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Critique of Pure Reason

Lecture Notes: Space

G. J. Mattey

The first part of the Transcendental Doctrine of the Elements is the Transcendental Aesthetic. It is quite short relative to the rest of the book, but it has been extensively studied, partly because it lays the groundwork for the much lengthier Transcendental Logic, and partly because its doctrine is much less complex and easier to grasp than those of Transcendental Logic. The Transcendental Aesthetic is devoted to an investigation of space and time as well as a crucial consequence of the account of space and time that Kant adopts.

In ancient philosophy, the word ‘aesthetic’ signified primarily that which pertains to the senses, as opposed to what pertains to the intellect. Kant’s investigations in the Transcendental Aesthetic are directed at a power of the human mind he called “sensibility,” and so his use of ‘aesthetic’ is appropriate. Kant mentions in a footnote at A21/B36 that “the Germans” had given the word a new meaning, pertaining to the critique of taste. It has turned out that this usage has prevailed over Kant’s, so that ‘aesthetic’ nowadays is almost universally used in connection with the beautiful or more generally with works of art.


We saw in the last lecture that Kant divided cognitions, that is, objective perceptions, into concepts and intuitions. The logical difference between the two is that concepts refer in a general way to one or more objects, while intuitions refer specifically to a single objects. In §1 of the Aesthetic, Kant adds that it is only in intuition that objects are referred to directly, and that all thought “aims” at intuition as “a means,” apparently as a means to refer to objects.

A central Kantian assumption is that for human beings, all intuitions are given to the mind, rather than being created spontaneously by the mind, and they are given only as the mind is “affected in a certain manner.” The capacity of receiving presentations “as the result of the way in which we are affected by objects is called sensibility.” (A19/B33) So sensibility is required in order for the mind to be supplied with intuitions. Concepts, which are products of the understanding, are the ways in which objects are thought. They refer to objects through intuitions by means of characteristics (components of the concepts) which apply to the intuited objects themselves.

Sensations, it will be recalled, are subjective perceptions. In the Aesthetic, Kant describes a sensation as “the effect of an object on our capacity for presentation, insofar as we are affected by the object” (A20/B34). So an object may be presented as an intuition through sensation. In that case, the intuition is called empirical. Considered merely as that to which an intuition refers through sensation, and not with respect to any of its specific features, the object is called an “appearance,” or “the undetermined object of an empirical intuition” (A20/B34).

Having defined appearance, Kant distinguishes between its matter and its form. That in the object that corresponds to sensation is called the matter of the appearance. Generally, the matter of appearance consists of what in the object corresponds to a “manifold” consisting of a number of sensations given to sensibility. That which gives order to the manifold is called its form. It may help to recall that in the Amphiboly, Kant had said that in general, the matter is the determinable and the form is the determination. The sensations are the elements whose features are to be determined. That which gives order to them, or determines them, is what accomplishes this.

Now Kant makes his first substantive claim, though he does not advertise it as such. The form of an appearance must be distinct from its matter. “That in which alone sensations can be ordered and put into a certain form cannot itself be sensation again” (A20/B34). This way of distinguishing the matter and form pushes us in a certain direction in construing what the form is. That is, the form is some kind of thing “in which” sensations are placed into an order, rather than being simply a pattern to be found in the sensations themselves.

If there is to be something in which sensations are ordered, and it is not to be found in the sensations themselves, Kant concludes that it must be a product of the mind. The way Kant puts it is that something, the basis of order, remains when we strip off everything sensation and the understanding contribute to intuition. This he calls “pure intuition” (A20/B34). As an example of the process of isolating the form, Kant uses the intuition of a body. Sensation contributes impenetrability, hardness, color, and many other of a body’s features. The pure understanding conceptualizes the body as a substance that is a seat of force and is divisible. Strip these away, and you are still left with extension and shape. “These belong to pure intuition, which, even if there is no actual object of the senses or of sensation, has its place in the mind a priori, as the mere form of sensibility” (A21/B35).

