of God or absolute” – then all situations would be at best connected with an iron necessity of dependence and causal relations: in other words, freedom of an individual would be impossible. Thus, Hegel intervenes the flow of his thought by noting that such a state of affairs would still be pantheistic in the sense of having no place for the worth of individuality. Yet, as we saw earlier and as Hegel repeats here, the structure of measure is at least far more informative than a mere indeterminate and unrelated state of being could be. This final note serves then as a connection to the final point of the paragraph. Although the structure of measure already implicitly contains the idea that we can construct one context from another – that is, the qualitative aspect of a thing or situation from its quantitative aspect – this idea is still left implicit at this stage of Logic. Instead, we merely focus on the individual situations or states of being that have been arranged in this framework of measure: they are measured in the sense that they have been assigned a position in a quantitative series, which in one hand can be arbitrarily changed within certain limits, but which on other hand would change its place in a qualitative series if the quantitative change were repeated beyond the limits. This is the most complex framework for mere situations, Hegel maintains, but in the course of this division he tries to show that we have the ability to construct something beyond mere framework of possible situations.

7 8./576. The structure of essence can be discerned in the structure of measure: from an identical structure we can construct a series of multiple appearances in different situations from which this underlying structure can then be constructed anew. The appearances of the essence are independent in one sense, but in another sense they are mere aspects of the whole: similarly, a measured state can have many quantitative positions, which in another sense are mere inessential modifications of its essential position in a qualitative series. But the state of measure has not yet been constructed as an example of essence: the qualitative and quantitative positions of a measured situation are merely given as united in this situation. Our task is to first construct from this first stage a more complex structure where the qualitative and quantitative series become apparently independent and then to construct a connection between these series: thus, we show that a structure of measure can be applied in a more complex structure where it shows only one aspect of the whole.

The complexity of the text increases radically, when the style changes into Hegel’s official jargon. The content of the text is actually quite simple: measure resembles essence, but only in itself or according to its concept and thus it needs to be constructed as essence. A proper Hegel-scholar could merely on this basis invent the most extraordinary tale of what Hegel wants to say, yet, there are many questions that still need answering. What is the structure of essence? How does structure of measure resemble it? Why is measure in itself essence? How measure is then constructed as essence? Starting with the first question, we undoubtedly cannot yet give a proper answer, but at least certain inkling of what Hegel is saying can be discerned. Essence should be a self-identity in the immediacy of the state of being-determined. This seems actually rather simple: it is the old tale of how one thing can have many determinations in different situations or contexts – it is the same thing, although at every moment it is given (immediate) as being determined in a different manner. The problem is how this description differs from all previous forms of “identity within difference” or one thing having different characteristics in different positions. The keyword is mediation. Hegel suggests that in the structure of essence the many appearances or aspects of the thing or generally of the underlying basis are viewed as having been constructed from this basis: it is like a life of a human being, where the stages in life are not just aspects of the person, but also in a sense caused or explained by the person. Similarly, the underlying identity can be looked as something constructed from the different aspects: a person is just the stages of her life taken as a unity. Which of the two viewpoints is then the correct one, that is, is the true basis the identity or its aspects? Neither, Hegel answers, but the whole framework of possible constructions of difference from identity and identity from difference. Now, when Hegel moves on to discover the structure of essence within measure, his starting point is not the whole framework of situations or the structure of measure, but a single situation or

8 position within the framework: not e.g. the whole system of chemical substances, but a single chemical substance. Such a situation is now characterized by one quality – say, being a substance in a certain state, like liquidness – and many possible quantitative appearances – for instance, a liquid can be liquid, although its temperature would vary. Here the qualitative nature of the situation or thing is its essential identity, while the quantitative variability corresponds with the possible different aspects or appearances of this basis. Just like in the structure of essence, the essential in this structure is the interconnection between the aspects of identity and difference: one can always construct a series of several quantitative appearances for the same quality – e.g. different temperatures in which a stuff stays liquid – and one can also interpret the different quantitative appearances as mere facets of the same quality. The division of measure starts from a position where the qualitative and quantitative are, as Hegel says, united immediately. This means nothing more difficult than that for one quality there corresponds in this first structure exactly one quantity: one could think of an atomic weight of a basic element as an example of such measure. In this sort of structure, the interconnection of qualitative and quantitative aspects and hence also the structure of essence is obviously present. Yet, this is somewhat trivial example, as the quantitative and qualitative aspects of the situation or thing seem to be distinguished only by different interpretation: the exactly same division of situations or objects is interpreted first as quantitative and then as qualitative. Hegel proposes then that we should at first develop more complex forms of measure, where the quantitative and qualitative aspects would be more independent and only after that construct them as an example of essence – something similar was done at the end of the previous division, when we had to construct more complex examples of quantitative relations.

9./577. (1) At first we are given a structure in which a division of objects can be interpreted both as qualitative and as quantitative: a quantum can be interpreted as a qualitative determination. We should first construct a more complex structure where the quantitative and qualitative aspects become independent.

The preceding interpretation of the need for constructing measure is justified in the very next paragraph where Hegel describes the specific nature of the simplest structure of measure. There we have one quantum that at the same time has in another context a qualitative meaning: that is, it is one sense a position in a quantitative series and in another sense a position in a qualitative series, where the two series are actually the same series under different interpretations – where every quantity corresponds with a different quality and every quality corresponds with a single quantity. The first task of the division is then to distinguish these two moments, i.e. create or discover a

9 structure where the qualitative division is not exactly identical with the quantitative division: where one quality corresponds with many quantities.

10./578. (2) From distinct aspects of measure can be constructed a framework of separate measures: as these measures can be quantitatively mapped to one another, we can interpret them as states of a measureless.

The further task of this determination is not just to make the quantitative and qualitative aspects of the structure of measure more independent, but also to interpret these aspects as separate measures, or “whole measures”, as Hegel says: in effect, given one relation or function between number systems, we construct more of them. What we have then is a framework of measures, and the next task is to show that this framework can also be given a structure of measure: thus, we have ended up with a set of relations or functions between different number systems such that themselves form a number system. This complexification of the structures involved may seem rather mystifying and intended merely to model certain chemical structures – chemical matters are characterized by relation of their volume and weight or specific gravity, and different specific gravities form a number system – but it is actually familiar from mathematics also: e.g. functions are not just operations in some calculus, because functions of some calculus can be interpreted as being operated with some ”higher-order” calculations. The final construction in this chapter is the return to a unity: when these apparently independent measures have been embedded in a number system of their own, it is possible to introduce mappings of one measure to another, and hence, we can view the different measures as mere states of underlying unity. As this unity cannot be characterized by any single state or measure, we can say that it has no measure, that is, it is not bound to any single measure.

11./579. (3) The measureless is still only an abstraction out of the concrete measures, if they are indifferent to one another: when we construct them as having an indirect ratio to one another, we can interpret them as being constructed appearances of an underlying essence.

