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Second division. Magnitude. (Quantity)

Hegel began by showing that we can make divisions between different situations: they can be seen

as differently qualified. Then he proceeded to show that all such divisions can be overcome: we can

look at situations with apparently different qualities as aspects of a larger whole. Hegel also

applied this general method to the case of situations with objects. If we are given an object – and if

such is not given, we can always find one – then we are also able to find new objects. Yet, all these

objects can then be taken as aspects or parts of one object. We have thus gone through all basic

ontological structures or structures of mere situations and things: a situation or an object without

any relation, a situation or an object related to another and a situation or an object having aspects

within itself. The next step would be to study the methods by which these situations and objects are

produced, but Hegel does not yet make this step. First, he investigates what sort of more complex

structures one can build with this mere possibility of bringing up ever new situations and objects:

this is the realm of quantity, where Hegel shows great ingeniousness in constructing the content of

the mathematics of his times.

The partition of this division follows the schema of the previous division: thus, most of

the transitions are already obvious. First, we start from a quantity – a structure with a potentially

infinite progression of situations or aspects. At first, this quantity or potentially infinite progression

is unrelated to anything; then we relate it to another quantity or divide the first quantity to two

quantities or from another quantity – we have now many possible potentially infinite progressions.

Finally we see that the two quantities can be turned into each other by some sort of process – both

quantities are already potentially infinite progression of numeric values, thus, the relating process

is a function from one series of numbers to another.

1./281. Qualitative structure is a division into characteristics without any relation to another division. A quantitative

structure forms a division of situations that can be changed: it gives limits that can be moved or it in one sense separates

situations and objects, although they can be identified as aspects of a larger whole in another sense.

We may finally perhaps understand the definitions of qualities and quantities given already at the

paragraph 104. Qualities form a structure in which objects or situations are classified according to

these qualities: here are the red objects, there are the blue objects etc. This is actually the only

requirement that a quality has. Quality is immediate, that is, it is merely given and not constructed

or deduced from other characteristics of the object, like perhaps the reflexive characteristics later

are. Furthermore, the qualities are taken as essential determinations of objects or qualitative

structures truly differentiate between objects: a red object is not a blue object, and if the object

changes its colour, it has actually been replaced by another object.

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Hegel’s first characterization of quantities seems not as satisfying. Quantity should be a

determination that has become indifferent to the situations and objects or a limit that can be seen as

no limit. This characteristic shows one aspect of quantities. A quantitative structure does

differentiate between objects or situations: this is two meters long and that is three. Yet, the

quantitative characteristics are not essential to the objects or the situations and therefore an object of

one quantity could be identified with an object of another quantity: they are merely the same object

in different situations, e.g. when a road of two meters wide is changed into a road of three meters

wide. Hegel seems not to notice that any inessential or extrinsic characteristic (Beschaffenheit) of an

object conforms to that description: if we interpret colours as inessential to an object, then the

change of the colour does not change the object.

The further characterizations of quantity reveal its full nature. Quantitative structure of

situations consists of a state of infinity – we can see all the situations as aspects of a larger whole or

their apparent difference is indifferent – but this state can also be seen as determinate or related to

other states – we can always produce new situations. Or in the case of objects, they are organized

quantitatively, when we can repulse or produce ever new objects, but these new objects can always

be seen as parts or aspects of a continuous whole. Thus, we should not speak of a quantitative

structure, but of a series of quantitative structures: it is not just one division of objects or situations

which can be seen as no true division or only a division of aspects, but a series of possible divisions,

all of which can be seen as mere divisions of aspects. Thus, Hegel’s account of quantities already

presupposes method by which to find/produce and assimilate new situations and objects – repulsion

and attraction.

2./282. An independent object is constructed as an object that can be identified with another object, although we can

find apparent examples of different objects and situations related to one another. A characteristic of such an object is

not a given characteristic of a determinate object, but points naturally through some construction of repulsion to another

object: as these characteristics are indifferent to objects, we may say that such characteristics and their objects are in

some sense identical with what is in another sense extrinsic to them. Such limits and their indifference towards

themselves and their objects form quantitative structures.

Hegel treats us once again with a complex looking paragraph, but actually his message is fairly

simple. We are now in a position where seemingly independent objects – objects which are only

related to themselves – can be identified with other objects: this was the main idea behind

constructing attraction from repulsion. For now, the result is a mere single object continuing

through many different situations, but the difference of determinate situations and objects is bound

to come up again, because we can produce more objects and situations through some construction

of repulsion. Indeed, given any determination of an object, we are able to produce objects with new

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determinations: for instance, given a number, we can always construct an example of larger or

smaller number through addition or division. Furthermore, these characteristics have meaning now

only in relation to one another: their place in the space of characteristics is determined by how we

can construct situations with other characteristics from them – for instance, number four is

characterized by being a certain multiple of a given unit. Now, these characteristics are already

indifferent to the objects and situations that carry them and this indifference is just emphasized by

the possibility of finding more and more such characteristics: it is indifferent to a situation with

apples, whether it is a situation with four, five, six, ten, hundred etc. apples. Indeed, because the

characteristics are determined only by their relation to other characteristics, they are in one sense

indifferent in themselves or out of themselves: every characteristic could be identified with every

other characteristic, that is, there are functions which move all the places in the structure of

characteristics systematically, while still keeping the relations similar – e.g. it is indifferent where

we put the origin in the space. It is this combination of indifference to objects and mere relativity

that constitutes the essence of quantitative characteristics: any external characteristic might be

indifferent to an object, like clothing to a person, but there is no reason to assume that the nature of

such characteristic would be determined merely by its relations – we can say that a person is

wearing red clothes, even if we don’t compare them with other clothes, whereas the size of those

clothes must be determined in relation to some unit of size.

3./283.1) A quantity without a relation to other quantities must be separated from a quantity related to other quantities.

This pure quantity is a state of infinity that has been found again from the difference of independent objects and

situations: it is an unbroken, continuous and infinite unity.

The previous chapter began from a situation which was not explicitly related to any situations: a

primary example was the empty situation to which any situation could be related by abstraction, but

from which no situation could anymore be abstracted. Similarly, the study of quantitative situations

must begin from a quantitative situation without any relation to other quantities. As actually any

situation or state of being could be an example of a pure state of being, as long as any relations to

other situations were abstracted (although the empty situation was the paradigmatic example),

similarly we can now take any quantitative situation as an example of a pure state of quantity. Thus,

the only characteristics it can have are internal to it: furthermore, they should be implied by the

previous discussions. Pure quantity, then, is a situation where the number of the objects is arbitrary:

we could find more and more objects within that situation, but all of these objects can also be seen

as mere aspects of one object or unity.

4./284. 2) It is possible to relate the quantity or determine it, but this determination is arbitrary or only relative: quantity

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has then become a quantum. A determinate state of quantity is determined only by its relation to other quantities, that is,

other quantities could be constructed from it: thus, we can progress indefinitely far in producing new quantities or

making current quantities into new quantities. The progress taken as a potentially infinite unity is not arbitrary anymore,

but a new qualitative situation.

The task of the first chapter in this division is to show that any given quantity can be related to

another quantity and thus made into a quantum: this occurs easily because every quantity becomes

with a construction with which to copy itself. Now, because every new quantity can then be

multiplied or new quantities can be found by the same construction, we are once again on our way

to an indefinite progression: this time the progression can have the peculiar characteristic that the

constructed quantities or quantitative situations are always either larger or smaller, which leads to

the conclusion that there can be no largest or smallest quantity for Hegel. Yet, we can also identify

the original and the constructed quantity: then the whole indefinite progression of quantities would

actually be a progression of only one quantity, growing or diminishing in a series. How does this

result lead into a new emergence of a quality? The state of possibly-being-quantified-in -this-

manner,-possibly-in-that-manner-etc. is not as indifferent to the object as its current quantitative

constitution – although a field could be of another size, it would still be possible for it to be of this

size. Furthermore, the relativity of the quantitative determinations would be incorporated into the

quantity itself: we would not be talking of e.g. two compared to some unity, but of a quantity which

would be two compared to that unity, three compared to another etc., that is, of a variable with its

own “rate of change”. This idea will be developed in the final chapter.

5./285. 3) When we make quantum qualitative, it becomes a relationship of quantitative states. Earlier we just went

through different states of a quantity or made it indefinitely larger or smaller, now we can also relate it to another

quantum: thus, in one sense we can also take the result of the construction as identical with its beginning.

A construction or operation of creating new quantities from a given quantity, such as multiplication

and division, can be interpreted in two ways. Firstly, they may be thought as changing the quantity

itself, like we did in the previous paragraph. In such a case the progression can be extended

indefinitely, as we saw. Secondly, they may be thought as creating new quantities or relating the

original to new ones: given a quantity, give me a triple of it. It is the second interpretation that

interests Hegel in the third chapter and there are two things that Hegel emphasizes in these relations

between quantities. Firstly, as we already saw, there is a familiar case of being able to identify and

differentiate between two things: an operation in one sense creates new quantities, in another sense

it creates new aspects of one quantity. Secondly, the quantitative relations provide stability to the

quantitative situations or make them qualities in some sense. The determinate quantity of a

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quantitative situation depends on what is taken as the unit quantity to which other quantities are

related: in this sense all quantities are variables. Then again, the relations between the different

quantities should remain stable, although the unit quantity is changed: a quantity three times larger

than another is always three times larger than the other, no matter how their numeric value is varied.

This makes explicit the fact that quantities are determined only through their relations. Furthermore,

the operations by which quantities are related – three times, half of, etc. – are also quantities, but

differ from the quantities related in being more stable or quality-like.

6./286. The quantitative relationships are still based on the indifferent variability of the related quanta: thus, it unites

aspects of quality and quantity only formally. It can be changed into a more necessary unity of these aspects or measure.

All quanta can be seen as variables or operations by which to provide new aspects of one

quantitative state: for instance, operation +1 can be seen as variable going through phases of 1, 2, 3

etc. Such operations can be related to other quanta also: e.g. +1 to +3, or 1, 2, 3… to 3, 6, 9… These

series form then in one sense a system of differing quanta, related to one another by such operations

as x3, which may also be taken as quanta. Yet, the difference between +1 and +3 is in one sense

trivial: it is only the question of what unit we have happened to pick up. Thus, we may in one sense

say that the two quanta related by a function are actually aspects of the same quantum: the aspect of

the indifference is still in control, while the aspect of quanta being different seems unimportant.

