In what follows, I reconstruct two of the four arguments that Kant offers in the First Critique¹ that space is a priori. The arguments are stated in the section titled Transcendental Aesthetic. The first pair of arguments (1 and 2) claims that space is a priori, or independent of and precedes experience and sense perceptions;² and the second pair (3 and 4) claims that space is an intuition, an immediate, singular representation which allows objects to be sensed or perceived (cognized). Focusing on only the first pair of arguments (of a priority), I show how Kant’s second argument is the proof for his first argument. Next, I contend that the second argument seems rather problematic. The problem consists in Kant’s failure to distinguish what it means to represent empty space and not representing space at all. This will lead to my further contention that space can be conceived as non-existent, which I do not think is contradictory.

To fully grasp Kant’s arguments, a logical reconstruction must be in place. But I should point out that in reconstructing Kant’s arguments, we are confronted with several difficulties. Two of these difficulties will become clear as the reconstruction proceeds. It is therefore expedient that we identify and clarify these difficulties: (i) necessity, and (ii) the distinction between the metaphysical exposition and the transcendental exposition.

Necessity: Kant concedes that human cognition starts with or in experience, but he disagrees that it arises from experience (B1-2). He argues that there is a kind of knowledge that is derived from experience, but disagrees that it is the only kind of knowledge. Kant thinks there is a priori knowledge, i.e., not derived from and independent of experience. He proposes two criteria that determine a priori knowledge: necessity and strict universality (B3-4) Since both are logically equivalent (as many Kantian scholars including British philosopher Sebastian Gardner have suggested), we shall only discuss necessity. Kant’s claim that necessity is a valid criterion of the a priori fits into his view that what is empirical is synonymous with what is contingent (B5). Experience, Kant suggests, accounts for what is actual; the a priori, on the other hand, gives that which cannot be otherwise, that is, what is necessarily the case. An a priori cognition, knowledge or judgment does not allow any possible exception. It is valid, Kant would add, not only for humans, but for all other conceivable beings as well, even if they differ from humans in many or all respects. But it seems, the a priori has both a positive and a negative aspect: negatively, it is independent of or not derived from experience; positively, its ground is in reason, i.e., human reason. From this, it follows that necessity as a criterion seems deeply psychological, since its origin is subjective. But it doesn’t have to be only psychological. It can also be logical or formal (which is what Kant may want to show). Perhaps, there is something that allows us to do or not do something by logical necessity. 

The Metaphysical-Transcendental Expositions Distinction: It is difficult to say exactly why Kant divides the expositions into a metaphysical exposition and a transcendental exposition. In the metaphysical exposition, Kant explores the nature of the concept of space, or just space, to show that it is given a priori, and therefore, an a priori intuition (B38). This follows from his opening remarks that it is only through or by the means of the senses (sensibility) that we are able to know anything about objects (A19/B33), and that the sensibility has its form – i.e., space – to which the object (matter) of experience must necessarily be interfaced through. In the transcendental exposition, he goes on to show how synthetic a priori judgements are possible. Synthetic a priori judgements (A158/B197) are judgements which derive from the formal conditions of a priori intuitions and concepts as well as the unity of these, for which a cognizing subject is needed (what Kant might call the ‘transcendental ego’ or ‘transcendental unity of apperception’).  In short: in the metaphysical exposition, Kant claims and proves his theses about space, that it is a priori and intuitive, and hence is the ground for synthetic a priori judgements; and in the transcendental exposition, he goes on to show the applicability of his claims. Synthetic a priori judgments are possible only if we assume and accept that space is a pure form of the sensible intuition (which the metaphysical exposition establishes). Kant, in the transcendental exposition, puts this to the test, using geometry, which he claims is “the science that determines the properties of space synthetically and yet a priori” (B41). 

All of these will become clear as I restate the arguments and how that it has a logical or formal structure. The two argument that Kant makes for the apriority has an argument structure which can be stated as follows:

P1: Space is either a posteriori (derived from relations among representations or experience), or a priori (independent of sense perceptions or experience).

P2: Space is not a posteriori (derived from relations among representations or experience). 

Conclusion: Space must be a priori (independent of sense perceptions or experience).

From this, we can obtain an argument structure that is a disjunctive syllogism, and thus has the logical form:                 

As the argument goes, Kant denies that space is a posteriori, or not derived from experience. His denial is a direct refutation of the Leibnizian view of space as merely relations among objects or things. From the argument structure we just obtained, we can reconstruct and explain the argument as shown below. (Note that we reconstruct the argument in terms of the second premise, P2, and the Conclusion that follows. P2 is the first argument that space is not empirical, and the Conclusion is the second argument that it is necessarily a priori).