The third chapter is not just an end to the division of measure, but to the whole book on being: its starting point is actually a structure that has occurred in different guises in many occasions, namely, that of one underlying identity being present in many different situations and contexts with different characteristics. Until now, it has actually been contestable whether the interpretation of seeing different objects and situations as mere aspects was the more natural one: in most cases it could have been argued that the objects in question were actually independent and only in some abstract context identical with one another – like Aristotelian prime matter could be said to be a mere

10 abstraction out of different concrete matters. The task of this chapter is to show that this construction of identity can also be made from the viewpoint of an independent object or situation. A similar need for a double transition was shown in the case of constructing unity from multiplicity. There the second transition or construction of unity according to the viewpoint of an individual object applied the fact that the repulsion or exclusion of other objects was supposed to be an actual construction according to an object: we could see the other objects as mere background for this one object. Here a similar construction occurs. The structure of measure is not just a relational framework between different objects and situations, but an actual method of constructing one object or situation type from another: a method of changing one sort of “matter” to another. Thus, we can literally change and example of one sort to an example of another sort, thus truly creating a unity out of apparently different objects or situations – the result being that it is possible and natural to interpret the different situations as mere aspects or appearances constructed by an underlying essence.

12./580. This division should develop the general structures that could be applied to quantitative relations between different phenomena of nature: the application of these structures belongs to real philosophy. The different abstract structures correspond to different levels of nature: in mechanics, the quantitative framework can be very complex, because there are only few qualities involved, while in physical and organic spheres the number of qualities simplifies the quantitative framework. Furthermore, the relationship of different number systems becomes subordinate to more complex structures: e.g. the quantitative relations of organs are simple and the movement of animals is governed by simple mechanical rules.

The usual judgment on the division on measure is that here Hegel at least implicitly uses empirical material in his transitions: this has been seen either as a lack of Hegel’s presentation or then as a proof that even Logic requires experience. Most of these interpretations undoubtedly think Hegel is trying to argue something, whereas we have seen that Hegel is merely constructing examples of certain structures. As we shall see, nothing more is required for these constructions in this division but capability of constructing certain mathematical structures, such as different number systems: thus, no recourse to any more concrete experience is required. Another issue is that Hegel desires that the structures constructed could be applied to experience and especially natural phenomena. In this application the experience must undoubtedly have the upper hand: we cannot know beforehand whether this structure applies to that phenomenon, Hegel admits. Of course, we do know that the structure has some possible application – this is just the point of Logic, in which such applications or models are constructed – but these applications remain at very abstract level. Interesting is that Hegel still is willing to insist that our ability to

11 construct models is on the whole more important than the arbitrary fact whether they happen to correspond with a determinate concrete phenomenon: this attitude implies that theoretical search for “correctness” is for Hegel subordinate to our practical search for ways to manipulate things – the human practice with its viewpoint is far more important than any external standard of truth. It is against this backdrop of the preference for human standpoint that we have to evaluate Hegel’s often cryptic appearing statements on nature: usually we see that they accord with what one would be inclined to say of the state of the science of Hegel’s days, once we have at first transferred the standard from the human side to the side of the “world”. For instance, Hegel here says that mechanical phenomena involving mere space, time and weight can have a fairly complex quantitative structure, while the amount of qualities in more concrete cases admit only a fairly simple quantification. The point Hegel is making is that in the case of few qualities it is quite easy for us to make such complex quantification, while the introduction of more qualities makes it harder for us to discern any quantitative pattern in their relations: a statement on our capabilities is given a seemingly objective look through the change of viewpoint. The higher structures to which the quantitative models must be subordinated in the organic realm are obviously meant to be functional-teleological. Good or proper functioning of animal organism may undoubtedly require or at least be benefitted of some quantitative proportionality between its parts: furthermore, the best means are usually the simplest, and thus the proportions of a well-functioning organism are probably of a simple variety. While in the case of the structure of organism we can speak of no awareness of a desire or want of some goal, in the case of animal movement this may accepted: the bird wants the worm and therefore flies to catch it. Here, the animal does not desire movement for its own sake, but uses it as means: it tries to get from one place to another in an optimal time. Thus, the animal movement does not have any necessary complex pattern, but at most is regulated by simple laws of mechanics.

13./581. Spiritual structures can even less be described quantitatively. The size of a state might have some connection with the best constitution and the optimal relations of different classes, but no laws of this connection can be stated. In psychology, we may speak of strength of character etc., but no proper quantification can be given.

The inappropriateness of quantitative description in human affairs is already a familiar proposition of Hegel’s. In matters we could nowadays call sociological, the question is not so much of impossibility of finding something to quantify, but more in the meagerness of the results of such quantification. As in the examples from the organic level, the laws required would be connected with normative matters: they would indicate what sort of sociological structure would be most

12 suitable for a given situation. Hegel mentions the old idea that the size of the country is somehow connected with the constitution it should have: only small countries could be democracies etc. Such rules of thumb are very context-dependent – we cannot e.g. determine any exact size in which a state becomes too large for Athenian democracy. Similar “lack” Hegel would undoubtedly find in the modern sociology, which at its best describes some uniformities of a given society, but cannot state any general or necessary laws – at least of quantified variety, although it may give general qualitative models. The field of psychology is even less capable of providing us with quantified laws. Here the problem is, according to Hegel, that even the quantification seems difficult. In addition to the dubious example of intelligence, there seems to be no reliable way to quantify faculties and characteristics of human mind. How would one decide e.g. how imaginative a person was?

First chapter. Specific quantity.

The purpose of the chapter is to develop more and more complex measures in which the aspects of qualitativeness and quantitativeness become ever more independent. This otherwise rather tedious affair is broken up with remarks on how to apply these complex structures to physical issues: in fact, it has often been thought that here Hegel’s thinking requires a reference to empirical. We have seen that this need is overstated: Hegel’s constructions rely mostly on the possibility of constructing certain mathematical structures, namely, frameworks of related powers. Still, this is one section where Hegel’s work would have benefitted from the later developments of a science – not physics, but mathematics: certainly it is possible to construct much more interesting examples of related number systems than mere functions between variables of different powers. The division of this section seems somewhat artificial, although all the levels are sufficiently independent and equally essential: we face first a classification which is both quantitative and qualitative, then a relation of a merely quantitative variable to a quantitative variable with its own quality and finally a relation of differently qualified variables or number systems. The air of artificiality arises more from the fact that the chapter itself seems somewhat truncated when compared with all the potential issues it could tackle.

1./582. (A) In first stage of investigation we have a quantity that is given as corresponding exclusively with a specific quality.

We start with a quantity that should also have a qualitative significance. Hegel makes now only few

13 specifications for this rather meagre description. Firstly, the quantity in question should not be unrelated, but quantum – that is, part of some number system or quantitative classification, no matter how imperfect or open-ended. Secondly, the quantity should be specific. This says almost nothing else but that it should also be part of a qualitative classification: it should define some species of objects. Yet, the immediate connection of the quantum with this specificity or qualitativeness tells us something more. Quantum should not just have a quality – like two meters contains a qualitative aspect when related to areas – but it should be a quality: what is in one sense a quantitative classification should be in another sense qualitative classification.

2./583. (B) We can relate such a qualitative number system to another number system: we have then a regulating function between a qualitative number system and a mere number system – which has thus been integrated into a structure with qualitative elements.

In the next phase we should construct an example of structure where the quantitative and qualitative aspects latent in the first phase are clearly set out as independent: that is, we need an example of a relation between purely quantitative number system and a number system that has also the meaning of a qualitative classification. Example of such a structure has actually occurred already in the previous chapter: we only have to look at any number system related to the series of different powers – a relation between a variable and its potencies. Thus, for instance, the variability of a straight line is essentially connected with the variability of squares and cubes. Here the “merely indifferent” quantum is sublated or specified: that is, the number system is given an almost qualitative meaning through the functions that regulate the quantities of different qualities.

3./584. (C) We may further give both systems a qualitative meaning: the regulating function is then a relation between differently qualified number systems – a measure connecting separate measures.