Still, we can find a more substantial difference between some quanta. For instance, if we think of

the operation x3 as a similar series – series of +1, +3, +9… – we notice at once that compared with

e.g. the series 1, 2, 3 it grows in a different rate: the difference between these series is not just one

of choosing a different unit. Thus, given some system of quantities, it is also possible to build

another system of quantities, related to the first as one quality to another: this is how Hegel thinks

he can construct measures.

Remark

Hegel has presented his definition of quantities, but it is still unclear whether this definition

coincides with what we usually understand as quantities. One important note of the Hegelian

quantities must be made. When we think of quantities, the primary example are perhaps the natural

numbers, which describe amounts of groups, packs, shoals etc. of discrete units. As we shall later

see, a Hegelian quantity can always be divided into further Hegelian quantities: similarly, a part of

a piece of mass should be a similar, but smaller piece of mass that can also be divided into further

pieces of mass. Only against this background does Hegel’s definition make sense: natural numbers

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are not determined just by their relations, for instance, any other number cannot take the place of

the number one or the first positive term of the natural numbers, because then the one itself

couldn’t be mapped to the place of any other number.

1./287. A quality is essential to an object, and its change changes the object: change of a quantitative limit, on the other

hand, does not change the object – field remains a field, if it grows, and red remains red, if it becomes brighter, but the

field and the red are destroyed with the change of quality. Every other example of a quantum shows that it consists of

something being indifferent to its characteristic.

2./288. Magnitude means usually quantum instead of quantity, and for making the difference explicit I use the Latin

words.

It is strange how Hegel clings to the idea that quanta are defined by their inessentiality to the object

or situations they characterize, while they are differentiated from all other inessential or extrinsic

characteristics only by their complete relativity: a rose could be red or white, while red rose could

not be taken as a white rose. Yet, there is some sense in Hegel’s strategy. The essentiality of every

other determination depends on the viewpoint we have assumed: redness is not essential for rose,

but it surely is for a red object. Because quanta are also inessential in themselves – that is, any

quantum could replace the place of the every other quantum, supposing there would just be suitable

rearrangement of all the other quanta – it would make no sense to take them as intrinsic to an object

or a situation: situation could well be said to consist of a hundred objects, but also of a thousand, if

the unit objects would be picked differently. Note that the inessentiality of determination is only a

characteristic of quanta: in the stage of mere quantities, there has yet been no relating of different

quantitative states and thus no determination.

Kant mentions also the difference between quantities and quanta, but it is not the same

difference that Hegel has. For Hegel, quanta is a quantity related to some other quantity: the

difference is thus one of a length of a wooden stick and the same length compared to a measure of a

meter half the size of the stick. Kant, on the other hand, refers to the difference between a variable

or indefinite quantity and definite quantity: it is the difference between algebra and arithmetic. For

Hegel, both algebra and arithmetic require quanta in order to work: in the case of algebra, the

relation of the quantities towards one another and towards any definite unit is just left implicit.

3./289. In mathematics magnitudes – by which are meant quanta – are defined as something, the magnitude of which

could be increased or decreased: this means that the magnitude of a magnitude could become something else than it is.

The definition is clumsy because it uses the term to be defined: better would be to speak of a possibility to affirm or

negate a quantity in an external or indifferent manner – it is indifferent to the magnitude how large it is taken to be.

It is not known to me whether Hegel’s reference to the mathematical definition of a magnitude

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refers to any specific writer or book or was it truly common in text books of mathematics: Euclid

defined the possibility of one magnitude being the part of the other through the possibility of

creating the other through growth of the one, but not the concept of magnitude itself. Yet, it is likely

that the definition refers to the fact that an object quantified by a magnitude could always be larger

or smaller, while e.g. the length of an object seems always of a certain size, whenever it remains

constant. Hegel sees something deeper in the definition, namely, that every magnitude is actually

always a variable in the sense that we have fixed its size only when we have fixed some reference

size, that is, what is constant is only its relation to other magnitudes. What makes us see the

magnitudes or quanta as constant is the fact that we ourselves or our bodies have a certain

magnitude to which we must always relate the magnitudes we see: we cannot arbitrarily turn into a

size of mouse and view the world from this quantitative reference point.

First chapter. Quantity.

As the first chapter on qualities began with a state of pure being, the first chapter on quantities

begins with a state of pure quantity. Hegel will shortly give many examples of such quantities, but

actually there is a very simple answer to the question what is a pure quantity. A pure state of being

was exemplified simply by any state of being or situation as long as it was abstracted from its

relations to other situations (although an empty state of being was paradigmatic case of such a

situation). Similarly, a pure state of quantity is exemplified simply by any quantity when it is not

related to other quantities. Of course, this being unrelated is only contextual and it is the task of

this chapter to show that any quantity can be related to other quantities: the construction is fairly

simple, because quantities come with a method of multiplying them. The division of this chapter

falls almost into the pattern of “introduction-description-transition”. The first section does also

describe pure quantities or the inner structures of all quantities, second section then defines two

species of such quantities and the third section finally shows how to find other quantities.

A. Pure quantity

The first stage of the story on quantities is not very exciting. All we hear is what sort of things

quantities are and what aspects are contained within their structure. We can repulse – multiply or

divide – them, but still all of these repulsed structures can be seen as aspects of the original

quantity. All of this was clear already at the end of the chapter on Fürsichsein, and the repetition

serves merely to remind us what we are now studying.

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1./290. In a state of quantity the state of independency of units has been integrated into a whole: formerly units

excluded one another, but now they can be identified or they attract one another. Repulsion of one can always be

eliminated within this quantity, but it always remains a possibility and thus the unity behind units consists of seemingly

external units seen as identical: this is the aspect of continuity in a state of quantity.

An example of a state of quantity has been constructed by noticing that all given unities can be

identified with one another or that they can be seen as aspects of one unity. Hegel thus begins his

study of quantities with this aspect of having united many unities or state of attraction. In a state of

quantity many independent unities have been idealized: unities that previously repulsed or excluded

one another – that is, were differentiated from each other – are now being identified. All this is

familiar from the account of attraction in the previous chapter: it is like seeing points of a line as a

movement of one point or stages of my life as aspects of my person. Hegel offers a new description

of the previously independent unities in their aspect of being independent by calling them brittle.

Hegel refers to the fact that brittle substances, such as sand, exist in a state of a collection of

external points that do not form any stable wholes by themselves. This brittle stuff of unities has

now been, as it were, melted into a unity: all these pictorial descriptions give mere analogies of the

more strict account of the previous chapter.

Novel is also Hegel’s description of this state or aspect of attraction as a state or

aspect of continuity. Hegel is anxious to reconstruct some basic concepts of the mathematics of his

times – to show how they can be understood in light of his investigations. One of them is the idea of

continuous magnitudes contrasted with discrete ones: we shall return to this attempt in the next

section. For now, we need only to say that Hegelian continuity is not the same thing as the

continuity in our days. When modern mathematicians speak of continuity they mean either 1) a

characteristic of a continuum or an infinite set of the cardinality of the real numbers or 2) a

continuity of functions described in the delta-epsilon formula and satisfiable only when they are

applied to some continuum. For Hegel, on the other hand, continuity means the similarity or

identity of certain objects or aspects, the collection of which never reaches infinity, as we shall see:

the fact that every one of the units of the collection could replace any other. Further difference is

offered by the fact that the continuity in modern mathematics is supposed to be an exclusive

characteristic of only certain sets or functions, while for Hegel continuity characterizes a certain

aspect of all quantities.

2./291. Aspect of continuity is then an uninterrupted identity of aspects that in some sense can be taken as independent:

a state of plurality is implicit in it. The plurality is constructed as what it implicitly is, that is, as a similarity without

differences: differing unities can be identified as one unity in different situations.

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In a state of continuity we have only one true object: all the seeming differences have been

idealized through some construction of attraction. Yet, as we should be well aware by now, an

idealization of differences does not mean that they would be completely wiped out. We have merely

interpreted the different objects anew: what seemed to be independent objects are now taken as one

object in many different contexts. Hence, it is still possible to discern the many independent unities

in the continuity: for instance, if the continuity in question is one of a line, we can divide the line to

its parts – or more properly, interpret the parts of the line as independent lines. Note the use of the

description “beside one another”: it is a common description of quantities in Hegel, especially

spatial ones, and refers to the relative indifference of unities in a quantitative structure. Despite this

possibility of dividing the quantity, there always remains the contrasting possibility of unifying the

parts left loose by the division: we may see the small lines within the line as mere aspects of the one

continuous line. In a state with many unities this possibility was not yet explicit and we had to show

that the unities had nothing to differentiate themselves: we could easily change the places of the

unities without any apparent difference. Now this result has been constructed as a state of quantity

and we may use it as a given fact: all the differences within a state of quantity can be seen as mere

differences of aspects.

3./292. From an aspect of continuity it can thus be found an aspect of discreteness or repulsion as an aspect of a state of

quantity: continuity is a similarity between many different aspects, or only repulsion makes identity into a state of

continuity. In a state of discretion unities are not connected only by the fact that they are alternatives to one another or

that they share the same situation, but also by [transworld] identity or being aspects of the same object.

If the aspect of attraction in a state of quantity was called an aspect of continuity, the aspect of

repulsion Hegel names an aspect of discreteness: this is a clear reference to the difference between

continuous and discrete magnitude of which we will speak more later. Once again, as we shall see

in a more detail, a discreteness of a quantity does not mean the same thing for Hegel and the

modern mathematics, although here the difference is perhaps harder to discern: the question is more

of what is not seen as discrete. In any case, the discreteness is for Hegel also a mere characteristic

of an aspect instead of a species of quantities.

