One of the major philosophical disputes in the seventeenth and eighteenth centuries concerned the issue of space (and, also, time—though I’m mostly going to talk about space in what follows). What is space? By the mid-eighteenth century there were two major competing positions. On the one hand, following Newton, many philosophers and scientists construed space as a kind of subsisting entity. On this view space (and correspondingly time) exists independently of any possible object that may occupy space, and independently of any possible relations between such space-occupying objects. Space is, on this view, a kind of container in which objects are placed. Spatial relations (e.g. position, relative motion) are determined by the relations which objects have to this container space, and the relations of these spaces (or spatial regions) to one another. Call this the ‘absolute’ conception of space.
In contrast to the conception of space as absolute, Leibniz proposed that space is merely a set of relations that hold between independently specifiable objects. This ‘relationalist’ conception of space says that space is not something that exists over and above the objects which occupy it. It is merely the order of possible relations between distinct existing substances. So while we could, on the relationalist view, conceive of empty space, insofar as we conceive of the possible relations which substances might enter into, space could not exist as something independent of any such set of possible objects and their (possible) relations.
In the Critique of Pure Reason Kant argues against both of these theories of the metaphysics of space. His suggestion is that space is neither a subsisting container entity nor a set of relations between independent objects, but rather something else—a product of our subjective apprehension of objects that are distinct from us. It is, in Kant’s phrase, a ‘form of intuition’.
The Metaphysical Exposition of Space
Kant’s arguments concerning the nature of space and the nature of our representation of it occur in the ‘Metaphysical Exposition’ of the first Critique. Kant makes four arguments here, which raise a variety of questions concerning their proper interpretation. Part of the complexity of interpretation is due to the difficult nature of Kant’s arguments. He is trying to do several things simultaneously. First, he is intervening in a debate concerning the ontological status of space—is space a subsisting thing (absolutism) or is it an order of relations (relationalism)? Second, he is concerned with answering the question as to the origin of our representation of space. Space is different from the objects that are in it, and in some ways seems to be a condition of representing spatial objects. So where does this representation of space come from? How could it come (if we endorsed empiricism) from an experience of objects? Third, Kant is concerned with the content or nature of the representation of space. Is it a fundamentally intellectual (conceptual) kind of representation or is it a sensible one?
Kant’s first two arguments (at A23-4/B38-9) run as follows:
1.) Space is not an empirical concept that has been drawn from outer experiences. For in order for certain sensations to be related to something outside me (i.e., to something in another place in space from that in which I find myself), thus in order for me to represent them as outside
one another, thus not merely as different but as in different places, the representation of space must already be their ground. Thus the representation of space cannot be obtained from the relations of outer appearance through experience, but this outer experience is itself first possible only through this representation.
2.) Space is a necessary representation, a priori, that is the ground of all outer intuitions. One can never represent that there is no space, though one can very well think that there are no objects to be encountered in it. It is therefore to be regarded as the condition of the possibility of appearances, not as a determination dependent on them, and is an a priori representation that necessarily grounds outer appearances
There are a variety of possible targets for these arguments, but at least one target relates to our second question above concerning the origin of the representation of space. Specifically, Kant is querying the claim that the representation of space is drawn from the content of our experience. This is particularly clear from the first argument, as Kant seems there to be denying the Lockean claim (Essay, II.xiii.2-4) that the representation of space is derived from the representation of the distances between objects. Kant’s point (or a point) in the argument (1) is that we cannot represent one object as distant from another without already representing them as being in one space in which they are related. More generally, Kant’s claim seems to be that we cannot represent to object as spatially related without presupossing a more general (and distinct) capacity to represent the space in which those relations stand (on this, see especially Warren (1998)).
Kant’s second argument is perhaps less persuasive, for it argues that we cannot represent a lack of space. Certainly, this fact would seem to go against the empiricist view of space as something we learn from experience. But it seems that the empiricist might counter that once the representation of space is gained via experience, we cannot part with it psychologically, and so cannot represent its absence. Kant’s “cannot” needs to be more than just a psychological “cannot”, and I don’t see that he provides the argument necessary to show that the empiricist position founders.