Final phase of this chapter begins when we make the previous structure even more complex by letting both related number systems have a qualitative meaning. This construction can happen at least in two manners. Firstly, we can interpret the number system that was supposed to merely quantitative as being also qualitative: for instance, we may suppose that lines are distinct qualities in comparison with squares or cubes. Secondly, we may also relate to one another different qualitative number systems that were formerly related to the mere number system: e.g. we could compare squares and cubes. Now, the function relating such number systems is obviously a function between different qualitative number systems, that is, a sort of measure. Then again, the two number systems involved have also both quantitative and qualitative aspects: thus, they too are measures, albeit

14 simpler than the function connecting them. Such a “measure of measures” is the somewhat artificial ending point of this chapter and topic of a new chapter.

A. Specific quantum

We begin the chapter on what Hegel calls specific quantum, and the name already reveals what sort of structure we are dealing with: it is a situation or position in a classification that can be interpreted both quantitatively and qualitatively – specifically. We have already mentioned one possible example of such a structure, namely, that of atomic weight. On the other hand, this is also an example of a far more complex structure: atomic weight is essentially a relation between two measures, that of the weight of matter and that of the number of its atoms. A simpler example is provided by dimensions: every dimension has a quantitative meaning and still is represented by a single quantum. This example is good also because it could be easily constructed from the final result of the previous chapter, i.e. the relation of variables or number systems with different powers. The task of this section would be to construct an example of a structure where merely quantitative number system would be related to a number system with a qualitative meaning: here Hegel uses the already familiar construction of ignoring the aspectual nature of differences within some structure. The division of the section deviates from the ordinary: the first sub-section already analyses the structure of specific quantum, while the second subsection investigates the special problematic of apparent paradoxes that appear when quantitative and qualitative structures are connected.

1./585. 1. Introduction and analysis of specific quanta. In a measure structure, a quantum has a qualitative meaning: in the simplest stage of this structure, quantum in one sense has some definite numeric value and in another sense is a part of qualitative classification. The same situation can be seen either quantitatively or qualitatively, and neither description differs from the other.

In the previous chapter we had managed to construct a structure where differently qualified number systems were related to one another: this structure provided us an example of a structure Hegel calls measure, that is, a structure connecting both quantitative and qualitative aspects. The new chapter does not begin with this particular example of measure, but as it is Hegel’s manner, from the simplest structure exemplifying measure. We do not have many number systems qualitatively related to one another, but only one number system or quantitative classification of objects that at the same time can be interpreted qualitatively: for instance, the dimensionality 2 describes a specific

15 group of geometric objects, that of areas, which are distinct from both lines and solids.

2./586. In this sort of structure everything is measured, that is, every situation and thing has a quantitative determination that is an essential characteristic of the situation or thing: change of quantity changes the quality or even destroys the situation or object.

The statement “everything has a measure” has two meanings. Firstly, it may describe the current stage of Logic: everything in the sort of structure we are now investigating has measure, that is, quantity that has qualitative significance. In this manner Hegel does not state that everything in general would have measure, just that some things might do. Secondly, Hegel appears to mean this statement also in a more general way. Yet, he is not saying that everything has measure in all situations. Instead, we have become aware of a method by which we can construct for any object a context where it has measure: that is, we can e.g. construct some qualitative classification for the object and then quantify this classification. Hegel also mentions the fact that in structure of measure the change of quantity of a thing can lead to a change of its quality also: this particular issue is investigated more closely in the second subsection and then again in the next chapter. Here we may note that in this particular structure the change of quality is always attached to the change of quantity, because the classification of qualities and quantities is exactly same and differs only in interpretation. Thus, if we changed a line into two-dimensional, the result wouldn’t be a larger line, but something that was not line at all: similarly, change of atomic weight of matter leads to change of the matter itself.

3./587. Usually measure is understood to mean an absolute unit to which all amounts can be related, but although such unit may be a specific quantity of some natural object, it is external to measure other objects by it: there is no natural system of measure, but only conventional.

The subject matter of this paragraph was undoubtedly a heated issue at the time when the French revolutionaries had among other things changed the old systems of measures to a more rational system, supposedly on some natural units of measure. The unit of length or meter – word originating indeed from the Greek word for measure – was defined alternatively either as a certain part of Earth’s meridian or as the length of pendulum having a period of certain length of time. Although both definitions are connected with certain natural quanta, they are still somewhat artificial: in the first definition it is problematic why just this part of the meridian was chosen, while the second suffers from the fact that the time of the period of a pendulum varies in different places on Earth due to differences in gravity. Generally, Hegel notes, while our measuring unit may be

16 natural quantum of some species of objects, there is still no necessary reason why we should choose just this quantum to measure other species of objects: this criticism is especially strong on the Hegelian assumptions that there is no necessary upper or lower limit for possible quanta of natural entities. Hence, the choice of a measuring unit can be only arbitrary and conventional. Indeed, we should have some common measuring unit in order to facilitate trade etc., but the measuring unit is more likely to be picked out because of its practicality: meter is a good unit for measuring the socalled “middle-sized” objects, while it is too large for atomic distances and too small for astronomical distances.

4./588. Some concrete objects, like organic beings, have specific quanta, while all of them must have some quantum in order to exist among other objects: because of this quantum, they can change, but because this quantum is essential to their existence among other objects, its change can affect their relation to other objects.

The previous paragraph already approached the question where to apply the structure of specific quanta or “natural measures” – or more correctly, showed that it couldn’t be applied to measuring – and the same trend continues in this paragraph. Hegel does not pick a very good example of specific quanta. Of course, in one sense foot is always a foot long: yet, in another sense organisms usually do not have a fixed size for feet, legs etc. nor is there even a fixed quantitative relation between different members of an organism, but the size of the members of an organism belonging to a species admits of a slight variability. Still, in a sufficiently inaccurate level of investigation we might say that these objects fit the structure of specific quanta. Hegel is also in quite a hurry to move on to the general structure of measure and its applications. He seems to repeat the statement that everything has a measure: actually he says in more detail that all existing things – presumably the concrete spatio-temporal objects – have measure-like structures. Firstly, they all must have quanta in order even to “be there”, that is, to exist in a position of some classification of objects: in a classification of spatio-temporal objects we can obviously measure e.g. the size of objects or species of objects by comparing them to some reference point. Secondly, the quantity of these concrete objects is not arbitrary, but regulated within some limits: e.g. a tree couldn’t grow too large, because it couldn’t bring water to its uppermost leaves quickly enough.

5./589. When the quantity of a measured object is changed, its quality could suddenly change: explaining the change through the gradual nature of the quantitative change does not explain the apparently problematic qualitative change.

The consideration of the general structure of measure continues in this paragraph by an introduction

17 to the issue of the next subsection or the apparent paradox of measure. This paradox arises from the fact that measure is a connection of two structures, one quantitative and another qualitative. The quantitative change seems inessential, because mere quantities are positions having a definite value only in relation to some arbitrary reference point or conventional unit. Still, as all quantitative values are functionally related in this structure to some definite qualities, the quantitative change may affect some qualitative change also: and this change may seem sudden if we are not consciously out to cause such a change. Hegel is against any explanations that would reduce the qualitative change to gradual, quantitative variations: such an interpretation would actually change the context by eradicating the whole qualitative classification. In fact, Hegel is endorsing the idea that “sudden”, qualitative changes and even generations of completely new objects are possible. The favourite interpretation of some scholars that Hegel would here state the possible vagueness of conceptual classifications finds little support in Hegel’s actual words: on the contrary, the whole “paradox” of quantitative and qualitative change is based on the possibility of very strict qualitative classifications.