The aspect of discreteness is necessary for the aspect of continuity: we are not speaking of a

mere undifferentiated unity within one situation, but of a unity that can be at least virtually divided

into apparently independent unities or of a unity existing in many different situations. Similarly, the

aspect of discreteness is necessarily connected to the aspect of continuity. The unities are not just

externally related to one another, but they can be seen as aspects of one whole: parts of a line are

unified by being parts of that line. Furthermore, every part of a quantity can also be identified with

the whole quantity: they are both quantities. This means some obvious limitations on what can be

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taken as a quantity. A pack of wolves is clearly not similar with any single wolf, because wolves

cannot be divided to more wolves. It is not collections or sets which are the primary examples of

quantity for Hegel, but mass terms: a piece of matter can always be divided – at least according to

the Hegelian philosophy of nature – and the result is another piece of matter. These are the sort of

quantities Hegel is interested in.

4./293. The state of quantity is the unity behind the aspects of the continuity and discretion, but it is explicitly still in the

aspect of continuity, because it is the result of uniting independent unities: in the state of quantity it has been

constructed what is merely implicit in the state of infinity. A state of infinity was a continuous possibility of producing

copies of itself, which could then be identified as aspects of the original: because of the sameness of the produced, there

is only an uninterrupted continuity, because of the possibility of producing, there is an implicit possibility of

multiplicity.

The state of quantity forms a possibility of a series of alternating states of continuity and

discreteness. First all we have is one object with no differences; then we can copy or divide this

object into a multiplicity of objects; finally we can interpret these multiple objects as versions of the

same object. The same process can then be begun anew and the result is an indefinite progression of

one object multiplying or dividing itself. Why is the aspect of continuity the first state in this

implicit series? Hegel refers once again to the fact that the state of quantity has been produced by

noticing the similarity of apparently different units. Furthermore, states of continuity can be

abstracted from every state of discreteness, in so far as continuous or unified quantities are what

discrete quantities are made of: a collection of many separated pieces of matter have as their parts

continuous quantities of matter. Thus, a state of continuity can be in some sense more abstract than

a state of discreteness.

Hegel continues by showing that a state of quantity is nothing more than a state of having-

constructed-what-is-still-implicit-in-a-state-of-infinity: an obvious conclusion. In a state of infinity

or in a state with one independent object we already had the opportunity to make more objects – to

move to the aspect of discreteness. Undoubtedly, it then seemed exclusively to be a multiplying of

the original unit instead of dividing of it: in the state of quantity this difference can indeed be in

some sense put aside, because the division produces another state of quantity, and because of the

possibility of an indefinite repulsion or division of quantities, we can in some sense say that the

whole and the part are of the same size, that is, if the whole has actually some number of parts, the

part could have as much through a suitable amount of divisions. Furthermore, the state with many

independent units could then be attracted or interpreted into one unity, like the state of continuity

can always be found from a state of discreteness.

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Remark 1.

Here is one of those rare places where the remarks give actually more information than the sections

they are attached to. While the section on pure quantities merely repeated the results of the

previous chapter, the first remark tries, firstly, to show that the structure described actually is the

structure of quantities – as long as we remember that it is still silent about the relations between

quantities – and secondly, to provide some examples of quantities in general.

1./294. A pure state of quantity is not yet limited by other quantities, and even then these limits are external to it: the

essence of a state of quantity is having states of independency integrated within it. It has an aspect of discreteness, that

is, we can find independent unities in it, but then again all these unities can be interpreted as continuous.

The object of this chapter is the general structure of all quantities, in so far as they are not related to

other quantities: we are looking at a quantity in itself or its inner structure, one could say. When we

regarded a state of quality in itself, we were actually left with nothing else but an unrelated state of

being. A state of quality receives its meaning only by its relation of incompatibility towards other

qualities: for instance, a quality of red is characterized just by its difference from blue, green etc. A

state of quantity, on the other hand, has some structure even without the relation to other quantities:

indeed, it has many possible structurations, according to how many unities we happen to discern in

it: a piece of matter can be taken as a unity or as a combination of two, three etc. pieces. Thus,

Hegel clearly admits that it is the inner indifference which characterizes quantities most: a state of

quantity is such a state where the number of objects is arbitrary. Hegel calls this possibility of

discerning indefinitely many units a real possibility. In the book on essence we shall learn that a real

possibility in Hegel refers to some actual method by which to change a situation at hand. Thus, we

have always the possibility of finding another independent unit within the state of quantity,

although all these units can then be seen as aspects of a larger whole.

2./295. In an unstructured thinking one may confuse continuity with combination, in which units stay exclusive and

independent, but we have seen how units themselves contain a possibility of attraction. The externality of continuity is

an assumption of atom theory: mathematics, on the other hand, denies that e.g. space would consist of spatial points etc.

True, it speaks as if a line would be a sum of infinitely many points, but the infinity in a limited space can only refer to

the identity of apparently discrete objects.

In the previous chapter Hegel presented his low view on the atomistic theory: objects or units are

not just externally united into wholes, but they already share something common and at least

through that fact are intrinsically united. Although Hegel enlists the help of mathematics, the

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modern mathematics is actually closer to the ideas of atomism on the question of continuity:

continuity does not mean a stronger connection between units, but merely the existence of a

different amount of units compared to a discrete set. Both Hegelian and the modern ideas of

continuity could perhaps be developed from the mathematics of Hegel’s time, when it was not clear

what continuity actually meant. The idea of an infinite number of points making up a line was

actually taken in earnest by the modern mathematics: only that the infinity in question must be of a

larger sort than the infinity of natural numbers. In light of his investigations concerning infinity, for

Hegel, infinity of e.g. a line can mean only two things: a quantitative infinity of the modern sort

would be quite out of his reach as a contradictory or even meaningless concept. Firstly, it could be

infinity of the bad kind. A line would be infinite in this sense, as long as it wouldn’t be related to

any other lines, that is, as long as it could be seen as the biggest line there is. Yet, here the line

should be related to other lines, that is, it should be taken as limited in one sense by them. Secondly,

it could be what Hegel calls the true infinite. A line would then contain a difference within itself,

but integrated into a unity, that is, its separated parts – smaller lines or even points – could be taken

as mere aspects of one whole. As we already know, this is what Hegel means by continuity: he is

hence merely consequent when he equates the infinity of a line with its continuity.

3./296. Spinoza refers to the difference between the pure quantity and its representation when he speaks of the

difference between quantity imagined to be divisible and quantity conceived as unified substance.

Hegel is anxious to find some allies for his idea of continuity against the atomism and Spinoza is a

natural choice. We seem to be able to e.g. divide matter, but actually no division has been made,

Spinoza says. The only true or primary entity for Spinoza is the world or the one substance, and the

world remains a unity even after the so-called division: it is impossible to cut the universe into

pieces. The divisions we see are only apparent – it is how the substance affects itself when it is in

the mode of human understanding – but a true sort of knowledge would reveal that there is only one,

indivisible substance. Although Spinoza thus is an upholder of unities, he perhaps goes too far into

the other direction: the divisions, or in general, differences between entities can be only apparent for

Spinoza, whereas Hegel admits at least the possibility of dividing matter to smaller pieces.

4./297. More definite examples of pure quantities – not quanta – are e.g. space and time, matter, light and even

consciousness: these are possibilities of constructing or finding further instances of themselves, but any qualitative or

independent units cannot be found in them. Space is absolute state of being side by side, in which every situation is

similar to the others. Time is a process of creating new states or nows, but all these states are later idealized into the past

time.

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Hegel wishes to provide also some definite examples of pure quantities, although it is clear that any

quantity would actually do, as long as it is not compared to other quantities: these quantities share

the property that they could be divided or multiplied in one sense, although they in another sense

remain unities. Similarly, the examples Hegel provides here could as well be examples of quanta, as

soon as they were compared with examples of quantities of the same kind. For instance, when

Hegel speaks of space as a pure quantity, he is speaking of any space, when it is not compared with

other spaces, but if it were compared, it would be taken as a definite space. Notice how Hegel

describes pure quantities as flowing: this does not suggest that all quantities were somehow

temporal, but implies their incompleteness: for every quantity there is a possible larger or smaller

quantity or for every quantitative series there is a further member to be found or constructed. In the

case of pure quantities, all these further quantities are taken as mere further stages of one quantity:

the other interpretation would make these quantities into quanta.

Of the exemplary quantities, Hegel first mentions space: here, then, is a question not of any

“total space”, which would be a non-entity for Hegel, because space can never be total, but of space

in general, that is, of any space unrelated to other spaces. Such a space Hegel describes as “state of

being-outside-itself”. Seemingly complex description is actually quite simple. In one sense a space

is always outside something else, namely, another space: the word ”being” means here that we only

find this new space which has still existed even when we were not aware of it. Yet, in another sense

this new space is merely a part of the old space: we can in no way differentiate between these

spaces, because one space is as good as the other.

Time, on the other hand, should be “state of becoming-outside-itself”: and here it is once

again question of a time not compared to other times. In the case of a time, we are constantly

finding new times, but these times were not there, before we found it: on the other hand, they are

not also our free productions, but something that happens to or is inevitable for us. At first, this new

time differs from the previous time: it is the current time compared to the past time. But then it

becomes past time also and can thus be once again seen as a mere aspect or part of the original time.

5./298. Leibniz has identified quantity and matter: indeed, what quantity is in thought, matter is in external existence. A

consciousness or I is also continuous repulsion that remains a unity uninterrupted by limits of sensations. These

examples provide intuitable examples of the concept of quantity.

Whereas space and time offer a quantitative structure with situations – places and times – matter

provides us with a quantitative structure consisting of concrete objects: of space and time we can

only perceive that something is here or now, but of matter we can perceive that it is. The ways how

matter can be said to be a quantity are exactly same as those of space and time. Matter is spatial,

thus, for every piece of matter we can find another piece of matter outside it. Matter is also temporal,

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that is, we can sense changes in the position of matter relative to other pieces. Furthermore, as a

quantity in general, matter is also infinitely divisible, in a similar manner space and time were also,

without thereby consisting of many pieces of matter: these pieces are only potentially within the

piece of matter.

Hegel mentioned light in the previous chapter as one example of quantities, and its place

should be around here, according to the general structure of Hegel’s philosophy. Light can be

divided into pieces – light particles or rays – but these are according to Hegel only abstractions of

the whole light. Light then consists actually of lighted areas, which thus, as it were, create another

space in which the dark or not light-emitting objects exist. If the light resembled space in being a

container for objects, the consciousness or I resembles time in being a container for processes. I is

constantly being multiplied, or its content or object is changing: now I see a house, now a lake etc.