Since the form of sensibility is a priori, its treatment is “transcendental” rather than empirical. So the Transcendental Aesthetic is the treatment of the a priori forms which give order to what corresponds to our sensations in empirical intuition. There are two forms of intuition that apply to appearances: space and time. Kant claimed that appearance has no other forms. “This is evident from the fact that all the other concepts belonging to sensibility presuppose something empirical,” as even motion and change do (A41/B58).

The Concept of Space

Section I of the Aesthetic is entitled “Space.” In the second edition of the Critique, it is divided into three parts, a “metaphysical” exposition of the concept of space, a “transcendental” exposition, and a very important set of conclusions drawn from them. These conclusions are supplemented by a section containing “General Comments” on the Aesthetic as a whole. This section occurs after an examination of the concept of time, which has the same three-part structure as does the examination of the concept of space.

An exposition of a concept is a “clear (even if not comprehensive) presentation of what belongs in a concept” (A23/B38). Thus it seems that in giving an exposition of the concept of space, Kant is giving an analysis of that concept, so that the judgments he makes here with the concept of space as a subject would be analytic. (For example, two of the characters of space are its being infinite and its parts being simultaneous, see B40.) A metaphysical exposition is one which “contains what exhibits the concept as given a priori” (A23/B38). One question we must ask at the outset is how “what belongs in a concept” can exhibit the way in which the concept is given. On the face of it, the way the concept is given is not one of the characteristics that makes it up.

Kant begins the exposition of the concept of space by making some uncontroversial claims. We present objects as being “outside us” by means of “outer sense.” Objects that are presented as outside us are presented as “in space.” It is because objects are in space that they have some of the properties we attribute to them, such as “their shape, their magnitude, and relation to one another” (A22/B37). These are not claims about what belongs in the concept of space, but rather about how other presentations, presentations of objects, relate to space.

The Ontology of Space

The first fundamental question to be answered by an exposition of the concept of space (as well as of the concept of time) concerns the ontology of space and time: “What, then, are space and time?” (A23/B37). Here we are given two main choices, the second of which branches into two further choices.

The second fundamental question depends on the answer to the first question. If space and time are not actual beings, and hence are dependent in some way on things, then there are two possibilities.

Regarding the first question, it should first be noted that Kant here refers to space as being singular. He asks his ontological questions about space rather than spaces. Kant is presupposing here that the concept of space has only one referent rather than many. He will later claim that spaces are simply parts of one all-embracing space (A24-5/B39). So is this one space an actual being, a property of things, or a relation between things?

This question lies at the heart of the conflict between Leibniz and Newton over the nature of space, as played out in the Leibniz-Clark correspondence. Newton had given the first answer to the question: “absolute” space is a real being. He thought that he had to postulate the reality of space in order to account for real, rather than relative, motions of bodies.

Leibniz contested the need to postulate the existence of real space. We have already seen that Leibniz, as Kant understood him, considered space to be a form or determination of things whose matter or determinable is monads. His view seems to have been that space is a relation among bodies, an “order of co-existence.” For (Kant’s) Leibniz, then, things in space are metaphysically more basic than space itself. Strictly speaking, the monads which are ordered by means of space have no real spatial properties. Rather, as substances, all their properties are intrinsic, which could only mean that the real properties of are mental states of some kind.

The ontological questions Kant has posed at the beginning of the section on space are not answered immediately. Rather, Kant proposes that, “In order to inform ourselves on these points, let us first give an exposition of the concept of space” (A23/B38). It will turn out, somewhat surprisingly, that the concept of space is the concept of a certain component of intuition.

Space as A Priori Presentation

Kant begins his exposition of the concept of space by claiming that it is “not an empirical concept that has been abstracted from outer experiences” (A23/B38). This claim is purely negative: it does not assert that space is an a priori concept. Note also how broadly Kant construes what does not “belong to” the concept of space (as is required in an “exposition” of the concept). As noted above, the way in which a concept is given counts as something that “belongs to” it.