6./590. 2. Paradox of quality changing with quantity. These paradoxes were noticed also by ancient philosophers: they ridiculed people by asking whether e.g. a loss of hair could make one bold and then noted that a sum of such losses would make it so. 7./591. Such examples are not mere jokes, but show that one shouldn’t abstract the quantitative classification from its qualitative connections.

Hegel had an enthusiasm for paradoxes invented by such ancient philosophers like Zeno Eleatic and Gorgias and he believed that the paradoxes usually showed that conceptualization of some topic was inadequate: paradoxes arose from a restricted grasp of situations. Thus, the apparent paradox of measures was discovered by the so-called Megaran school and presented in Aristotle’s Sophistic elenchi as one form of erroneous argument: nothing of consequence should happen with a loss of one hair or an addition of one grain, but when these small additions are multiplied enough, all hairs are gone or the grains make a heap – or as in the moralistic version Hegel presents (perhaps in order to show what worth such paradoxes are), small financial losses add up to a bankruptcy. These seeming paradoxes teach us, according to Hegel, that the mere quantitative conceptualization of certain changes is inadequate: the loss of hair involves also the qualitative classification to hairy and hairless. Hence, everything is not reducible to mere quantities, or at least such a reduction loses some information

8./592. Some situations can be cunningly changed through their quantitative aspect, although e.g. the change may at

18 first seem positive.

Hegel refers to the strange idea of the cunning of reason or concept that we shall meet later in a more substantial context. The word “cunning” may raise the idea that Hegel supposes reason or concept to be some supra-individual or even supernatural entity that uses its wits to affect the history of the mankind: Hegel’s example of the fall of states just enforces that image. As we shall see in a more detail later, the concept to which Hegel refers to here is a general term for all methods and strategies that could be used in a goal-seeking activity. Cunning of concept would then merely refer to the cunningness of some method – such and such an activity made some goal come about in a very effective manner: note that the event in question might also be only such that would resemble purposeful action, as long as it would have been called “cunning” if it were purposeful. Thus, if one wanted to reform a state, it would be cunning if one would let it increase and then reform naturally. Not so outlandish example would be the feeding of an animal for the sake of fattening it: an apparently fortuitous event of getting more nourishment will change the quality of being inedible to the quality of being edible. 9./593. 3. Transition to specifying measure. The simplest form of measure is a state of quality that has its own quantitative value. We must now separate the state of quantity as an external position from a specific quantity: in the current structure these are mere aspects of the same situation or object, but they can also be separated as independent – then we get a relational framework where qualitative classification specifies merely quantitative classification.

After the detour to seeming paradoxes inherent in all measure structures, Hegel returns to the structure of specific quanta only to construct the next stage from it. The supposed topic has been a classification that could be interpreted both qualitatively and quantitatively: a position in this classification has had a numerical value, but has also determined a distinct species of objects or situations. Thus, we could say that the numeric value of any position has here two tasks: it is, firstly, a merely indifferent and arbitrary number, but secondly, it is also an essential nature of some species – these two tasks are, as it were, different aspects of the same whole. Now, as we should know by now, in any case where a situation or an object has many different aspects, we could abstract from the supposed identity between the aspects and assume or interpret them as independent entities: e.g. stages of my life could be seen as distinct object or slices of life. Similarly, we can here assume that the two aspects of a state of quantity are actually two different quantities and parts of different classification: this is at least a possible interpretation. The result of this “ignorance” is a construction of a new structure, where the former aspects are now independent situations: we have, firstly, a position in a merely numeric or quantitative classification, and

19 secondly, a position in a classification with qualitative significance. Despite the difference, there still is some connection between the former aspects: this should still be some sort of measure. Hence, there is a function from the arbitrary quantities to the specific quantities, although now we cannot say that there would be one quality for every quantity. Still, the qualitative classification at least regulates the arbitrary quantities in some manner: it appoints some group of quantities to one species or “specifies” them.

B. Specifying measure

We have investigated number systems where quantitative determinations have immediately a specific qualitative meaning, and we saw that it is possible to separate the merely quantitative and specific aspects of such systems, thus producing independent, but related number systems: in effect, we are then duplicating one number system. The result of the construction is a relation between a merely quantitative number system and a specific number system, and in this section we are aiming to construct an even more complex structure where both related number systems have a specific or qualitative meaning. The middle position makes the status of this section somewhat problematic. It has its own, peculiar subject matter, that is, relations between qualitative and merely quantitative number systems. Yet, Hegel introduces here also issues that belong more to the other sections of this chapter. This is reflected in the division of this section where the first subsection could well have been integrated into the first section, while the third subsection is very close to the issues of the following section.

1./594. (a) In the first stage, we have a regulating function that externally measures quanta.

The immediate result of the previous construction was duplication of one system of quantities into a specific and merely quantitative version. If no other – for instance, mathematical – difference is introduced between the two systems, we may well ask whether the two number systems are not still mere aspects of one number system. Even the introduction of mere direct ratio between the number systems might not be enough, as we have seen earlier: two variables having a direct ratio to one another can be understood as mere aspects of the same variable number. At most, we can say that we are here using one variable as an external measure of the other: we choose some quantity as an arbitrary reference point, to which we compare other quantities.

2./595. (b) We may then construct a relation where quantity with a qualitative significance regulates the merely external

20 number system.

While the first structure investigated in this section feels more like something belonging to a previous phase, here we find the proper topic of this section: a structure where merely quantitative number system is regulated by a number system with a qualitative meaning. Question is what more do we need to add to the relation between number systems in order to ensure that it is an example of specifying quantity instead of a mere external measure. Hegel’s suggestion is, as we shall see, that by supposing one number system to be a power of the other we make it clear that at least one of the number system has qualitative meaning in comparison with the other.

3./596. (c) Finally, we may interpret both sides of the relation as qualitative, and furthermore, view them as aspects of one measure.

The specifying quantity can be replaced by an even more intricate structure where both of the related number systems have qualitative significance: we can affect this construction either by a mere change in interpretation of specifying quantity or by direct construction of a framework of e.g. different powers of one number system. Like the first subsection, this feels somewhat out of place, as it investigates issues belonging to the next section: the only possible difference being that here we have perhaps not explicitly interpreted the related number systems as aspects of a larger structure or measure.

a. Rule

We now return to an issue familiar from the previous section, that is, the possibility of measuring one object through another: e.g. when we measure a height of a person by comparing it with the length of a measuring stick. Everything substantial that Hegel has to say of the issue has already been said earlier and thus the occurrence of this topic here seems rather artificial and forced. Even Hegel himself is quick to pass on to the primary issue of this section, that is, the truly specifying quantities. Note that Hegel’s transition is somewhat incomplete: he merely says that this structure does not yet live up to the standards of specifying quantities, but does not actually provide us with an example of a structure fulfilling those standards – this construction is postponed till the end of the next subsection.

1./597. We may use a natural quantum of one situation or object to measure the size of another situation or object: it is

21 arbitrary to choose particularly this reference point, because it could also be measured against another measure. We may also construct a more complex structure where one number system specifies another.