Yet, the I itself is always the same person: remember that we are not speaking of many egos, but

only of one unrelated consciousness.

Why does Hegel feel the need to give these examples? The Logic provides us, firstly, with a

method of recognizing quantitative structures, and secondly, with a method of constructing an

exemplary linguistic model of quantities: this model is provided by our capability of producing new

linguistic signs and identifying them with one another. What Logic cannot give is a sensuous or

intuitable and given example of such model: the linguistic model itself is dependent on our

constructing it. Now, spatial, temporal and material phenomena are instances of quantities not

created by us: even the basic structure of our consciousness must be independent of our actions in

some sense. These examples then show that the concept of quantity has applications outside what

can be produced by us: an important task indeed.

Remark 2.

This is Hegel’s first investigation of Kant’s antinomies in Logic, and one of two proper ones: Hegel

promises to study all of them, but merely describes shortly the third antinomy and leaves the fourth

antinomy unmentioned, perhaps because it is a repetition of the third antinomy in Hegel’s opinion.

As the first investigation this serves as a model for the rest and is thus the fullest of them all.

Hegel’s strategy with all Kant’s antinomies is same. Kant made a great discovery in finding out the

antinomies – that e.g. matter can be seen as both divided to the last limit and still divisible into

further pieces – but his explanation of this antinomy was wrong: it is not the result of the nature of

our cognition, but of the incompleteness and imperfectness of all natural totalities – e.g. of the fact

that no division of matter is final, but that an infinite division is impossible.

15

1./299. Because quantities are discrete in one sense, but continuous in another, the divisibility of space, time, matter etc.

seems antinomical.

2./300. The antinomy arises, because both aspects are essential for the state of quantity. If we concentrate on the aspect

of discreteness, we must conclude that the division of quantities could end; if we concentrate of the aspect of continuity,

we must conclude that quantities could be divided indefinitely long.

A state of quantity is for Hegel a structure with a situation or an object that, firstly, can be divided

or multiplied into many situations or objects, and secondly, where all these new situations or objects

can be interpreted as mere aspects of the original situation or object: the results of the division are

structured similarly when compared not just with one another, but also with the original undivided

quantity. Both aspects can be found in every sort of quantity: every spatial position, temporal

moment and material object is divisible, but the results of the division are further spatial positions,

temporal moments and material objects. Suppose first that we abstract from the second aspect and

interpret the state of quantity in light of its discreteness: then it seems that the constructed division

has revealed the ultimate constituents of the original quantity – atoms, in the case of matter, points,

in the case of space. But if we secondly abstract from the first aspect and interpret the state of

quantity in the light of its continuity, the result is just the opposite: the seemingly indivisible

constituents have a similar structure as the divided quantity – that is, they are quantities that can be

divided into as many parts as the original quantity has been currently divided.

3./301. Kant introduces four cosmological antinomies, second of which concerns the two aspects of quantity.

4./302. The value of Kant’s antinomies lies in showing that categories of former metaphysics describing states of finity

are inadequate in their content: the subjective idealism sees their fault merely in the fact that they describe the thought

of the subject. Yet, Kant’s presentation is full of unnecessary complexities: furthermore, Kant presupposes mistakenly

that the cognition can describe objects only through the finite categories.

The one aspect of Kant’s philosophy that Hegel respected highly was the discovery of antinomies.

The metaphysics before Kant had worked with conceptual pairs that were meant to be applicable to

every possible object: finite-infinite, simple-complex, free-determined etc. The metaphysicians

presumed that one and only one of the two predicates would apply to a certain object. Question was

whether a certain object or class of objects was characterized by the one or the other predicate. Kant

argued that in some cases it seemed equally probable to characterize an object with any of the two

predicates: e.g. matter could be described as divisible and indivisible. Hegel interprets Kant’s

discovery in such a manner that the categories in question can describe only some partial aspects of

the object, but not its true structure. In this manner Kant would have shown that these categories

were unfit or inadequate for describing certain objects or structures and their area of application

should thus be restricted. Note how Hegel contrasts Kant here with subjective idealism, although it

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usually seems that Hegel classes Kant as a subjective idealist. Hegel is saying that a subjective

idealist merely assumes that categories describe only the way how subject appropriates objects: this

assumption already presupposes that categories could not be used to describe objects as they are

without the subject or experience. Kant, on the other hand, at least tried to show the inadequacy of

categories without resorting to this sort of argument.

Kant’s endeavour wasn’t still a complete success for Hegel. First of all, Hegel accuses Kant

of unnecessary complexity: as we shall see, Kant aimed at providing proofs acceptable to both sides

of the argument, but ended up with circular proofs. The far graver mistake in Hegel’s opinion is that

although Kant had the opportunity of avoiding subjective idealism, he used antinomies merely to

underline the dependence of categories on the subject. Antinomies tell that the subject cannot

structurize or model totalities of objects correctly, because it can use only means fit for

characterizing only certain aspects of these totalities. Hegel wants to ask what more we could

possibly want to tell of such topics, except that they truly have aspects describable by the two

categories, which both then apply in a limited fashion to the supposed totality of objects: the two

aspects described by categories and the fact that these two aspects are mere aspects cover all the

necessary information for describing something through these categories.

5./303. Kant believed the four antinomies were all the antinomies that there could be, because of his schema of

categories: actually all concepts are antinomical or connect exclusive aspects, for instance, there could be antinomies of

becoming and being-here. The ancient skepticism aimed at showing antinomies in every scientific concept.

The fourfold division of the antinomies follows the fourfold division of categories into groups of

three, and as Hegel is not very convinced by Kant’s schema of categories, he is bound to feel a

disappointment when looking at his division of antinomies. As it is part of Hegel’s general strategy

to try to find connections of different aspects in every structure, he is surely against any attempt of

restricting such antinomies by some preconceived schema. For Hegel this is one step in trying to

find natural changes of context among different situations and objects – the dialectical part of it,

which helps in connecting seemingly different or even opposite structures to one another – although

it at first sight – especially when it is a case of a same object being structured in different manner

according to different situations – seems to be a mere attempt of disproving that these structures

truly fit to these objects: indeed, it was a method of ancient sceptics and academicians to show that

contradictory opinions had been held concerning the question investigated. Whereas for sceptics the

only result was one of indecision, for Hegel the discovery of such antinomies is itself revelatory of

the nature of the object: it is something that can be categorized in this way and also in another way.

Hegel mentions few examples, but does not explain what sort of antinomies one should find

in them. First of them is state of becoming or a process in which states of being and nothing change

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to one another. We might state the antinomy in question in the following manner: in one sense we

can always abstract from all that exists, and thus we may conclude that a state of nothingness exists

or is possible; on the other hand, we can always find something that exists – at least a situation in

which nothing exists – and thus we may conclude that there does not exist any state of nothingness.

The state of being-here or a structure, in which a state of being is connected with another state of

being, provides us with another antinomy even more easily. In one sense, a state of being-here is the

reference point – that is, according to itself – in another sense, it is a mere dependent background –

according to another situation. Another antinomy arises from the question of the relation of states of

being-here: in one sense all of them are similar in being states of being-here, in another sense they

are all different in having different qualities.

6./304. Further mistake is that Kant has mixed his antinomies with cosmological determinations like matter, space and

time, while the antinomies arise merely from the nature of the categories.

Kant’s antinomies were supposed to discredit the cosmological studies of the previous

metaphysicians by showing that they could not find any plausible answers to their primary

questions – or better, that they could find equally plausible, but contradictory answers to them. This

is not enough for Hegel. Antinomies – contradictory ways to interpret things according to different

contexts – are not the result of the thing interpreted, but of the structures or categories that the

things are supposed to instantiate. It is not just matter that can be seen as divided and divisible. On

the contrary, anything that is in some sense discrete is in another sense continuous and vice versa:

that is, anything that can be divided into pieces similar as it itself can be always divided further. In

fact, it is the structure of quantity that results in this antinomy: every number is already indefinitely

divisible and thus antinomical.

7./305. Kant says correctly that reason necessarily bumps into antinomies: “antinomies will always seem strange,

although their solution helps us not to despair over them”. Kant’s solution is to suppose that the antinomies depend on

the subject, which leaves them completely unsolved: true solution is that the contradictory characteristics can describe

only some limited aspects.

Hegel comes back to the themes of the paragraph 303, namely, the good and the bad sides of the

Kantian idea of antinomies. Kant did a service when finding antinomies: and this was not just a

contingent discovery – like if someone bumped into a possible misunderstanding that would be easy

to correct. Quite the contrary, the antinomies are something that one must constantly fight against,

in order not to be blinded by the light from either of its branches: it is too easy just to choose one of

the sides of the antinomy and hold dogmatically onto it. As soon as Kant then starts to discuss the

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solution for the antinomies, Hegel feels disappointed: indeed, it seems that according to Hegel Kant

just left the antinomy unsolved actually. Kant apparently hides from the awkwardness of the

antinomies by declaring that they are occasioned by the nature of how the things appear to us: the

things surely are not antinomical as they are in themselves, but we cannot say what they were in

themselves, because we see them only as they appear to us. Hegel suggests that we can resolve the

antinomies already at the level of what appears – and then we wouldn’t have to suppose any

characteristics of things in themselves that wouldn’t appear to us. Hegel’s solution is quite simple:

the two seemingly as convincing structures are equally true, that is, they describe some aspects of

things, but none of them describes the whole adequately: the whole structure of the thing would

then be given by a clause “it is this in one sense and that in another, both necessarily”.

8./306. Kant in fact merely assertorically states both sides of the antinomy, and hides this assertoriness with seeming

proofs.

9./307. Here we should discuss the antinomy of the divisibility of matter, which is based on the aspects of continuity

and discreteness in the state of quantity.

It may seem that Hegel’s criticism against the arguments of Kant is not very interesting – why

would he want to fight over some petty details? – but actually there is a clear reason why Hegel

would want to show that they are not true proofs. If Kant’s proofs would be acceptable, then the

experience of e.g. matter and its divisibility would contain some truly contradictory elements:

matter would be experienced as having contradictory predicates in one and the same context, and

thus we would have to assume that the correct account of matter would be something not appearing

to us. Hegel aims to show that Kant’s so-called proofs are actually constructions: Kant changes the

context by introducing some statement not true according to the context of one branch. Thus, we

could, firstly, accept Hegel’s solution that the branches of an antinomy describe different contexts,

and thus secondly, that we can with clear mind leave uninvestigated the possibility that the things in

themselves would have some further characteristics that wouldn’t appear to us – because it would

still be a mere empty possibility.