So why is space not an empirical concept? The reason is that the presentation of space is required to make “outer experience possible in the first place” (A23/B38). The argument is rather tricky, and here is a way of reconstructing it as a reductio ad absurdum.

  1. Suppose space is an empirical concept.
  2. If C is an empirical concept, then anyone with the concept C abstracts it from experience of objects to which it refers.
  3. So, anyone with a concept of space abstracts it from experience of objects to which it applies.
  4. I have a concept of space.
  5. So, I abstract the concept of space from my experience of objects to which it applies.
  6. If I abstract the concept of space from my experience of objects to which it applies, then I abstract it from my experience of objects presented as standing in spatial relations to me and to one another.
  7. So, I abstract the concept of space from my experience of objects presented as standing in spatial relations to me and to one another.
  8. If objects are presented as standing in spatial relations to me and to one another, then they are presented to me in relation to space itself.
  9. If objects are presented to me in relation to space itself, then there is a presentation of space itself independently of the objects of experience presented in relation to it.
  10. If there is a presentation of space itself independently of that of the objects presented in relation to it, then the presentation of space is not abstracted from the objects of experience presented in relation to it.
  11. So, the presentation of space is not abstracted from the experience of objects related to it.
  12. So, space is not an empirical concept.
Given that this argument is valid, the question is whether it is sound. The substantive claims in the argument are found in steps 2, 6, 8, 9 and 10.

While any of these five conditionals might be disputed, step 8 seems the least obvious. In effect it precludes a relative concept of space which is based on internal characteristics of objects, as on Leibniz’s view. Kant appears to be presupposing that space itself is the basis of all spatial relations. It does seem that such a concept of space is not an abstracted one, but instead is a theoretical concept invoked to explain the spatial relations we discover in experience. Another conditional that might be questioned is step 2. One could object that the concept of space is abstracted from experience of spaces. In that case step 8 might be denied, as the objects presented to me could be related to spaces rather than space itself.

The first argument has a negative conclusion. It tells us that space is not a concept abstracted from objects of experience, but it does not tell us what the origin of the concept might be. The second argument is intended to show that “space is a necessary a priori presentation that underlies all outer intuitions” (A24/B38). A reconstruction of the argument for this conclusion will again be presented in a step-wise format.

  1. If D is presented as a determination of object O on the basis of experience, then O can be presented without determination D.
  2. So, if being in space is presented as a determination of an outer object O on the basis of experience, then O can be presented without being in space.
  3. But no outer object can be presented without being in space.
  4. So, being in space is not presented as a determination of an outer object O on the basis of experience.
  5. Outer objects O are presented as being in space.
  6. If an object O is presented as having a determination D, then D is presented as a determination of O either a priori or on the basis of experience.
  7. So, space is presented either a priori or on the basis of experience.
  8. So, space is presented a priori.
We can take premises 3, 5 and 6 to be relatively uncontroversial. This leaves us with premise 1, which is a kind of Humean principle. The idea is that experience can never guarantee the presence of any determination of an object, as it is always possible to conceive of an object without that determination. Another way of putting the matter is that determinations of objects based on experience are not “essential” to them. However, this does not seem to be true in general. It does seem as if some characteristics of things are essential, given the manner in which they are presented. If a body is presented to me in experience as being extended, can I really say that it can be presented in experience as not being extended?

Space as Intuition

Now Kant has ruled out the presentation of space as being an empirical presentation of any kind, whether concept or intuition. His next task is to show that the presentation of space is not an a priori concept, which requires him to show that it is not a concept.

A concept presents an object discursively, by attributing to it characteristics that can be shared by other objects. So a concept of space would have to describe space using common characteristics. There is nothing about this procedure that would rule out the possibility that a concept could apply to a single object. Even Kant himself seems to have held that the concept of a supreme intelligence would apply to only one object, if any (see A672-3/B700-1, for example).