We have begun from an example of a classification with both quantitative and qualitative aspects: a classification where every distinct quantity has some qualitative significance. Now, picking one specific example from this classification as a reference point, we may abstract from the qualitative nature of the number system and merely compare the quantitative value of other positions with this reference position: in effect, we have picked one entity with a natural quantum as a provider of a measuring scale for all other entities – the quanta of other positions are then multiples of this reference point. Only thing worth mentioning of such a structure is that the choice of a reference point is quite arbitrary: we might as well pick out the object that has been measured as a new reference point against which the former reference point shall be measured. Hegel points out that such a structure is still not a proper example of a specifying measure. True, the structure relates a specific number system to a merely quantitative number system, but it seems very plausible to interpret both as mere aspects of one number system: it is the same scale of quantities which is first used as specifying a certain sort of situations or objects and then is used as a mere scale for comparing quantities. Hegel only asks that we should find better, more concrete examples of specifying measures: hardly a proper transition yet, but we shall see at the end of the next section what sort of structure would provide the required example.

b. Specifying measure

The structure of measuring quantity of one object with quantity of another, arbitrarily chosen object was revealed to be an improper example of a specifying measure, because of the externality of such a measure. What we are looking for, then, is a structure where the relation of measuring one quantity or number system with another would be more essential or not so arbitrary. Intriguing in this subsection is that we are not yet aware of what sort of structure could live up to these standards and hence we do not have any example of such a structure: it is only at the end of this subsection that we finally discover a method of constructing examples of specifying measures. Because of this concentration on finding specifying measures, the transition to next phase is left to the following subsection, although certain hints of how to discover the next structure are given here and in the immediately following remark.

1./598. A proper specifying measure is a structure where a qualitative number system is related to a merely quantitative

22 number system. Because of its place in the quantitative classification an object can change its quantity arbitrarily: the quality of the object gives it a proper measure by relating it to an object in the same qualitative classification, but with a merely quantitative number system.

The introduction to the structure of specifying measure reveals yet nothing to differentiate it from the previous structure of external measure. We learn that the structure consists of two related number systems, one of which should be merely numeric, while the other has qualitative significance in specifying the quantity of the other number system. The only thing that is added to the previous structure is the requirement that the number system used as a reference point should be immanently or naturally measuring: thus, we are here presented with a mere assignment – find an example of a structure where one number system naturally measures or specifies another one. Hegel also gives hints of the next structure and how it shall be discovered: he mentions that the apparently external or merely quantitative number system is actually also qualitative – it determines certain “species” of objects having the same genus or higher quality than the species represented by the specifying number system. Yet, Hegel admits straightaway that at this moment we abstract from the fact that both number systems are qualitative and treat the other one as merely quantitative. 2./599. The change of quantity [of the external number system] is external to an object specified by the quality: the quantum of the object is not just assumed by the object, but altered in some manner – this shows the independency of the object. Yet, the possible quanta of the object form a number system dependent on the first number system: still, the value of second number system varies in a specific manner in relation to the first number system, and measure is a function regulating between these number systems.

Hegel speaks here of “somethings” or objects that are supposed to be characterized by the different qualitative number systems: he is probably thinking of the following example of temperatures of different bodies. Yet, it seems that this restriction is unnecessary and we could speak also of situations: even in the case of temperature one could say e.g. that different regions have different manners of absorbing warmth. What is important is that these situations or objects – actually only one situation or object is needed – is in Hegelian terms for itself, that is, independent and separable from the rest of the situations and objects. This independence is guaranteed by the fact that the qualitative number system or variable is some particular function of the external or merely quantitative number system. This function should apparently be something more than a direct ratio, because the variation of the qualitative number system does not just arithmetically follow the variation of the external number system – here we finally have a more definite criterion for a structure being an example of a specifying measure.

23

3./600. Intensive and extensive quanta are aspects of the same quanta: the value of quantum does not change in these aspects. Here, on the other hand, the value of a quantum in external number system differs from the value of the respective quantum in the qualitative number system.

The comparison of specifying measure with the difference of intensive and extensive interpretations of quanta may seem arbitrary: the only information gathered from this comparison is that the difference of number systems in specifying measures is not just aspectual, but this could have been learned from comparison with any number of structures containing mere aspectual differences. The point of this comparison becomes evident only in the next subsection, where Hegel states that difference of intensive and extensive quanta is one facet of a framework of related number systems: i.e. one number system is more like intensive in comparison to the other. Hegel’s comparison has thus a purpose of showing that the difference of extensive and intensive quanta is still not a sufficient condition for a structure being a specifying measure. Note, by the way, that Hegel admits we could still view the difference of number systems as aspectual in some level of abstraction: we just have to ignore the specific values of the variables, as they are identical in being variables. 4./601. The function between number systems cannot be a mere direct ratio, because then the function would be only an external quantum: instead, it could be a relation between powers. The relation of powers is here connected to a mere arbitrary quantitativeness: one number system varies in an external manner, while the other number system specifies the variation by being some power of the first system.

We finally find the answer to the question that has haunted us in this subsection, that is, we find a method for constructing an example for a specifying measure. Hegel begins by saying that a direct ratio between number systems can never be a true example of a specifying measure: what he should have said perhaps is that a mere direct ratio cannot by itself guarantee that a framework of related number systems is truly a specifying measure – for instance, we may measure one line by taking another as the reference point, but this does not tell of any essential connection between the lines. The relationship of a variable to its arbitrary power is different, on the other hand. While a direct ratio might not be a qualitative relation, power relation can always be interpreted in such a manner: although one of the number systems would be interpreted as merely quantitative, the other number system varies incommensurably compared to the first – a familiar example is the relation of lines to their squares. Note that despite what Hegel says, there is nothing to prevent specifying measures having different shape from the power relations: the power relations are merely the simplest numeric structures where the structure is exemplified, while even direct ratios could be interpreted

24 as specifying measures, if only they have some more complex properties.

Remark

The section on measure should be instantly applicable to all sorts of physical phenomena: indeed, it often feels that Hegel concentrated on certain structures merely to provide a conceptualization for certain interesting phenomenon. The example Hegel presents here of the specifying measures – the relation of a temperature of a medium and an object within that medium – is rather interesting, as it differs somewhat from Hegel’s usual examples: the movement and the chemical reactions. The worth of the example is somewhat reduced by the fact that it could actually be conceptualized in different manners, and the specifying measure is not even the most complex of them: in fact, it makes sense to ask whether there is any phenomenon that could exemplify the specifying measure, but wouldn’t exemplify also the following structure of related quantitative qualities.

1./602. The relation of the temperature of an object to the general temperature is an example of a specifying measure, because one changes with the other, but the two temperatures are not in a direct ratio. Actually, because the general temperature is also a temperature of an object – the air – this can also be seen as an example of a relation between many qualitative number systems.