10./308. Thesis of Kant’s antinomy is that every complex substance of the world is a complex of simple substances and

that there are only simple substances and complex substances with simple substances as constituents.

11./309. Here simplicity is compared with complexity, which is an abstract structure compared with continuity: the

substances in the world mean here things that can be sensed or perceived, but they could be replaced with spaces or

times. Because the question is of complex things, the thesis is in one sense a tautology: complex thing is by definition

something that is combined out of other things [which are ontologically prior to it]. A complex thing can be

differentiated [by its complexity] only from simple things: thus, complex things must be made of simple things –

another question is how we know that an object is a complex. If the constituents of complex things are only contextually

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simple, then they are in some other context complex and we can identify the constituents with their complex: the

question would be unanswered.

It would be expected that Hegel would admit that thesis would work in some contexts: yet, it seems

puzzling when he says that it is even a tautology. The first part of the thesis seems to be easily read

as a sort of tautology: every complex entity consists of simple entities – whether the entities in

question are material substances, areas or events. We have here a twofold division. Firstly, we have

the entities that are called simple – we could say that they are ontologically prior – and entities that

are externally combined from them – we could call them mereological sums of the simple entities.

Like Hegel says, the combination and consisting relations – one is the reverse of the other, as

“atoms combine into molecules” correspond with “molecules consist of atoms” – demand that the

terms of the relation are different: it would not do to say that ink consists of ink or that ink

combines into further ink. If then in some level of abstraction this difference is not evident, then the

combination relation does not hold – at that level of abstraction at least. Now, there must be at least

one difference between the relata of the combination or consisting in order that the relation would

hold.

Supposing now that we start to abstract from all properties, what is the last property, besides

the property of being-combined-or-a-mereological-sum, that must remain in order that there could

be combination relation and thus something combined? Clearly there must be something

uncombined or something that is not a mereological sum of anything – in this context, that is. Note

that the possibility of entities in an intermediary state of combination is still possible. We might

have e.g. three levels of uncombined entities, entities combined from uncombined and entities

combined from the second entities – atoms, molecules and bodies, for instance. Here the entities of

the middle state would be uncombined relative to the further combined entities or uncombined in a

context. Yet, such a difference of combined and relatively uncombined cannot be the only one

separating levels of combination at this situation, because the difference between combined and

relatively uncombined arises only from the relation to the strictly uncombined entities of this

situation: there can be no bottomless relation of combination in any particular situation.

The latter part of the thesis seems to be more problematic to state as a tautology. The

problem arises from the fact that Hegel introduces yet another sort of determination, namely, the

continuity. The continuity seems to differ from both simplicity and complexity: whereas a complex

entity consists of something not similar to itself and a simple entity does not consist of anything, a

continuous entity has a similar, but not equal relation as consisting to something – a similar,

because these things can be taken as its parts, but not equal, because these parts can be identified

with the continuous thing. How then the existence of mere complex and simple things could be a

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tautology? The solution is actually quite simple. In every particular state of division there can be

only simple and complex entities: everything is either combined from something else – a

mereological sum – or it is uncombined, primary entity. Yet, it may well be – and in case of

quantitative things, it must be – that the simple thing of this situation could be further divided: the

division changes the context by making simple into a complex and creating new entities. The

continuity refers then to the identity or actually similarity between the simple entities of this state of

division and the possible further state of division: both are simple entities in their current state of

division, but could be divided further.

12./310. Kant’s proof of the thesis uses a fake reductio ad absurdum.

The apagogical form of Kant’s deductions is not arbitrary, but a statement that even an advocate of

the opposite opinion should be convinced of it. Thus, Kant begins with the assumption that the

thesis is wrong – that everything is infinitely divisible – and ends up with the result that the thesis is

still right – that there are simple entities. As we have already seen, Hegel’s tactic will be to show

that Kant actually changes the context in the course of his reasoning. When we are speaking of the

divisibility of substances, we relate different states of division to one another. Now, Hegel suggests

that Kant has forgotten that for every state of division there can be found a further state of division

and concentrates on one arbitrary state of division, leaving its connections to further divisions

unexplored.

13./311. Kant begins “supposing that complex substances do not consist of simple parts, then thinking all combination

away makes all substances vanish.”

14./312. This is a correct deduction, but this tautology could be left out and Kant could begin with “either we can think

away all combinations or then there exists something simple.”

The beginning of Kant’s proof could be read in two ways. Firstly, we can say that the complexity

here refers to the complexity Hegel spoke of in the paragraph 309. Then a situation with only

complex situations would consist of an infinite series of combined objects with no end at sight – an

impossible situation for Hegel and for Kant a situation impossible to cognize by human beings.

Clearly, if all combined things or mere mereological sums were wiped out, there would be no things

left at all. Secondly, we may assume that the complexity is actually also a misnomer for what Hegel

calls continuity, that is, a possibility of division. Then the situation itself might be a finite one, but

with a possibility of further division. If then all the objects that were either combinations or possible

to be divided were wiped out, no objects would also in this case clearly be left.

The next sentence of Kant’s deduction suffers also from this ambiguity: in fact, it is correct

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only in some senses, but in other senses it isn’t. The thinking away of differences could refer either

to the thinking away of complexity or to the thinking away of divisibility: furthermore, the

simplicity might either refer to something not complex or to something not divisible. Now the two

correct possibilities to read the sentence are, firstly, “either all complexity cannot be thought away

from a situation or there must be something not complex in that situation”, and secondly, “either all

divisibility cannot be thought away from a situation or there must be something indivisible in that

situation”. Kant, on the other hand, seems to interpret the sentence in the following way: “either all

divisibility cannot be thought away from a situation or there must be something not complex in that

situation”. Clearly the two sub-sentences of this disjunction can actually be true at the same time:

what is simple can still be divisible.

15./313. “In the first case complex entities would not consist of substances (because combination is not an essential

relation to them), thus, the complex entities should consist of simple entities”.

16./314. The ground of the argument has been hidden in parentheses: either the complex or the simple entities are

substantial, but because complex entities are mere arbitrary combinations, the simple entities must be the substantial

ones.

The ground of the proof, hidden in the parentheses, speaks clearly of thinking away all complexity –

combination of substances is contingent in the sense that the situation should remain essentially

same even if all the mereological sums were abolished. Thus, Kant is arguing for the conclusion

that there must be non-complex entities: the situation should remain essentially same, if all the

mereological sums were thought away – if we interpreted them as non-existent – thus, there must

still remain something which clearly cannot be a mereological sum, that is, a simple entity in the

sense of a primary, non-complex entity.

17./315. The thesis could be straightaway grounded in the statement that all combinations are contingent, which was

assumed in parentheses like an insignificant detail. True, all combination is contingent and the thesis is in this sense a

tautology, but we should be talking of continuity.

18./316. The proof of the thesis goes in the following manner. Suppose all substances are complex. But we can think

away all combination (because it is contingent), and then there would be no substances anymore. But we have assumed

the existence of substances, thus, there must still be simple substances.

Kant’s proof allows him to deduce correctly that there must be simple or non-complex substances in

every situation: indeed, Hegel has accepted this even as a tautology applying to all situations with

objects. Yet, Kant seems to think that he has proven that these simple substances are also indivisible:

if they are in this situation simple, they are simple in every situation. This identification of the two

statements would be correct only if there already were no divisible simple substances: if division

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would only unmake existing combinations, but would not create new substances. This was a

presupposition of Leibniz’s monadology and Wolffian philosophy, which Euler had already

challenged: we do not have to assume that parts exist before the wholes or that divisibility involves

compositionality.

19./317. Kant’s conclusion is that “there are primary simple substances to which combination is merely an arbitrary

condition”.

20./318. The arbitrariness of combination is here presented as a result, while it formerly was used as a hidden ground.

It is, in fact, strange that Kant does not here admit the possibility of divisibility without

compositionality, while in the Metaphysical first grounds of the natural science he explicitly states

that the monadological presupposition identifying divisibility and compositionality is not correct

when it comes to experienced matter. Kant apparently thought, like Hegel, that the thesis was

correct in any single situation or state of division. Now, he perhaps also endorsed the idea of a sum

or final limit of all states of division, at least as a conceivable possibility, whereas for Hegel such a

limit state would be meaningless because of the constructivist nature of his philosophy. In a limit

state, which we are naturally inclined to think as the final true state, there could be no further

division, yet also no indivisible simple substances, but only complex substances: while for modern

mathematics this would perhaps be acceptable as all infinite series are, for Kant and Hegel such a

bottomless series of complex objects or mereological sums would be unstable and perhaps even

incoherent, as we saw in paragraph 309. It is perhaps to account for this apparent contradiction why

Kant in the First grounds suggests the possibility that things in themselves would consist of simple

substances. Hegel has a simpler solution of denying that we could even think of any final state for

divisions: all states of divisions we know are such that could always be divided further.

21./319. Kant says he has not used any tricks in his proofs, but actually he has not even proved, but only assumed.

If we forget the for Hegel outlandish idea of a limit of all states of division, we must conclude, like

Hegel, that Kant has forgotten something. Kant says he is starting from the assumption that

everything is divisible: in Hegelian terms this means that for every state of division we can find

further, more divided state of division. Yet, Kant at once forgets this possibility and thus changes

the context of discussion. Kant shows that within a state of division there must be some simple

substances, but then ignores the assumption that there might be further states of division: he has

abstracted from all but the one state of division.

22./320. Kant’s antithesis says that no complex thing consists of simple parts and there are no simple substances.

23

23./321. Kant’s proof of antithesis is again a fake reductio ad absurdum.

If Kant in the proof of the thesis forgot the supposed divisibility of all states of division and only

pretended being in a context of indefinite divisibility, here Kant on the contrary, makes a false

statement of supposing the simplicity of pieces of matter, but then again changes the context by

ignoring this simplicity. Kant concentrates on the fact that every state of division can be divided

further. Yet, he either forgets the fact that every division results only in a further state of division

with simple substances or then he supposes the possibility of some final state of division in which

all divisions would be actualized.