Thus, with these first two arguments Kant attempts to refute a particular position concerning the source of our representation of space—viz. experience. The representation of space is not to be found in the content of our experience, rather, outer experience presupposes that its content will consist of spatially related items. Spatial representation is thus presupposed by our having any representation of anything distinct from ourselves or other things at all. Thus, if the first two arguments are correct, the representation of space is a priori and not empirical. This still leaves open our first question concerning what space is. It also leaves open our third question concerning whether the representation of space is sensory or intellectual in nature. It is with this issue that Kant’s next two arguments are concerned.
Kant’s third and fourth arguments (B39-40) run as follows:
3.) Space is not a discursive or, as is said, general concept of relations of things in general, but a pure intuition. For, first, one can only represent a single space, and if one speaks of many spaces, one understands by that only parts of one and the same unique space. And these parts cannot as it were precede the single all-encompassing space as its components (from which its composition would be possible), but rather are only thought in it. It is essentially single; the manifold in it, thus also the general concept of spaces in general, rests merely on limitations. From this it follows that in respect to it an a priori intuition (which is not empirical) grounds all concepts of it. Thus also all geometrical principles, e.g., that in a triangle two sides together are always greater than the third, are never derived from general concepts of line and triangle, but rather are derived from intuition and indeed derived a priori with apodictic certainty.
4.) Space is represented as an infinite given magnitude. Now one must, to be sure, think of every concept as a representation that is contained in an infinite set of different possible representations (as their common mark), which thus contains these under itself; but no concept, as such, can be thought as if it contained an infinite set of representations within itself*. Nevertheless space is so thought (for all the parts of space, even to infinity, are simultaneous). Therefore the original representation of space is an *a priori intuition*, not a concept.
Kant’s interest in these two arguments is with our third question, and his aim (or one aim) is to deny the view that the content or nature of the representation of space is fundamentally intellectual (in the sense of originating in the intellect). According to Kant, spatial representation is not fundamentally intellectual or conceptual, but rather ‘intuitive’. This is not to say that spatial representation is sensory. We do not have a sensory representation of an infinite space! Kant is trying to carve out room for a position according to which we have a non-sensory but also non-conceptual/non-intellectual representation of an infinite space, which makes possible our representation of particular spatial expanses. There are similarities here between Kant’s argument and previous rationalist arguments concerning the priority of the infinite over the finite. Kant’s third argument above is that the single infinite expanse of space is prior to, in the sense of being presupposed by, the existence of any particular spatial region. Kant’s fourth argument reinforces this by arguing that if spatial representation were conceptual (on Kant’s understanding of conceptual representation) then we would have to represent an infinite number of spaces or spatial regions. This would present us with an infinite task, which we would obviously be unable to accomplish. Since Kant thinks we clearly do represent space as an infinite expanse, we must be doing so non-conceptually.
The Transcendental Exposition of Space
The central argument of the TE, the so-called ‘argument from geometry’ was originally part of the ME, located between the two pairs of arguments discussed above. Kant moves this argument to its own section in the second or “B-edition” of 1787. It is not clear, however, what the argument actually is. Is it an argument concerning the status of the representation of space as ultimately an a priori intuition? Is it an argument concerning the dependence of the science of geometry on space so understood? Is it an argument concerning something else entirely?
The Argument from Geometry
The argument proceeds as follows,
Geometry is a science that determines the properties of space synthetically and yet a priori. What then must the representation of space be for such a cognition of it to be possible? It must originally be intuition; for from a mere concept no propositions can be drawn that go beyond the concept, which, however, happens in geometry (Introduction V). But this intuition must be encountered in us a priori, i.e., prior to all perception of an object, thus it must be pure, not empirical intuition. For geometrical propositions are all apodictic, i.e., combined with consciousness of their necessity, e.g., space has only three dimensions; but such propositions cannot be empirical or judgments of experience, nor inferred from them (Introduction II). (B41)
It is not altogether clear what Kant is arguing here. He is commonly interpreted along the following lines.1
- We have synthetic a priori cognition of Euclidean geometry Or: Euclidean geometry is necessarily true
- Such cognition is possible only if space is a pure intuition Or: pure intuition of space is a necessary condition of our synthetic a priori cognition of geometry
- ∴ Space is a pure intuition.