The structure of specifying measure consisted of two related number systems – one merely quantitative, but other with a qualitative meaning. One possible example of such a structure was the relation of lines and their squares – or generally, any relation between number systems of different dimensions would do – but if we are looking for an example where the function connecting the number systems has some physical meaning, it is not very satisfying. Hegel himself suggests the relation between the general temperature and a temperature of an object. Now, this is not a paradigmatic example of a specifying measure (a simple relation between a variable and its higher power), as it involves somewhat more complex structure: the temperature of an object tries to equalize itself with the temperature of the environment, and the progress of this equalization depends e.g. on the heat capacity of the object – a quantity determined by the internal constitution of the object or matter involved. Of course, this is no criticism, because any physical example would do, no matter how complex it would be. A more pressing problem is the fact that Hegel himself admits at the end of the paragraph. The so-called general temperature can be interpreted as a mere external quantity, but this is actually an abstraction, because the general temperature is also a specific temperature of an object, namely, the medium surrounding the other object. Thus, the

25 relationship of the two temperatures is actually a relationship between qualitatively different number systems, which is the structure to be studied next.

c. Relationship of both sides as qualities

A measurement of one quantity as a multiple of another quantity taken arbitrarily as a reference point did not live up to the standards of being a specifying measure. We needed something more complex, and the appropriate example was found when we constructed a structure of a number system related to its power: here a mere number system was related to a number system with a qualitative significance. We still haven’t constructed an example of the next structure, that is, a framework of many related qualitative number systems, although we are about to do it in a short while, in the first division of this subsection: the construction involves a mere change of interpretation. The place of the current subsection is somewhat awkward, which is reflected in its internal structure. It is divided into just two parts, which cover the first two steps of the familiar “introduction-analysis-transition”-sequence. Transition to the next structure is lacking, and in fact, we cannot find one in the following section either. Indeed, this subsection already deals with the same structure as the following section – Hegel even mentions the “Fürsichsein” of measure in passing – and it would be reasonable to place this subsection as a part of the next section.

1./603. 1. Introduction of a relationship of qualitative number systems. In a specifying measure, a function with a qualitative meaning specifies an externally quantitative number system: yet, this quantitative number system can also be interpreted as qualitatively differing from the specified number system – both number systems have external and specific aspects. The number systems have qualitative meaning only in relation to one another: they are, therefore, inseparable in some sense – we have an example of a structure of immanently related qualitative number systems.

We begin our construction from an arbitrary specifying measure, that is, two related number systems, one of which is merely quantitative, while the other of them has a qualitative meaning because of the function specifying its variation in relation to the first number system: examples of specifying measure include the relation between lines and squares – generally all relations of a number system to its power – and the relation of specific temperatures to the general temperature. Now, it is only the choice of viewpoint that has made us call one number system merely quantitative. For instance, suppose we are studying the relation of a number system to its squares: we could as well think of the squares as the external reference number system, related to which the other number system would consist of square roots – the roles of merely quantitative and qualitative number systems would have been reversed. A similar result was discovered in the previous remark,

26 in which we noticed that the so-called general temperature is as well specific, being the temperature of the surrounding medium. Only thing we must still do is to “posit” this insight, that is, construct a structure where the qualitativeness of both number systems is accepted: we merely have to change the interpretation of the specifying measure. We have then two qualitatively different number systems related to one another. Now, both of these number systems have two aspects: they are in one sense mere number systems, but in another sense, namely, in their relation to one another, they determine some species of situations or objects. As we can see, if we abstract from the fact that the number systems are related to one another, we cannot think them as qualitative. Conversely, if we have the possibility of viewing the number systems as qualitative, we have the possibility of constructing one number system from the other, that is, the possibility of relating them to one another. Hence, in a context where the number systems have a qualitative meaning or aspect, the number systems can be “immanently” related: we can map one number system to another and vice versa. Therefore, we can interpret the number systems as mere aspects of a larger whole, that is, of a structure where two qualitatively different number systems are functionally related to one another: we have so already constructed an example of a measure “being for itself”.

2./604. 2. The analysis of a relation of qualitative number systems. In a structure of measure the variability of quanta becomes evident, because here a quantum has also the aspect of being qualitative: this is exemplified by difference of powers. In the simplest measures the variability has not been constructed, because all quanta have an immediate connection to a quality: in a specifying measure both aspects of quantum appear embodied in different number systems – general and specified. Even more complex is a structure where both related number systems are qualitative: here measure has been realized.

Hegel continues with a recapitulation of the development of measures thus far. Measures were introduced when we constructed an example of relations of number systems or of functions between variables: the paradigmatic example of such a structure was the relation of different powers, although it isn’t the only example. Here a quantum has a dual role to play, that is, in addition to being a quantum, it has also a qualitative meaning: this is the abstract definition of measure, and the task of this chapter has been to realize this definition in the sense that we should show that such a structure could be applied to quite complex situations and that we should also construct examples of such situations. In the simplest possible form of measure, we merely have a quantity combined with a qualitative meaning, e.g. a dimension is represented by a single number: here, the quanta are yet no variables, because they are so tightly connected with a quality. Now, the realization of the measure begins by loosening the tie between the quantum and its quality. We construct a separate

27 system of number where quanta can vary without any immediate qualitative significance. Still, in order to be a measure this structure must have some connection with qualitative matters: the externally variable number system is related to other number systems through a complex function that specifies the variability of the specific number systems in relation to the “general” number system. Finally, we can interpret even the so-called general number system as specific or qualitative: thus, we end up with a relation of many different number systems which have a qualitative meaning in relation to one another – every number system represents a variable quantity of one specific quality.

3./605. A mere quantum can change, but this change is merely a change of reference point within a number system: true quantitative change can be noted only in relation of many qualitatively different number systems.

Hegel’s use of the term “variable” or “changing magnitude” may seem rather perplexing: why couldn’t we speak of a variable in abstraction from other number systems? If we look at all quantitative changes within a number system, without any reference to another number system, we notice that all such changes can be interpreted as no true change: only the arbitrary position of the quantum has changed, but not the quantum itself. How is this fact different when the number system is related to a new number system? The change of variable Hegel refers to as true or genuine is still not the change of quantum within number system, but the “change” of one sort of number to another: that is, the operation of calculating the value of one quantum from another. Thus, all true changes are changes of qualities, Hegel is saying: what happens within a number system is always arbitrary, but the alteration of the number system itself is important, even if this change is a mere change of viewpoint.

4./606. The related number systems have arbitrary qualitative meaning, e.g. they are space and time coordinates: one of them is taken as external and the other is determined as a number in relation to that reference point. The qualities of the number systems do not determine which of them should be taken as the reference point: square is as good a unit as a square root.

A measure in general involves a function or operation by which to count the value of one variable in a number system, when the value of another number in another number system is assigned, and the measure we are studying now makes no exception: we have an operation by which to count the value of a variable of one quality when a value of a variable of another quality is given – for instance, we can count the area of a square, when the length of a line is given, or in the favoured example of Hegel’s, we can measure the distance travelled by a falling rock, when we know what

28 length of time it has been falling. Now, all functions Hegel is investigating are quite well-behaving in that they are bijective, that is, reversible – presumably he would be interested only of such functions, even if he knew of any counterexamples. We can count the area from the length, but we can also count the length from the area, and we can also count the time variable from the space variable in the case of falling object. In the previous structure we had a natural candidate for the role of the given variable, namely, that from the external number system, but as in this case both number systems are equally qualitative, we seem to have no clear reason for preferring one over the other.

5./607. Qualitative number systems can have a distinct role: one of them is more extensive or represents a collection of distinct parts, while the other is more intensive or represents a change of one unity through many situations – first is naturally interpreted as an amount, divisible or power, while the second is then a unit, divisor or root. The external change of quantity is given for the intensive quantity, while some function specifies the extensive quantity: a direct ratio suffices only for abstract situations and even the relation of a power and a root is not sufficient – the structure is best exemplified by a relation of different powers.