24./322. “Suppose complex substances consist of simple substances. Because combination as an external relation

requires space, a complex substance consists of as many parts as the space it takes up: because space consists of spaces,

the parts of substance must take up space.”

25./323. “We supposed complex substances to consist ultimately of simple substances.”

Kant begins his proof by noting some general properties of matter. Firstly, there is the assumption

that all complex substances consist of simple substances. We noted in the proof of the thesis that

this can actually mean two things: either that all mereological sums in a given state of division

consist of substances which are not mereological sums, or that in some state of division there are

substances which cannot be divided further. The former is in some sense a tautology, while the

latter was the assumption of Leibnizian monadology. It is in fact not very clear which of the two

possibilities Kant is supposing here, and this is one of the reasons why his proof ultimately fails.

Secondly, Kant assumes the spatiality of everything complex as a given. Furthermore, he

accepts an apparently evident, but on a closer look ambiguous statement of space: space does not

consist of simple parts, but of spaces. This means either that a division of space does not consist of

undivided spaces – a clear falsity – or that space can always be divided into further spaces. On a

closer look this is actually the same ambiguity that made it difficult to interpret the other

assumption of the simple substances.

26./324. “Thus a simple substance would be spatial.”

27./325. “But everything spatial is a combination of parts that are substances and a simple substance would hence be

complex.”

Now, Kant produces a contradiction from the set of his assumptions: a) complex substances consist

of simple substances, b) everything complex is spatial and c) space consists of spaces. All parts of

spatial things are also spatial (c): thus, any simple substance which is a part of a complex substance

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or a mereological sum is also spatial (a, b). Now, everything spatial consists of further spatial parts

(c), thus a simple substance should consist of parts, which is a contradiction. Hence, because b) and

c) seem obvious (to Kant), we must deny the assumption a). The deduction depends on the

ambiguous statements a) and c), thus the result itself can be read in two ways: stating either that no

division of a substance would have any simple substances or that there would always be a

possibility of dividing substances further. Indeed, even the correctness of the deduction depends on

how we interpret the assumptions, as we shall see.

28./326. This deduction is a nest of mistakes.

29./327. a) Kant merely assumes that all complex things are spatial and that space does not consist of spaces.

Kant’s proof is based on three assumptions, one of which must be discarded because the implicit

contradiction derived from them. Now, Hegel makes the simple suggestion that we could discard

one of the other two assumptions, instead of the assumption of the simple parts of complex entities.

The suggestion seems even more tempting when we remember the ambiguous nature of all these

assumptions. For instance, the statement that space consists of spaces could be interpreted in two

ways: either as a statement that in any or some primary state of division space would be infinitely

divided – that the division of spaces would have no bottom – or as a statement that any division of

space could be further divided. Of these statements only the first one is clearly in contradiction with

the assumption that Hegel calls tautology, but it is also suspect in both Hegel’s and Kant’s

philosophy: it would be correct only in some final state of division.

30./328. b) Kant’s proof begins with the assumption that all (spatial) combinations are external, but then forgets it and

declares that spatial things consist of further spatial things: but simple substances should not be interpreted as spatial

[when spatiality is understood as a mere external combination].

Kant begins with a state of division, for instance, with substances a and b and their combination or

mereological sum ab. Now, Kant first defines spatiality as the contingent relation between

combinations and their parts: in this case, the relation of ab to a and b. This relation is contingent in

the sense that ab is only a secondary construction from a and b: the state of division wouldn’t

change essentially if we didn’t talk of the ab at all. The next step in Kant’s proof is to say that the

parts of space are also space or that parts of spatial things are also spatial things. But if spatiality is

interpreted as in the previous statement, a would in fact not be a simple substance, but a contingent

mereological sum of, say, a1 and a2. Yet, in this state of division it isn’t: we should change the

context before we could say that a is a combination, because it is a simple substance according to

this context.

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31./329. c) Kant supposes that space does not consist of simple parts, because it is an intuition instead of a discursive

concept, that is, it is given with a single object: the role of intuition in cognition has been overstated by Kant’s followers,

but even space must be conceptualized, if we want to conceive it. Even if space as intuited would be a simple continuity,

it can still be conceived as consisting of equally simple parts: thus, space, like time, is a victim of the same antinomy as

matter.

By intuition Kant meant a state of mind in which we apprehend some object or phenomenon as a

whole, instead of combining it from many partial or aspectual apprehensions. If space is an intuition,

it should thus be primarily a whole and only secondarily has parts: hence, it is perhaps divisible, but

not because of that a combination of simple parts. Now, Hegel contests, firstly, the value of such an

intuition, and secondly, the special nature of space because of it being an intuition. Intuition was

estimated as the ideal form of cognition among many post-Kantian philosophers, for instance, by

Schelling, because it apprehended wholes directly. Hegel notices – following Kant, actually – that

such an apprehension of a whole is only the beginning of cognition, because it still must be

analyzed in order to truly know it: thus, even space must be analyzed or conceptualized, before its

structure can be determined. Indeed, all cognition begins from an undetermined apprehension of

some whole, which must then be further determined by analysis: hence, space is no different from

any concrete object or phenomenon. Matter is at first an undifferentiated whole, but we can divide it

– in thought or in reality – and interpret the original piece of matter as a mereological sum of the

now existing parts. But the same procedure can be applied also to space: a continuous space can be

seen as combination of still undifferentiated parts. Hence, the space, as a quantity, is actually

antinomical in same manner as matter: if there is no problem in the divisibility of space, there

should then be no problem in the divisibility of matter.

32./330. In Kant’s proof the two aspects of space – space as external combination of parts and space as divisible

continuity – are used in one context, which results in an incorrect deduction.

If Kant in the proof of thesis ignored the relation that a state of division necessarily has to further

states of division, he now instead concentrates his attention so strongly to this relation that he

forgets that there actually are any determinate states of division. While doing this he confuses the

two aspects of space that were outlined in the two previous paragraphs. Kant begins by looking at

one state of division in which all mereological sums consist of simple substances: this division of

matter forms also a division of space into undivided, and in this sense, simple areas. In the

following sentence he then reminds us that the areas and therefore also matter within them can be

further divided: thus, he forgets that we were supposed to look at only this state of division and so

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changes the context. Kant has thus not been able to prove the divisibility of matter, but only to

assume it. He has also forgotten that after the division we are still just in another state of division

where all mereological sums still consist of simple substances, although these simple substances are

completely new to this state of division. The only place where this second fit of forgetfulness could

be allowed is the supposed limit of all states of division, which neither Hegel nor Kant could accept.

33./331. Kant reminds us that according to critical philosophy bodies can be conceived only as a phenomenon and thus

as spatial, but there is more to substances than what can be perceived. Kant suggests that reason should accept the non-

existence of simple substances because we haven’t empirically discovered them.

In the eyes of Kant Hegel would perhaps seem to suggest the unacceptable impossibility of reason

being able to determine what things are beyond their appearance for human consciousness: for Kant,

the space is the necessary way how external things appear to us and therefore we cannot conceive

how matter would behave without space. But Kant would have forgotten the context in which

Hegel’s discussion is presented, namely, the question of the divisibility of matter. Hegel has

suggested that a material object has two aspects: first, it is a simple object instead of a mereological

sum or a construct out of smaller objects; secondly, it is divisible into smaller objects. Now, the

spatiality as presented by Kant’s proof of antithesis is especially connected with the second aspect

of matter. Kant’s suggestion that all matter as it appears is always spatial implies then according to

Hegel that matter as it appears is primarily divisible: any state of division is always inessential in

relation to or at least as essential as further states of division. The primariness of the divisibility is

apparently based on nothing more than the fact that we can always divide matter into smaller pieces,

without any noticeable loss. Yet, Hegel asks, couldn’t it be possible that a division would take

something away, thus suggesting that one state of division would contain essential elements not

contained in a further state of division. One possible example would be a state of division with

living beings as simple substances, where a further division would not just create new substances,

but also destroy the living things, which most certainly are no mereological sums of pieces of

matter. What Hegel is suggesting is that we can recognize such natural wholes, and this recognition

happens within the experience, not outside it, although it requires conceptual interpretation in

addition to mere sensing of objects.

34./332. The proof of antithesis supposed the spatiality and thus continuity of matter, while the proof of thesis supposed

the inessentiality of the combination of matter and thus its discreteness: as discrete matter would be divided into units,

as continuous matter would be only divisible and units would always remain within an integrated whole. But continuity

as a divisibility implies the possibility of finding units, while discreteness as a similarity of units implies the possibility

of connecting them into one whole: the continuity and the discreteness are only aspects of matter.

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After having gone through a detailed criticism of Kant’s proofs it is quite easy to miss the true

solution Hegel offers for the antinomy. In the proof of thesis Kant abstracted from the divisibility of

a state of division and discovered that in any singular state of division all substances are simple; in

the proof of antithesis he abstracted from the particular structure of a state of division independent

of its relation to other states of division and discovered that a state of division is always further

divisible. Together with a confusion of indivisibility and simplicity these results led Kant to declare

that something was wrong, while there actually wasn’t anything wrong. Simplicity and divisibility

are different aspects of matter, or for that matter, any quantitative entity, like space and time. In one

sense matter has always been divided to simple, discrete parts; in another sense these parts are

always further divisible. Furthermore, the two aspects are essentially related to one another. The

divisibility of parts means that there is a method of taking this continuous unity as consisting of

smaller parts. On the other hand, in some level of abstraction we can interpret the parts born from

the division as versions or aspects of the original undivided quantity – the structure of the original

and the partial quantities are identical, that is, as the original quantity was divisible, so must also its

parts be. The contradictoriness is only apparent, because there is no final stage of division – the

stage of infinite dividedness – in which the two contexts would finally coincide into one.

35./333. Zeno’s paradoxes of motion show much better the antinomical nature of quantities than Kant’s antinomy: they

should be dealt in connection with space and time or in history of philosophy. The paradoxes result in Parmenidian state

of being and thus work as a method of Heracletian becoming: they are thus not to be discarded through sensuous

evidence, but solved through thinking.