One problem with this interpretation of Kant’s argument is that it goes against the methodology Kant is supposedly following in the CPR as opposed to the Prolegomena. Kant puts the distinction this way in his logic lectures:
Analytic is opposed to synthetic method. The former begins with the conditioned and grounded and proceeds to principles (a principiatis ad principia), while the latter goes from principles to consequences or from the simple to the composite. The former could also be called ‘regressive’ and the latter ‘progressive’ (JL 9:149)
With the synthetic method one sets out from first principles/conditions and derives from them what one can. With the analytic method, one starts from some given claim and from it proceeds via a regressive analysis to its conditions. The synthetic method is constructive while the analytic method is deconstructive. Kant take the CPR to be arguing from first principles regarding cognition to the existence of particular kinds of synthetic a priori knowledge. In the Prolegomena Kant is starting from the existence of such synthetic a priori knowledge (in natural science and in mathematics) and giving a regressive analysis of its conditions (4:263, 274-5).
Unfortunately, Kant’s use of these methods in each of the works is not altogether clear. To take just two examples, Kant seems to give regressive analytic arguments both in the Transcendental Exposition of Space, above, and in the Transcendental Deduction of the Categories. It is not clear whether this is really compatible with the synthetic method supposedly being followed in the Critique.2 So appeal to the distinction between methods cannot, of itself, ground acceptance or rejection of any particular interpretation of the argument in the TE.
Lisa Shabel (2004) interprets Kant’s argument as synthetic, and thus as arguing from the notion that the representation of space is a pure a priori intuition to the claim that such a representation of space is the foundation for our cognition of geometric objects and relationships. According to Shabel the argument from geometry functions as a ‘bridge’ from the arguments of the Metaphysical Exposition, which purport to show that space is an a priori intuition, to the arguments concerning the transcendental ideality of space— i.e., that it is nothing other than a pure a priori intuition.3
At least three things work in favor of Shabel’s interpretation. First, it accords with Kant’s claim that the method of argument in the first Critique proceeds synthetically rather than analytically, though, as was remarked above, this cannot be a decisive issue. Second, it makes sense with respect to the position of the TE in the Aesthetic as a whole. It would be surprising if Kant meant the TE to be an argument for construing the representation of space as a priori and a pure intuition, since this seems to be what he intends the Metaphysical Exposition to accomplish. Third, as previously noted, the argument of the TE was, in the first edition of the CPR, located in the ME. A plausible explanation as to why it was moved would be that Kant did not in fact take it to be an argument for the status of the representation of space as originally a pure a priori intuition, and thus decided to give it its own section instead. This is perhaps made clearer by Kant’s emphasis on the role of the TE in the Aestehtic.
I understand by a transcendental exposition the explanation of a concept as a principle from which insight into the possibility of other synthetic a priori cognitions can be gained. For this aim it is required 1) that such cognitions actually flow from the given concept, and 2) that these cognitions are only possible under the presupposition of a given way of explaining the concept. (B40)
So, on Shabel’s reading, the TE does not give a new argument for the status of spatial representation as a priori and an intuition, but rather provides reason for construing spatial representation as the only explanation of geometric cognition. On Shabel’s reinterpretation the argument proceeds as follows.
- The representation of space is a pure a priori intuition (from the ME).
- Geometry is a science of synthetic a priori propositions concerning
the structure of space
- We need an explanation of (a) the a priori representation of space as an object and (b) the status of the propositions of geometry as synthetic rather than analytic.
- Since (according to the ME) the representation of space is a priori, we know that the geometer’s representation of its properties is likewise a priori.
- Since (according to the ME) the representation of space is originally an intuition, we know that the status of propositions concerning the nature of space is ultimately synthetic.
- ∴ The status of the propositions of geometry as synthetic and a priori is explained by the pure a priori intuitive representation of space — geometric cognition depends on the representation of space understood in the terms set out in the ME.