Kant’s argument was that the presentation of space is not an empirical concept, on the grounds that “we can present only one space; and when we speak of many spaces, we mean by that only parts of one and the same unique space” (A25/B39). However, the difference separating concepts from intuitions is that intuitions present objects directly, while concepts present them indirectly through common characteristics. So it is irrelevant how many objects fall under the presentation of space, i.e., whether or not “space is essentially one.”

A second point Kant made in this regard is that the parts of space cannot be considered as constituents from which space is, so to speak, assembled. Space is given first, and the parts of space are thought in relation to space itself. But again, even granting that what we call “spaces” are parts of one space rather than objects falling under a general concept of space, it does not seem to follow that we can have a general concept of space under which exactly one object falls: space itself.

Kant had a further argument in the B edition that might be thought to suffice to establish his conclusion. It begins with the premise that we present space as infinite given magnitude. That is to say, we think of space “as containing an infinite multitude of presentations within itself” (B40). Perhaps what Kant had in mind here is that we present space so that any quantity of it is infinitely divisible. He went on to claim that a concept could never contain such an infinite multitude of presentations within itself. Instead, it can only comprehend infinitely many presentations “under itself.”

This argument seems to have the same problem as the original ones. Suppose we do present space as an infinite given magnitude. The most plausible reading of this claim is that space itself, what is presented by the presentation (whether concept or intuition), is infinite in one way or another. Although each of the infinitely many parts of space can be presented individually, this does not mean that the presentation itself contains infinitely many partial presentations. So the fact that a concept is not divisible in the way that space is does not mean that it cannot present a single object which is divisible in this way.

Perhaps a better argument could have been based on the phenomenon of incongruous counterparts, which is invoked in the Prolegomena but is not found in the Critique. This argument is of a sort one finds frequently in contemporary philosophy, where a concrete counter-example is offered against a general claim. The concept of a right hand and the concept of a left hand are indistinguishable conceptually: their common characteristics are exactly the same. But they are distinguishable by experience. We can see that the two are different. So, at least some of the spatial features of the object are presented in intuition, not conceptually. This conclusion might be generalized to the claim that all spatial features of objects have their origin in intuition only.

Space and Geometry

Kant thought he had established that the presentation of space is an a priori cognition, an intuition to be precise. The concept of space is the concept of the object which is given a priori in intuition. Kant now begins the “transcendental” exposition of this concept. Here his aim is to show how the concept of space functions in synthetic judgments that are made a priori. This task in turn has two parts. The first is to show that the judgments of geometry are validated by the concept of space. The second is to show that this is the only way they can be validated.

It is assumed that the judgments of geometry are indeed synthetic and a priori. Kant argued that if the presentation of space were a concept, rather than an intuition, then any judgments based on it would be analytic. That is, they would simply present characteristics of space already thought in the concept. On the other hand, if the presentation of space is an intuition, there is a basis for synthesis in the judgment of geometry, though Kant does not here say how this would work.

If the presentation of space were empirical, then, Kant claimed, it could never be the basis for a judgment of geometry. The reason is that geometrical judgments are necessary, which means (as argued in the Introduction) that they are made a priori. On the other hand, if the presentation of space is itself a priori, we could give an account of the a priori character of the judgments we make about space.

There are no details given of how the a priori intuition of space validates the judgments of geometry. Of particular concern to later commentators is the fact that there are many alternative geometries, any one of which might be validated by our intuition of space. All Kant tells us is that if space is the form of sensible intuition, then the concepts of objects presented in sensible intuition must be determined by that form. He concludes that this is the only way in which the character of geometric judgment can be preserved.

The Ideality of Space

At this point, the account of the presentation of space that Kant has proposed does not seem extraordinary. Whether or not we agree with his arguments, the issues seem to be concerned entirely with the way in which space is presented to the human mind, whether empirically or a priori, conceptually or intuitively.