Zeno argued that we could never move anywhere, because the movement would have to go through

infinite places, which would clearly take an infinite amount of time. This paradox is clearly based

on infinite divisibility of space and thus resembles Kant’s antinomy: we can always find a more

intricate state of division for a motion – we have to go here, but first we have to go there etc. Zeno’s

paradox of a flying arrow is of a similar nature. Now arrow seems to be flying, but the now in

which it is flying is a time of some length and not a true now or moment and must still be further

divided: the conclusion is that an arrow cannot be flying in a single moment of time.

What makes these paradoxes so ingenious in Hegel’s eyes? Firstly, they are not limited to

mere matter as Kant’s antinomy was. Secondly and more importantly, their aim is not to discredit

the subject apprehending them, but the supposed object, that is, the motion. Zeno shows that motion

always contains many apparently contradictory or incompatible aspects: this connection between

these different aspects is what Hegel refers to as the becoming of Heracleitus. The result should be

the Parmenidian being, that is, Zeno suggests that we should give up all apparent structures in

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which incompatible aspects are united, and so only a single, static state of being would be left. This

result is, of course, as or even more unacceptable to Hegel as Kant’s result of antinomies. Hegel’s

suggestion, on the other hand, is that we should accept that motion, matter and such are incomplete

and aspectual according to every context: motion can be always further divided, but no final,

infinite state of division can be found and thus motion does not go through an infinite amount of

static moments in any context or state of division.

36./334. Aristotle solved Zeno’s problems correctly by stating that divisibility meant only a possible infinity: Bayle

insisted that divisibility implied actual infinity of parts, but ability does not mean actuality. Understanding is confused

by such non-entities as infinite collections.

The participators in the Zenonian problematic abound, so it is helpful to see who is arguing for what

side and especially who Hegel is arguing for. The crudest opinions are those of Diogenes who is the

representative of the senses. We sense motion as a unified phenomenon – this was there, but now it

is here – and according to such a sensation the non-existence of motion would be absurdity. But

nothing seems problematic before it has been analyzed, and Diogenes’ solution is actually a mere

denial of any problem and thus a step backwards. Analysis or understanding reveals then the

problem, but who is supposed to be the representative of the understanding? Hegel congratulates

Aristotle as having great understanding – a sense of analysis, that is – but this seems more of

remark: Hegel refers here to the discovery of syllogistic by Aristotle. It is actually Bayle with his

insistence that actuality of parts must follow of their possibility who is the true man of

understanding – and because of this, he cannot get out of the Zenonian problems. But it is the

Aristotle who has the true speculative answer to the problem: the aspects of motion analyzed by

Bayle and used against motion itself by Zeno are noticed to be mere aspects by Aristotle. Space and

time can be interpreted as both divided and divisible, but neither dividedness nor divisibility is the

only true aspect of space or time: furthermore, even space and time are abstractions out of the

concrete relations of material and moving objects.

The great invention of Aristotle is that infinite divisibility or continuity implies only a

potential infinity, but not an actual infinity: we have said the same thing by stating that a series of

states of divisions need not have any final limit which it would approach even asymptotically.

Indeed, this is a clear consequence of Hegel’s constructivism – nothing can be spoken of, before it

has been posited – that there can be no actual infinities: because we are capable of adding by

construction or discovery only finite number of objects to our current situation, we can never reach

a situation with infinite number of objects. We can never think of a quantitative infinity – although

here Hegel speaks of quantitative infinity as a mere thought object, he usually says that it is a mere

representation, that is, a word with no guaranteed or even possible object. Kant seems to be saying

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something similar by excluding such infinite from what we can cognize or know, but he still

suggests that we could think it as a possibility: here we see an example of how thought has in some

sense a wider area for Kant than for Hegel.

37./335. Kant’s solution is to return to the level of sensations and appearances: he does not provide the unity of quantity

to the sides of the antinomy.

Hegel’s judgment of Kant’s solution is perhaps a bit harsh. Both Kant and Hegel accept that matter

has two aspects of being divided and being divisible: the analysis of matter reveals similar facets for

both philosophers. What Hegel is missing is the acknowledgment that these two aspects are

necessarily connected in matter, and indeed, in any quantitative structure: the correct answer to

antinomy of matter in Hegel’s opinion would be that matter is in some sense still divisible and in

another sense divided to simple parts, but neither of these senses alone would be the whole truth of

the matter. Kant cannot accept Hegel’s solution, because he is still haunted by the possibility of the

final or limit state of division: after an infinite number of divisions there would be a bottomless

series of mere mereological sums, which would be absurdity in eyes of both Kant and Hegel. Kant

thus must have recourse to the idea that it is the cognition that somehow causes this apparent

contradiction and that the true nature of matter is hidden from us: while for Hegel the apparent

contradiction is caused by the belief in the absurd possibility.

B. Continuous and discrete magnitudes

This section is a bit, although not much more informative than the previous one: we learn the quite

obvious thing that a quantity can be regarded according to both of its aspects separately, although

even then there remains the possibility of interpreting it according to the other aspect. Thus,

although the title of the section speaks of continuous and discrete magnitudes, clearly any quantity

could be interpreted according to either aspect, depending on which context we are looking at

things (of course, there might be some quantities, which are more naturally taken as continuous

magnitudes, and others, which are more naturally taken as discrete magnitudes). There is a slight

development going on, as we notice the possibility of dividing quantity and thus multiplying both

the number of units and the number of quantities consisting of the units. This development makes

the original quantity then determined, although the true apprehension of this possibility is

artificially left to the next section. The section is divided in a rare manner to two subsections, yet

quite obviously according to the division of continuous and discrete aspects.

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1./336. 1. Of continuous magnitudes. A state of quantity has aspects of continuity and discreteness, and it can be

interpreted according to each of these. At first we are given a unified state of quantity, thus it is already regarded as

continuous.

We have already stated the primary differences between the modern and the Hegelian notion of

continuity: for Hegel continuity refers to that aspect of any magnitude where it is taken as a unity,

while in the modern mathematics the continuity refers to a certain sort of quantities or sets having

more than a discrete infinity of members. We may trace both notions to their common ancestor, that

is, the undeveloped idea of continuity in pre-modern mathematics. What properties a continuous

magnitude, a primary example of which could be a line, seems to have, compared to a discrete

magnitude? A continuous line seems, firstly, to be unbroken: there are no holes in it, but it covers

all the points within its range. Secondly, it seems to be not a mere collection of points, but a more

tight unity. The modern mathematics has emphasized the first aspect by saying that a continuous

magnitude is a certain sort of infinity: the problem is that at the same time it has made a line into a

mere collection of points, which makes it discrete in the eyes of Hegel. Hegel, on the other hand,

has concentrated on the unity of the continuous magnitude: when we call a quantity continuous, we

mean that all its apparent parts are mere aspects of the whole. The problem in Hegel’s definition,

when related to the original idea of continuity, is that then the difference between continuous and

discrete magnitudes is not one of species, but of different interpretations or states of quantitative

structures. Furthermore, because a Hegelian continuous magnitude is a unity, it is not even an

actually, but only a potential infinity, thus making it discrete in the eyes of modern mathematics.

Why is the continuous magnitude introduced in the first place instead of the discrete

magnitude? The first reason is obviously that we are then not introducing anything new, but only an

old friend with a new name, that is, the pure state of quantity from the previous section: a pure state

of quantity is a quantity that can be divided or multiplied, although it is currently taken as a unity.

But there is also a more substantial reason behind Hegel’s choice of order. We have begun by

describing a quantity in abstraction from all other quantities and our aim is to relating this originally

quantity to other quantities; the other possible direction, in which we first had a quantity in relation

to another quantity and then separated the quantities from this relation, would be a case of mere

abstraction. Now, as we shall see in a while, a discrete magnitude already contains a reference to a

continuous quantity, as a multiple of some continuous quantity, and it thus implicitly is a quantity

related to another. Hence, it is then more natural to begin with a continuous magnitude and only

later on construct an example of a discrete magnitude.

2./337. Continuity is one aspect of quantity together with discreteness. But these aspects are distinct, and we can thus

emphasize one or the other as the actual or reference aspect. If we take the aspect of continuity as designated, then we

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have a unity of potential discrete units: the result of this construction is a continuous magnitude.

The previous paragraph noted that it is already natural to regard the pure or unrelated quantity

according to its aspect of continuity. This paragraph emphasizes that like in a state of becoming we

have to have the possibility to take either of its aspects as the reference point: in the case of

becoming we could either say that here is a state of being, but there could be a state of nothing, or

vice versa. Now we can say, for instance, that here is a state of continuity – or that the quantity in

question is taken as a unity – but it could be taken as discrete – that the quantity could be divided

into further quantities: for instance, a continuous line can be seen as a combination of three different

lines. This is another definition of a quantity interpreted as continuous: it is a unity of potential new

quantities or a whole divisible to further divisible wholes.

3./338. 2. Of discrete magnitudes. A quantitative situation in its first state is continuous, but a quantitative situation

cannot be exemplified merely by one state, but only by a series of states. We should thus construct a situation according

to which the previous state seems like an external state of combination, or we should divide it to a discrete combination

of units.

The construction of discrete quantities from a continuous quantity – or a construction of a context

where a continuous quantity of one context could be interpreted as a discrete quantity – seems

rather simple, because a continuous quantity contains the possibility of viewing it as discrete: given

a line, we can cut it to pieces and then we have a combination of many lines. Hegel uses another

strategy to introduce quantities interpreted as discrete, but on a closer look the difference of

methods is not very great. Hegel merely once again explains why a continuous magnitude must

have the potentiality of being viewed as discrete. A quantity in general, Hegel argues, has been

defined in such a way that it cannot be characterized by a mere one state of being or situation, but

by a series of such states: for instance, a line is not a quantity because it is unified, but because it

can be seen both as unified and as divided to further lines. Thus, we must be able to divide a

magnitude: we must be able to interpret the units, which were previously thought as mere ideal

aspects, as independent entities. Indeed, we must even actually divide the magnitude, in order that

we could really show that it is a magnitude.