The Ideality of Space
Thus far Kant has made arguments for two of our three topics. He’s argued that the source of our representation is not empirical, and thus that the representation of space is a priori. He has also argued that the nature or ‘content’ of the representation is intuitive rather than conceptual. But what about the ontology of space? Is space a subsisting thing, capable of existince independent of any and all objects or their relations(absolutism)? Or is it merely the set of possible relations which objects may stand in (relationalism)?
After Kant gives the arguments discussed above he draws two surprising conclusions (A26/B42). First, he argues that
a.) Space represents no property at all of any things in themselves nor any relation of them to each other, i.e., no determination of them that attaches to objects themselves and that would remain even if one were to abstract from all subjective conditions of intuition. For neither absolute nor relative determinations can be intuited prior to the existence of the things to which they pertain, thus be intuited a priori. (A26/B42)
This is surprising because, from what Kant has said in the above arguments, he has only been discussing our representation of space. But here he makes a claim concerning the metaphysical status of space—viz. that it is nothing ‘in itself’. He then goes on to explain what space actually is.
b.) Space is nothing other than merely the form of all appearances of outer sense, i.e., the subjective condition of sensibility, under which alone outer intuition is possible for us. Now since the receptivity of the subject to be affected by objects necessarily precedes all intuitions of these objects, it can be understood how the form of all appearances can be given in the mind prior to all actual perceptions, thus a priori, and how as a pure intuition, in which all objects must be determined, it can contain principles of their relations prior to all experience. (A26/B42)
So, according to Kant, space is nothing in itself but rather only a representation—viz. the form by which we represent distinct objects. How does he get to this conclusion?
First, Kant rejects a presupposition he thinks both the absolutist and the relationalist about space hold in common—viz. that space is something independent of the mind in general and our perception (or intuition) in particular.4 In Kant’s terminology space is ideal in virtue of being mind-dependent. Kant makes this point clearer later in the Critique.
I understand by the transcendental idealism of all appearances the doctrine” that they are all together to be regarded as mere representations and not as things in themselves, and accordingly that space and time are only sensible forms of our intuition, but not determinations given for themselves or conditions of objects* as things in themselves. To this idealism is opposed transcendental realism, which regards space and time as something given in themselves (independent of our sensibility). The transcendental realist therefore represents outer appearances (if their reality is conceded) as things in themselves, which would exist independently of us and our sensibility and thus would also be outside us according to pure concepts of the understanding. (A369)
This perhaps makes clearer why Kant thought he was making a move with respect to the debate between absolutists and relationalists about space—both positions assume that space is ‘transcendentally real’. But why think that space is ‘transcendentally ideal’ in the first place? What licenses the move from a claim about space as an a priori intuitive representation to the claim that space is only the form of outer intuition?
According to Kant, intuition is an ‘immediate’ and ‘singular’ awareness of an object. This contrasts with the conceptual representation of an object, which is ‘mediate’ and ‘general’. The role of intuition, Kant says, is to present us with objects, while concepts allow for thought of such objects (A320/B377; see also A19 and A50/B74). Kant’s argument for the ideality of space proceeds from this definition of intuition (see Prolegomena §§8-10; cf. Allais (2010))
- Intuition necessarily involves relation to an object
- We have an a priori intuition of space (and time)
- A priori intuition cannot, by definition, be a relation to an existing object
- ∴ Space is nothing other than a ‘pure’ a priori intuition
Kant’s thinking here is that the role of intuition is to present its subject with an object. Intuition is what makes the difference between merely thinking or daydreaming about something and actually having it present to one’s mind. But, if this is right then there cannot be some existing thing that is present to one in an a priori intuition, since the definition of the a priori requires that the intuition be independent of experience and thus of any existing object.
Thus, (1) follows from Kant’s definition of intuition. (2) follows from the arguments of the Metaphysical exposition concerning the a priority and intuitive status of our representation of space. (3) is the crucial premise, but this seems to follow from the fact that a pure intuition must present something, and that something, in the absence of any experience, cannot be an actual object, but merely a feature of the mind itself. This gets us (4), that space is nothing other than a form of mental representation and not something subsisting in itself, or an order of possible relations between objects, independent of the mind.