But if the thesis is merely that our presentation of space is an a priori intuition, then its application to geometry is quite limited. That is, the only geometrical propositions that would be validated are those concerning the presentation of space. This is to say that when we present objects as in space, our presentations must conform to geometrical propositions. We can legitimately ask whether these presentations represent something in the objects which affect us in intuition. Do the objects themselves conform to the propositions of geometry?

Kant’s answer is portentous. He tells us that “space” (not the presentation or intuition of space, but space as itself an intuition) “represents no property whatsoever of any things in themselves, nor does it represent things in themselves in their relation to one another” (A26/B42). Here, a “thing in itself” is understood as a thing considered apart from the way in which it is presented to human sensibility. Thus, “space represents no determination of such things, no determination that adheres to objects themselves and that would remain even if we abstracted from all subjective conditions of intuition” (A26/B42). As such, a thing in itself could not be an object of empirical intuition, and hence it is not an “appearance.”

Kant’s denial that space represents properties or relations to things in themselves is so important that we will reconstruct the argument for it in detail.

  1. Suppose space as an intuition represents a property or relation of things in themselves.
  2. If an intuition I represents a property P or relation R of things in themselves, then I must be intuited on the basis of P or R.
  3. So, space is intuited on the basis of a property or relation of things in themselves.
  4. Space is an a priori intuition.
  5. If I is an a priori intuition, then I is not based on a property or relation of things in themselves.
  6. So, space as an intuition is not based on a property or relation of things in themselves.
  7. So, space as an intuition does not represent a property or relation of things in themselves.
The substantive premises here are 2, 4 and 5. We have argued earlier that 4 has not been adequately established. We grant premise 5, so the only premise left to examine is 2.

We first note that the conclusion of the argument seems to be too strong. That is, if we take ‘represents’ to indicate some kind of structural similarity between an intuition and a property or relation of things in themselves, it seems at least possible the intuition does represent a thing in itself. If the conclusion is rejected and the other premises are granted, then 2 must be rejected.

Step 2 says that any intuition representing a property or relation of things in themselves must be based on the presentation of the property or relation in that intuition. But why should we think that the properties or relations of things must be given to the mind in order for the mind to represent it successfully? It seems that the best we can say is that the properties or relations of things must be given to the mind in order for us to have any reason to think that our intuitions represent them. Thus, it seems that a skeptical conclusion is in order here. Kant’s later statement, “We cannot make the special conditions of sensibility to be conditions on the possibility of things, but only of the possibility of their appearances,” (A27/B43) seems less dogmatic than the conclusion of the argument we have just examined.

Even though such a skeptical conclusion seems rather shocking, Kant was not bothered by it. He was willing to give up any claims to knowledge of things in themselves in favor of knowledge of appearances only. So he tells us that, “Space is nothing but the mere form of all appearances of outer senses” (A26/B42). This is a central claim of the Critique, and it requires delicate handling.

The thesis of the metaphysical exposition of the concept of space is that the presentation of space is an a priori intuition. Now presentations are supposed to refer to objects (as Kant noted in his 1770 letter to Herz). The presentation of space has appearances as its ultimate objects. It allows us to “determine” those objects with respect to their spatial relations, such as their size, shape, and position. But it only determines objects in this way insofar as these objects are themselves related to human sensibility. “The proposition, All things are side by side in space, holds under the limitation: if these things are taken as objects of our sensible intuition” (A27/B43).

The limitation of the reference of the presentation of space to objects of sensible intuition is called by Kant the “ideality” of space. Space is ideal in the sense that without the relation to the mode of presentation of human subjects, space is “nothing” (A28/B44). This ideality is “transcendental,” as opposed to empirical. We shall understand that term here in a negative way: transcendental ideality is a status that has no significance within experience itself. Things in themselves, to which spatiality does not apply, are never “at issue in experience” (A30/B45).