Although the previous paragraph suggested that the qualitative number systems can be discerned in no manner, Hegel now reverses his position: interesting question is what the criterion for the distinction is. The distinction itself is simple to understand: all quanta have the possibility of being understood as intensive or extensive – as representing order or size – and here we merely assign one role to one number system. Hegel adds also a further description that in the extensive number system the quanta are to be understood as representing “externality in itself”, that is, collection of diverse and independent objects and situations. Similarly, quanta of intensive number system are to be “negative” in comparison: in light of our knowledge of Hegelian terminology this means that these quanta represent process of objects that recur throughout many situations, which thus are seen as its aspects. Problem is how we should determine which of the number systems is to have e.g. the role of intensive quanta. In case of Hegel’s favourite example we might say that space is more natural to interpret as consisting of separate parts, while time is more like a continuing process going through many aspects. Yet, this gives us no general criterion: we cannot say beforehand which qualities are better suited for some role. Instead, Hegel suggests a criterion based on the quantitative roles of the number systems. Depending on the nature of the function connecting number systems, there is always one that is somehow simpler or “smaller”: in the case of direct ratio this is obvious, and in the case of relation of powers or dimension it is the number system with the smallest dimension. Thus, the “smaller” number system, Hegel suggests, is more natural candidate for the role of intensive number system, which is also the one that has the simply given

29 value, while the “larger” number system has the extensive role and is the one the value of which we have to calculate. Hegel makes also a small remark on what would be paradigmatic example of the structure investigated here, that is, what are the simplest instances of it that would be naturally modelled by it. Mere direct ratios are once again rejected, because sides of the direct ratio can as well be understood as aspects of one number system. The relation of a root to a power is also rejected, because it could as well be interpreted as a mere specifying measure. Hence, the remaining option is then the relation of different powers of the same number system.

Remark

Hegel continues his quest of finding physical examples for different kinds of measures. This time it is Hegel’s favourite examples – that is, different sorts of movement – that are brought to the limelight. The particular example seems rather dated: nowadays, we do not believe that the movement of planets is somehow qualitatively different from the ordinary mechanical movement. Indeed, we might say that Hegel’s theory was dated even in his own days, because Newton’s theory of gravity was already common knowledge. Hegel’s mistake can be understood if we interpret him not as looking for the explanation of different sorts of movement, but only a description of them: although Newton would have discovered a common explanation for different sorts of movement, these movements would still follow different functions.

1./608. The previous structures are exemplified in different relations of space and time in velocity: in velocity as such, the roles of space and time are reversible, although space is more naturally extensive and time more naturally intensive, in fall, space is a power of time, and in the movement of planets, space is a larger power than time.

If we forego the idea that Hegel would be attempting some sort of explanation of movement and accept this paragraph as description of different sorts of movement, there is nothing to complain: it only becomes doubtful when Hegel says that these descriptions are not arbitrary, but follow from the nature of space and time. The simplest form of movement is that where velocity is constant: s = vt – here the roles of s and t could be changed without the form of the mathematical structure being altered, as in t = (1/v)s, because, Hegel conjectures, the further qualities of time and space do not affect the structure of mechanical movement. The case is supposedly different in the examples of fall and planetary movement. Generally, time is the quantity that is used to mark the succession of different situations, while the space – that is, the distance from some point, like the starting point or

30 the focus of the movement – is the quantity that can be calculated when the time factor has been given. In this case the determination of the given quantity from the nature of the qualities involved coincides with the same determination from the nature of the number systems involved: time happens to be one with the smaller exponent – a lucky coincidence we would perhaps say, but Hegel conjectures there is something essential of the whole relation of space and time hidden in these equations: at least it shows that the structurations found for the fall and planetary movement are natural. 2./609. Newton’s mathematical principles of the philosophy of nature merely showed a semblance of proving laws of movement which Galilei and Kepler had discovered empirically: what is truly required is the connection of the qualities of space and time to these quantitative determinations.

Hegel’s description of the mathematics of nature as a science of measures is actually quite apt: when we apply mathematics to natural phenomena, we are trying to discover mathematical structures and functions that fit the regularities governing the phenomena. The actual methodology of this science may seem rather hazy, but we can discern some stages of it in this particular paragraph. The first stage is simply the observation of empirical phenomena and especially quantities involved, such as the distances of planets from the Sun. The second stage is then the generalization of these quanta, that is, the construction of certain mathematical structures that the variability of the quanta follows: this is the stage where Galilei and Kepler supposedly made their contributions. The final phase should then be what Hegel calls proof or demonstration of these structures. It is easier to understand what is not involved in such a “proof”. It most certainly is not meant to be any regular sort of proof, used especially in mathematics: Hegel is quite well aware that such proofs cannot be valid, unless they already presuppose the existence of the structure they set out to prove. Hegel’s own preference for proof has apparently two sides. a) Hegel is looking for “proof” from concepts: this can then be nothing else, but a construction of a structure that was discovered empirically. This involves not just a construction of a mathematical structure which the phenomena instantiate, but also a construction of a connection between qualities connected in the phenomena or at least of a connection between structures involved in the physical qualities. Thus, in the philosophy of nature Hegel “discovers time” from space and vice versa: in effect, he seems to be constructing models for time in spatial terms and models for space in temporal terms. b) The second aspect of Hegelian “proof” should be an explication of quantitative terms involved in the structure from the structure of the qualities involved: we saw an example of such explication in the previous paragraph, where the relation of space and time should justify the exponents they have in the

31 structure of movement.

C. Being for itself in measure

The chapter begun from a simple structure in which every quality was represented by a single quantity. We felt the need to find more complex structures, and we then proceeded to construct relations of a power to a root as examples of what Hegel called specifying measure: here, the number system or variable that had the place of the root in the function was supposed to be merely quantitative, while the other number system had a qualitative meaning given by the function specifying the values of quanta in this number system in relation to the quanta of the merely external number system. The final alteration has been the discovery that both number systems could be interpreted as being qualitative, and this structure of related qualitative number systems could be exemplified by an even more complex relation of different powers. No further alteration happens in the step from the end of the previous section to the beginning of this section. Hegel merely investigates this familiar structure from two new viewpoints: he shows that we can either abstract the related number systems and their respective qualities from one another or we can look at the whole structure as a unity represented by the arbitrary exponent. These two viewpoints cover the first two subsections, while the third is reserved for the important matter of making transition to the next chapter: Hegel tries to show that we can multiply the number of measures or functions in general.

1./610. 1. The independency of the sides of the measure. In the previous structure qualitatively different number systems are aspects of one structure: but while the quanta have no meaning outside the relation, the qualities have. Qualities are given independently of one another, although they specify their quanta in relation to one another: for instance, space could exist without time and time without space.

In a relation of many different qualitative number systems – for instance, the relationship of areas and volumes or of space and time in motion – the number systems appear to be mere aspects of the larger structure: there is not just one qualified number system, but many that can be mapped to one another. The number systems or variables, as specified by their mutual relation, would lose their significance if they were taken apart from one another: we couldn’t speak of a quantity of a distance travelled in motion, if there were no time in which the motion had been completed. Yet, at least the qualities involved could be taken apart from this structure. True, they would not then be seen as qualities or related situations, but they would still be independent of one another: we could, for

32 instance, imagine that space would exist atemporally, as a changeless container, or that time would flow without any spatial changes. Of course, this is not the whole truth. Firstly, the existence of the measure already tells us that e.g. space and time can be connected to one another through the specification of their quantities. Secondly, Hegel appears to believe in a stronger statement that even from the state of independence it is possible to construct for any situation or quality a link to another situation. For instance, in the case of space it is possible to construct examples of structures resembling time – e.g. the unsymmetrical ordering of geometrical shapes – while similarly from an instance of time it should be possible to construct examples of structure resembling space – e.g. the unchangeable past. Despite these facts, it is still always a possibility to construct a situation where the qualities have been separated, if in no other way, then by abstracting and ignoring the connection of the qualities.