In §13 of the Prolegomena, which was a sort of summary version of the arguments we’ve been looking at so far in the Critique of Pure Reason, intended to introduce Kant’s readers to the main themes of that work, Kant makes an argument concerning what he calls ‘incongruent counterparts’, perfectly similar objects that are mirror images of each other, such as left and right hands. This argument, he thinks, shows us that space cannot be mind-independent, but must be an intuition.
Kant was highly enamored of this argument. He used it first in 1768, to argue that space was Newtonian (absolute) rather than Leibnizian (relational). In 1770, he used the argument for a different conclusion, that our knowledge of space is via intuitive not conceptual representation. The Prolegomena argument of 1783 is for a still different conclusion, that space is ideal. So what exactly is the argument and how could it serve all these conclusions?
Let’s look briefly at the first two uses of the argument (cf. Van Cleve (1991b); Van Cleve (1991a)). Kant’s argument against the Leibnizian relationalist position hinges on the fact that according to the Leibnizian relationalist, all spatial relations are a function of the intrinsic features of the relevant objects. So any change in spatial features is due to some change in the intrinsic, non-relational features of the object. In brief, Kant’s 1698 argument against Leibnizian relationalism says that incongruent counterparts like left and right hands are intrinsically exactly similar. This means that their orientation in space cannot be due to their ‘internal’ intrinsic, non-relational features, but must rather be due to something else—viz. their relation to some absolute space which contains them.
Kant’s 1770 argument takes up this point and applies it to our representation of space. If, Kant argues, our representation of space were ultimately purely intellectual/conceptual (as Kant construes the Leibnizian to argue), then we should be able, by means of concepts alone, to distinguish incongruent counterparts. But, he argues, conceptual representation of (e.g.) a left and a right hand will not differ since such counterparts do not differ (let us suppose) in any qualitatively intrinsic way (e.g. neither figure nor internal angles would differ), nor in any quantitative way (they differ not according to size, etc.). So since there is no intrinsic difference between the two hands, there is nothing conceptually to distinguish one as the left and the other as the right. We must represent their difference by some other means, which Kant argue is by means of intuition.
The Prolegomena’s use of the argument to conclude that consideration of incongruent counterparts demonstrates that space is ideal is deeply contested. The conclusion is not the only thing that is contested, unfortunately, for it is not at all clear how the argument for it goes. The relevant passage is below:
If two things are fully the same (in all determinations belonging to magnitude and quality) in all the parts of each that can always be cognized by itself alone, it should indeed then follow that one, in all cases and respects, can be put in the place of the other, without this exchange causing the least recognizable difference. In fact this is how things stand with plane figures in geometry; yet various spherical figures, notwithstanding this sort of complete inner agreement, nonetheless reveal such a difference in outer relation that one cannot in any case be put in the place of the other; e.g., two spherical triangles from each of the hemispheres, which have an arc of the equator for a common base, can be fully equal with respect to their sides as well as their angles, so that nothing will be found in either, when it is fully described by itself, that is not also in the description of the other, and still one cannot be put in the place of the other (that is, in the opposite hemisphere); and here is then after all an inner difference between the triangles that no understanding can specify as inner, and that reveals itself only through the outer relation in space…What indeed can be more similar to, and in all parts more equal to, my hand or my ear than its image in the mirror? And yet I cannot put such a hand as is seen in the mirror in the place of its original; for if the one was a right hand, then the other in the mirror is a left, and the image of the right ear is a left one, which can never take the place of the former. Now there are no inner differences here that any understanding could merely think; and yet the differences are inner as far as the senses teach, for the left hand cannot, after all, be enclosed within the same boundaries as the right (they cannot be made congruent), despite all reciprocal equality and similarity; one hand’s glove cannot be used on the other. What then is the solution? These objects are surely not representations of things as they are in themselves, and as the pure understanding would cognize them, rather, they are sensory intuitions, i.e., appearances, whose possibility rests on the relation of certain things, unknown in themselves, to something else, namely our sensibility. Now, space is the form of outer intuition of this sensibility, and the inner determination of any space is possible only through the determination of the outer relation to the whole space of which the space is a part (the relation to outer sense); that is, the part is possible only through the whole, which never occurs with things in themselves as objects of the understanding alone, but does occur with mere appearances. We can therefore make the difference between similar and equal but nonetheless incongruent things (e.g., oppositely spiralled snails) intelligible through no concept alone, but only through the relation to right-hand and left-hand, which refers immediately to intuition. (Pr §13, 4:285-6)
One possible reading of the argument (cf. Van Cleve (1999), 47) goes as follows:
- Incongruent counterparts are different in virtue of their differing relations to space as a whole.