In a passage not included in the second edition, Kant contrasts the status of space with that of the taste or the color of wine. This taste and color are subjective components of experience, as opposed to a property of the wine such as its volume, which we might say is empirically real. Space likewise is empirically real. As just stated, all things, taken as appearances, are side by side in space. It is space which allows us to determine the volume of the wine. To say that space is transcendentally ideal is to say (in this context) that space is not something underlying things in themselves, but only things in relation to the human faculty of sensibility.

The transcendental concept of appearances in space, on the other hand, is a critical reminder. It reminds us that nothing whatever that is intuited in space is a thing in itself, and that space is not a form of things, one that might belong to them as they are in themselves. (A30/B45)

For all Kant’s reassurances, this doctrine looks extreme. Consider the sentence after the one just quoted: “Rather, what we call external objects are nothing but mere presentations of our sensibility.” Here Kant does not say that what we call external objects are nothing but things as they are related to our sensibility. Rather, he says that external objects are presentations. That is, it seems as if external objects are something that are purely mental, rather than things as related to what is purely mental. Which of the two views did Kant hold, or did he hold both of them? We shall pursue this issue in more detail later.

Conclusions About the Ontology of Space

At the beginning of the section on space, Kant posed two ontological questions. The first had to do with nature of space itself, and the second with the relation of space to the mind. We now return to those questions.

It might seem that Kant endorses of the second alternative in the second question, that space is merely one of the “determinations and relations that adhere only to the form of intuition and hence to the subjective character of our mind.” But this would not be quite correct, since this account of space is based on the thesis that space is a determination of things. On Kant’s view, space rather determines the properties and relations of things. Does the fact that space is the form of appearances mean that space is a “real being?” This would be correct, so long as it is understood that space is “empirically” real and not a thing in itself.

As noted above, the dispute about the ontological status of space historically took place between Newton and Leibniz. Kant characterized the Newton/Leibniz dispute as being between “the mathematical investigators of nature” and “some metaphysical natural scientists,” respectively (A39/B56). The advantage of the mathematical investigators is that they can explain how we are certain that the propositions of geometry to apply to the objects presented to sensibility. Geometry applies to space, which is a real being which determines the spatial properties of the objects in space.

The disadvantage of the Newtonian position is that it is impossible to say what space is (at least within the framework of metaphysical categories which Kant would argue apply necessarily to objects of sensibility). The Newtonian would have to admit, absurdly,

two infinite things that must not be substances nor anything actually inhering in substances, but that yet must be something existent—indeed, must be the necessary condition for the existence of all things—and must moreover remain even if all existing things are annulled. (B70-71).
Kant correctly credited this criticism to Berkeley. It can be found in Berkeley’s anti-Newtonian treatise De Motu (On Motion), §§53-57.

The metaphysical natural scientists, headed by Leibniz, can make perfect sense of space by describing it as something “inherent” or dependent on existing substances. But their metaphysical gain requires the sacrifice of the certainty of the application of geometrical principles to sensible objects (A40/B57). The reason is that the properties of bodies are known only through experience of them, but experience can never guarantee that geometrical principles applies to unexperienced bodies. (This is the problem raised by Hume in Book I, Part II of the Treatise of Human Nature, although Kant does not seem to have been aware of Hume’s argument.)

Kant proposed an account of space that would overcome the dilemma by uncovering a presupposition that both sides share: that space is something “transcendentally” real (whether substance, property or relation). That is, they both presumed that objects in space are things in themselves, and that space is either a thing in itself or a relation between things in themselves. Kant’s aim was to incorporate the best features of the Newtonian and Leibnizian conceptions of space, without their respective drawbacks. He could explain how space applies to objects by making objects depend on space, but without making space into an unfathomable metaphysical rival of God. But he could do this without making space dependent on bodies, by making both space and bodies depend on mind.

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