2./611. The quanta of the qualities can also be abstracted from the relation of number systems or taken as merely given: because of this relative independency, the quanta can vary arithmetically. Even in their independency the qualitative and quantitative moments can be related to the larger framework: we can determine a direct ratio between the differently qualified quanta.

We can think of the qualities without determining the relation of their quanta through their variability in comparison with one another. Still, the qualities are not completely without quanta: line has a length although it is not compared to the area of any square. When the qualities are taken as merely given to us, they appear with some given quantum. Because this quantum is merely given, it seems completely arbitrary: it seems contingent how far from the Sun a planet is located. It is because of this arbitrariness that the change of the quanta seems so natural: it is inessential which quantum a quality has, and indeed, according to different scales it would have different quanta – thus, there are many possible lengths of a line and many actual distances from Sun in which there is a planet at this moment. Despite this apparently complete independency of the arbitrary quanta of the qualities, we may still compare them. We cannot provide any complex function between them in this singular situation – this would require the investigation of how one variable changes with the change of the other and thus study of many different situations – but we can say how much one quantum is compared to the other in this situation: we can find a direct ratio between the two variables in this single situation – thus, in the particular situation when a single unit of time has passed of the fall of an object, the object has moved g spatial units from its original location, where g is the gravitational constant.

33 3./612. 2. Exponent of the measure. This structure necessarily contains such a direct ratio, although its quantum is arbitrary: it is the other aspect in comparison with the qualities as independent – this aspect can be constructed from the aspect of difference, because we can always construct unities for related differences. This new aspect differs from the aspect of independence in being merely an arbitrary, empirically determined quantum: for instance, the quantity of space travelled in the first moment of time or the exponent of abstract movement. Such a quantum is empirically given, but because of its connection with measure we can construct differently qualified number systems from it.

It is easy to lose track of what Hegel is up to here, if one does not remember how the Hegelian terminology should be interpreted. The immediate or given magnitude of the direct ratio is negation of the relation of qualitative number systems: this means merely that the direct ratio is an alternative aspect of the whole structure in relation to the aspect of taking the number systems as independent. We can construct this aspect of the direct ratio from the aspect of independent number systems, because it is generally possible to relate differences through their common elements: this is a reference to an already familiar pattern of construction – if we have separate objects or situations having something similar in them, we can interpret them as modifications or aspects of one underlying object. Here we are looking at a common element between the number systems in one particular state – that is, not what is common for them in general or with all possible values of variables, but what is common for them, when we fix some particular values for the different variables: then the common element is the direct ratio connecting the two variables at that moment. The result of the construction is being-for-itself, that is, the direct ratio is the unitary aspect where the number systems are no longer taken as independent of one another – as in any case of direct ratio, we may even interpret the relata as being aspects of the same number system. It is only when we alter the values of the related variables that we truly see the variables as describing independent and in a sense incommensurable qualities or dimensions: the complexity of the function relating the variables guides us to interpret the whole situation in a more complex manner. Thus, the direct ratio is “being for itself” only “in itself”, that is, in abstraction from other possible states of the variables: when we change the values of the variables according to specifying function, the direct ratio between the variables changes also, hence, we can actually find examples of many different direct ratios within this structure. In addition to the abstract analysis of the direct ratios within the structure of measure, Hegel also spends some time in investigating them in one particular example, namely, the relation of space and time in fall. Here, the direct ratio of one particular state of the variables may be identified with the coefficient of the time factor: this coefficient is the direct ratio between the first temporal unit and the distance covered in that time. Hegel dismisses the idea that the value of the number itself would have some further relevance: we could choose another, smaller temporal unit

34 and then the coefficient would clearly be different. Yet, there is something “natural” involved in this coefficient: after the units of the variable number systems have been assigned, the coefficient of the fall is invariably same. This naturalness is, of course, related to the weight of the planet where the fall happens, as we now know: we would get different coefficient in Mars than in Earth.

4./613. 3. Realisation of measure. We have seen a structure where a specified function between variable number systems is connected with their direct ratio that forms one crucial moment of the whole structure: in the example of fall, the direct ratio was a mere abstraction. Here the structure of measure has been realized, as a relation between an external and specified measure: it has an aspect where the number systems are separated by their qualities and related by a specifying function, but we can interpret them also as modifications of a direct ratio and ignore thus the specific function, yet, we can also realize the direct ratio by developing the specific relation from it. These two different, but related measures can be seen as aspects of one underlying, truly independent unity: this unity can be multiplied into many different unities, depending on how we vary the aspects of the measure.

In the previous subsections we saw that the structure of measure we have been investigating had two different aspects. Firstly, there was the aspect with two qualitatively different number systems or variables: because of their qualitativeness, it seemed that the two number systems were completely independent of one another. Yet, because the number systems were number systems, we could at least compare the quantities of the qualities: we could thus move to the second aspect where we interpreted the variables as moments of one number system in a direct ratio. Because the direct ratio between the variables is itself variable, we could return to the original aspect: the two variables must represent different qualities, because they do not alter in the same ratio. Supposing we have went through the movement from one aspect to the other and back without at first knowing that they were or at least could be aspects, we would then be in a position to interpret the two aspects truly as aspects of an underlying unity: this unity is just the complex function regulating the variability of the two number systems. Hegel’s descriptions of this function should be familiar from the past. The function has “realized” the structure of measure, because it contains as aspects simpler measures. The function is the true “being-for-itself” in comparison with the direct ratio that was “being-for-itself” only in abstraction: the direct ratio connected variables into a unity only in some situation, while the complex function connects them in every situation. It can also be called something or it can be taken as an object: this is the result of the familiar construction that an underlying unity behind two aspects or “qualities” especially deserves to be taken as an object – mathematically speaking, we need not speak of functions merely as relations between number systems, but we can speak of functions as constituting elements in a system of their own. This point leads us to the following chapter, but first we need to have many functions as

35 objects: we couldn’t form a proper number or algebraic system out of merely one object. Hegel’s manner of constructing multiple functions from one function seems almost like a mere afterthought: he just notes that it is possible to “repulse” or multiply one function. Still, the point of the construction is obvious. We may vary the relations the two number systems have to one another, e.g. by assuming different coefficients: the results of this variation are new possible functions, which we require in the next chapter. Note that this construction might be essentially restricted to “mere mathematical objects”: we have no apriori certainty that the nature would show instances if such multiple functions.

Glossary:

Maaß = measure; structure connecting states of quality and states of quantity in the sense that a situations that are in one sense part of a quantitative framework are in another sense part of a qualitative framework and vice versa Specifische Quantum = specific quantum; a definite magnitude or a definite interval that characterizes some species of things or situations, that is, if a thing or situation moves beyond this magnitude or interval, it changes to another species or even is destroyed Regel/Maaßstab = ”measuring stick” or ”ruler”; something with a natural quantity used to measure quantities of others Specificirende Maaß = specifying measure; a number system characterizing some species of things or situations and related to another number system in such a manner that when this related system has the value x, the first system has the value f(x) such that f is a more complex function than mere direct ratio, for instance, f(x) = xa for some constant a Verhältniß beyder seiten als Qualitäten / realisirte Maaß = such a relation of two different number systems, which both characterize some species of things, that when one of these systems has the value x, the other system has the value f(x) such that f is a more complex function than mere direct ratio, for instance, f(x) = xa for some constant a Fürsichsein im Maasse = a function consisting of two different functions, one of which combines one type of number system to another, while the other is a ”coefficient” or direct ratio modifying the first function