- All relations among things in themselves are reducible to the nonrelational qualities of the relata.
- Therefore, if space and figures within it are things in themselves, one incongruent counterpart must differ internally from the other.
- But, in fact, the counterparts do not differ internally.
- ∴ Space itself and incongruent counterpart figures within it are not things in themselves.
According to this interpretation of Kant’s argument, Kant’s key move is with premise (2), which we might call the ‘Reducibility premise’. The idea here is that, according to many philosophers with whose views Kant was engaged (such as Leibniz), differences between distinct objects (or substances) were supposed to be grounded in differences between the intrinsic features of those objects. Here we see a connection to Kant’s 1698 argument. The Reducibility premise requires that spatial differences like left-handedness ground out in intrinsic differences between left and right hands. But there are no such differences (and the same goes for other kinds of incongruent counterparts) so space and the things in it must not be things in themselves, or more simply, they must not be real mind-independent things.
This argument may be successful, or at least valid. But there are worries about ascribing it to Kant. First, premise (2) is not explicitly stated in the Kant’s text. Second, we need not appeal to space as a whole in the argument. All we need is that the ‘handedness’ of the incongruent counterparts need not reduce to their internal features, whatever their relation to space as a whole. Third, it is not even necessary to point to incongruent counterparts. Such elements of the argument are wholly dispensable. All the argument needs is for there to be some set of relations that are not reducible to their relata. There are many spatial relations, such as distance, which do not seem to so reduce. So why then, if this was Kant’s view, did he focus on incongruent counterparts?
It’s obvious that Kant put great stock in the argument from incongruent counterparts, and used it to push a variety of different conclusions. Unfortunately it is not obvious exactly what the argument is supposed to be, nor how the fact of such incongruent counterparts could entail three such different conclusions—viz. that space is absolute, intuitive, and ideal.
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Messina, James. n.d. “Kant’s Stance on the Relationalist-Substantivalist Debate and Its Justification.” Journal of the History of Philosophy.
Russell, Bertrand. 1903. The Principles of Mathematics. Cambridge: Cambridge University Press.
Shabel, Lisa. 2004. “Kant’s ‘Argument from Geometry’.” Journal of the History of Philosophy 42 (2): 195–215.
Strawson, Peter Frederick. 1966. The Bounds of Sense. London: Routledge.
Van Cleve, James. 1991a. “Incongruent Counterparts and Things in Themselves.” In The Philosophy of Right and Left, edited by James Van Cleve and Robert E Frederick, 341–51. Springer Netherlands.
———. 1991b. “Introduction to the Arguments of 1770 and 1783.” In The Philosophy of Right and Left, edited by James Van Cleve and Robert E Frederick, 15–26. Springer Netherlands.
———. 1999. Problems from Kant. Oxford: Oxford University Press.
Warren, Daniel. 1998. “Kant and the Apriority of Space.” The Philosophical Review 107 (2): 179.
(Shabel 2004, 201). Various proponents of the following interpretation include (Russell 1903, 456; Strawson 1966, 57; Broad 1978, 45ff).[return]
- See (Gava 2013) for further discussion. [return]
- (Shabel 2004, 207). [return]
For further discussion see (Jauernig 2008; Janiak 2012; Messina, n.d.).[return]