4./339. An aspect of discreteness can be emphasized in a state of quantity, but even then it is possible to see the quantity

as continuous. In a state of quantity there are many apparently separate objects or situations; when we emphasize the

continuity, they are taken as aspects of one identical object or situation; if we emphasize the discreteness, some of them

are taken as independent units. Still, these units are not completely unrelated atoms: they can be taken as identical

because of their similarity in being instances of same unit type.

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The possibility of dividing a continuous quantity into more quantities seems quite natural compared

to the way how discrete quantities could be taken as continuous in Hegel’s opinion. Hegel uses here

the by now familiar idea that similar things can be taken as one identical thing in different contexts:

one meter is as good as the other, so we could see all lines of meter as the same line in different

places. We might say that Hegel uses here the idea that any meter could be mapped to another

through some function retaining the length, but not the position of a line. But Hegel’s idea implies

even more, namely, that a whole quantity could also be mapped to any of its parts, that is, any part

of a quantity could be taken as a quantity. Suppose for instance a line that we have divided to three

parts: we now regard the originally undivided line as a combination of these three parts. Then the

whole line can be mapped to any of its parts – not by a function that retains the relative size of the

original line, but certainly by a function that reduces the size of this line in a constant fashion. This

mapping still retains some essential features of the original line: for instance, any of the parts is

such that it can also be divided to three further parts.

Remark

The aim of this remark is to explain how the traditional species of continuous and discrete

magnitudes can be modelled through the constructions of Logic: the main aspects of this

explanation have already been introduced. As we have already seen, Hegel has to modify the

original idea of these species in order to fit them with his theoretical framework: species of

quantities become aspects or interpretations of quantities. Yet, it seems that some sort of

modification was necessary in order to make these imprecise notions more precise: the modern

mathematics did other modifications to make them compatible with the set theory using classical

predicate logic.

1./340. It is usual to ignore that continuous and discrete quantities contain both aspects of quantity. For instance, such

continuous quantities like space, time and matter can be divided in reality and not just in thought.

2./341. Similarly discrete quantity is continuous, in the sense that it repeats the same unity.

We have spoken of the differences between the informal, Hegelian and modern notion of

continuous quantities: informally continuous magnitude, like a line, is an unbroken unity containing

all possible objects or situations – points – within some area; for Hegel, continuous magnitude is

any quantity interpreted as an unbroken unity; and for modern mathematics, continuous magnitude

is a set of uncountable number of members or points. Interesting is Hegel’s separation of absolute

and merely empty possibilities: we shall later meet this division as the division to real and formal

possibilities. A tree could grow at this particular instant here, where there happens to be a rock, but

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there is no method of making this possibility actual. A continuous magnitude in the Hegelian sense,

on the other hand, is possibly a discrete magnitude in another sense: we can truly divide the quantity

and then interpret it anew as a combination of other magnitudes.

We may briefly indicate the differences between what discrete magnitudes mean for pre-

modern mathematics, Hegel and modern mathematics. If a continuous magnitude meant originally

an unbroken magnitude, a discrete magnitude, on the contrary, was a broken magnitude, that is, it

was not a unity, but a combination of many units. In modern mathematics, all sets are broken in this

sense – they are all collection of independent members or points – and discrete sets can be separated

from continuous only by the difference of their cardinality or size: a continuous set cannot be one-

to-one mapped to a discrete set. For Hegel, on the other hand, a discrete magnitude can be merely

an interpretation of possibly continuous magnitude: the question is whether the supposed parts of

the quantity are taken as independent or as mere aspects of the whole quantity.

3./342. We can take continuous and discrete quantities as species of quantities, but their division is not occasioned by

any external ground. Furthermore, they are not yet taken as different quantities, but as one quantity in different forms:

they can be called magnitudes, because they are more determined than an undetermined quantity.

The division of continuous and discrete quantities is, for Hegel, an example of what an ideal

division of a concept or structure should be like. In comparison, division of triangles to right-,

acute-, and obtuse-angled uses the sizes of the angles as a ground of division, taking especially the

right angle as a reference point, while there is no indication in the general definition of a triangle

that a right angle would have any more meaning than angles of other sides. In Hegel’s division of

quantities, on the other hand, we can at once see from the general structure of quantities – that they

have aspects of continuity and discreteness – that there is a possibility of taking either of the aspects

as the primary one.

Although Hegel speaks as if he had divided quantities, he at once admits that it is a rather

peculiar division he has made, because it does not divide quantities into two classes, but only

interpretations of quantities: every quantity could be taken either as continuous or as discrete. Yes,

it determines a quantity, when we interpret in either way, but this interpretation does not yet

separate the quantity from any other quantity. If we wanted to make a regular division of this

division of interpretations, we would have to decide – probably using some external criterion –

which quantities are to be interpreted in a certain way: this is how Hegel could explain the fact that

we usually classify e.g. lines as continuous magnitudes, but collections of pieces of matter as

discrete magnitudes.

C. The limitation of quantity

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The structure of a pure state of quantity has been introduced and we have seen two different aspects

of it – continuity and discreteness – either of which could be taken as the primary or reference

aspect of the whole state. The only thing left to do is to relate this given quantity to other quantities.

The task is fairly simple. While changing the interpretation of the quantity from continuous to

discrete we had to either divide or multiply the original quantity – depending how we interpreted

the change – and in both cases we must have created also another quantity to which the original

can then be related.

1./343. In one sense, we can find from discrete magnitude some unit; in another sense, we can find many other units; in

yet another sense, all these units can be seen as aspects of the same unit. Thus, a discrete magnitude can be seen as a

united state of quantity. Furthermore, in this particular interpretation it can be compared with e.g. its continuous unit

quantity: when it is not related to any other quantity, it does not look limited, but thus compared it seems limited.

Hegel begins with going through all the details in the structure of any discrete magnitude: we can

find a unitary situation or object, and from that situation or object we can find other situations or

objects and finally we can interpret all these situations or objects as aspects or copies of the same

unit – here is a meter, there is another, but they are both the same meter, that is, of the same size,

although situated differently. Now, the number of the units of a discrete quantity – as a quantity in

general – is not limited, just as long as we change the unit in question: two meters is twenty

decimeters. Quantities are infinitely divisible, therefore there is no natural unit that we should prefer.

Yet, we may fix the unit – which is then fixed as the continuous quantity, because it is taken as not-

divided – and this fixing of the unit is what Hegel calls positing the one: we choose one state of

division as the primary or reference state. Thus, we can separate the discrete from this continuous

quantity or the combination of units from the unit – which is then not an interpretation of the

discrete quantity, like one “double-meter” is an interpretation of two meters, but another quantity:

two meters can be separated from a meter. Hegel describes the situation by saying that it is the

chosen division of units that limits the quantities: without it we couldn’t separate the quantities from

one another. We are then back with states of being-here – with a “world” divided into different

situations. The only difference is that this time the separating characteristics are supposed to be

quantities: what is two meters is not a meter long.

2./344. The fixation of the unit separates quantities that could formerly be taken as continuous, but it also unites certain

units in states of quantities. We need not speak of any objects yet, but quantities can consist of mere situations [like

space does]. Unities in one state of quantity can be taken as aspects of the same unity: this makes the discrete quantity

into a state of quantity limited by others.

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The quantities have been defined as structures where we can always add new units and see those

units as part of the same whole. It has been left implicit whether the units in question are objects or

mere situations: indeed, because space and time are examples of quantities, we should allow the

possibility that quantity can be a structure consisting of situations. Hegel notices this when he says

that we are not perhaps limiting objects, but states of being. For instance, we have not related object

of one size – a two-meter long rod – to another – a meter long rod – but merely situation of one size

– a two-meters long place – to another – a meter long place.

Now, all such limits are indifferent in the quantitative case: there is no reason why we

should separate place of this size from place of that size and there certainly is no qualitative

difference between them. It is merely an arbitrary decision that has made us choose one state of

division over another and that makes it reasonable to speak of fixed quantities, but once this

decision is made, the sizes of the quantities are fixed. Yet, the arbitrariness of division and the

possibility of seeing different units as aspects of one unit remains valid also within the arbitrarily

differentiated quantities: both meters of a two-meter long place are completely similar as quantities.

Now, it is this possibility of interpreting the discrete units as a continuous whole, Hegel says, that

allows us to speak of a particular quantity as a unity: when we abstract from all the other quantities,

the aspects of a quantity can still be discerned within one quantity left to us.

3./345. It is indifferent whether the units of a fixed state of quantity are taken as independent or merely aspectual,

because both interpretations of the quantity are equally possible: thus, both the quantity fixed as continuous and

quantity fixed as discrete have been related to other quantities.

The quantity that was related in the previous paragraphs was interpreted as discrete, but the question

is whether continuous quantities can also be so related. Now, a related quantity can still be taken

either as continuous or as discrete, that is, we may arbitrarily change the state of division of our

system of quantities: the difference of continuous and discrete interpretation is merely the

difference of whether to take the quantity as a unified field or as a combination of units. A discrete

quantity can be related to its actual parts – a combination of two meters differs from those meters –

while a continuous quantity can be related either to quantities of which it is an actual part – a meter

differs from two meters – or then to its potential parts – a meter differs from a centimeter. Hence,

we may begin also with a continuous quantity, interpret it at first as discrete and relate it to its part

and then again interpret it as continuous. Thus, it is indifferent from which interpretation we begin

the relating, because the result will be same: we have a state of quantity that can be taken as a unity

or as a multiplicity and that has been related to another state of quantity.

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Glossary:

Quantität = any potentially infinite field of units, that is, a structure in which we could differentiate

indefinitely many units that still remain mere aspects of the whole quantity; Grösse (magnitude) is

sometimes a synonym, although often it means a quantum

Reine Quantität = any quantity when it is not related to or compared with any other quantity

Continuität, Stätigkeit = the aspect of quantity that any part discerned within the quantity is just a

dependent part of this quantity, and indeed, just another quantity

Discretion = the aspect of quantity that we can distinguish various seeming parts within the quantity

Continuirliche Grösse = quantity, in which the aspect of continuity is emphasized; that is, an

undivided continuum that could be potentially divided into further quantities, although these

quantities can exist only after such a division

Discrete Grösse = quantity, in which the aspect of discreteness is emphasized; that is, a collection

of units that are similar to one another (note that at least in paradigmatic case the units are also

quantities)