First Reconstruction: Space is not empirical or a posteriori

P→Q: Space cannot be an empirical concept supplied by outer experience or sensations.

Q→R:  For there to be experience at all, space must be presupposed.

R→S: If space is not presupposed, my sensations or experience cannot be distinguished from me, and hence, cannot even be distinguished or known by others. 

S→T: Likewise, without space external objects cannot be distinguished from and related to one another at all. 

T→U: Thus, the representation of space grounds all outer experience, which means it precedes experience in a temporal fashion. 

U→V: Therefore, space is not a posteriori, i.e., not derived from experience, since it grounds experience.  

The argument more precisely stated: Kant’s first argument serves a negative purpose. That is, Kant shows that space is not derived from experience, contrary to what Leibniz claims. Put another way: Kant’s conclusion establishes two important claims against Leibniz: (i) I couldn’t relate the objects of experience to one another if they are not spatially distinguished from me or “outside me”, since I occupy a particular location in space (R→S). It follows that space must be the ground or prerequisite for comparing or relating external things as different from myself; (ii) space is what allows the numerical distinctness of external objects. Two objects must be located in different spatial regions for them to be numerically distinct, that is, for there to be two objects rather than one (S→T). In both cases, space seems to be a matter of necessity. For experience to be had, i.e., for me to know relations among objects, space must be as necessarily constituted in my cognition. If space were not in my cognitive capacity, then I couldn’t possibly relate objects or have experience at all. However, the thesis of the first argument is to show that space is not empirical. In Kant’s second argument, we will show that space is necessarily given as such, and hence, impossible to be otherwise.  

Second Reconstruction: Space is a necessary representation

The second argument Kant makes for the apriority of space is basically the proof for and the conclusion drawn from his thesis in the first argument that space is not empirical. In reconstructing Kant’s two arguments as disjunctive syllogism above, we show that the conclusion follows from the premises. Since we reconstructed Kant’s first argument in terms of P2, the second argument can be reconstructed in terms of the Conclusion which follows from it:

C→D: Space is either a necessary representation, or it is not.  

D→E: If it is not a necessary representation, then it is contingent, which means it is a posterior or empirical. (We show this earlier while discussing “Necessity”). 

E→F: But space is what grounds experience (U→V), so it cannot be a posteriori. 

F→G: It is impossible to have experience, i.e., to know or sense objects of experience without the representation of space.

G→H: Likewise, it is impossible to imagine the complete absence of space, i.e., a world without spatial representation, though it is possible to imagine space without objects (empty space).

H→I: A representation which it is impossible to be without is a necessary representation.

I→J: Therefore, space as a necessary representation is a priori.  

The argument more precisely stated: The second argument reaches a positive conclusion. What Kant here seeks to show is that what can be given empirically, that is, objects, can be thought away. In other words, I can conceive space without objects. But that space cannot be eliminated or gotten rid of. Kant seems to suggest that for a cognitive subject, space is necessarily grounded in my subjective constitution or structure. This implies that space is a priori psychologically. Yet, this argument’s purpose is to establish the a priori necessity of space (logical condition), not its subjective origin or condition. Furthermore, what Kant implies here is, although we might be able to think of the absence of space, yet we are incapable of representing space as absent. It would then follow that I am or anyone is capable of thinking only in terms of the representation of space. Thus, the apriority of space follows from its necessary representation.

The two arguments taken together establish Kant’s proof that space is non-empirical and necessary, and therefore, a priori. Now, in describing the distinction between the metaphysical and transcendental expositions, we stated earlier that for Kant geometry is synthetic a priori.  Let us say something about this. Synthetic a priori judgements tend to make or establish objective claims, i.e., claims about experience or objects of experience. But such judgements are only possible if space is taken as a priori, and therefore, a form of sensible intuition (arguments 3 & 4, which we do not address here). On this note, if space were not a form of sensible intuition, no one would know anything about the truths of geometry regarding existing spatial relations. I would not be able to know what to do with or about space if it were derived from (empirical) relations among things or representations, nor can it be a thing in itself. 

Additionally, the propositions of geometry, which concern space, cannot be obtained from concepts, because such prepositions go beyond concept-containment. For example, although the concept of a triangle means the concept of a plane figure with three sides, one can derive a geometric proof to demonstrate that the sum of the interior angles of a triangle must be equal to 180 degrees.This resultant proof is true of all triangles. But this is not a proof that can be found in the concept of a triangle. I can derive this proof by the method of construction, which is not based on experience or has no sensory affection. Kant’s view of space as given a priori can be tied to his constructivist account when we consider this other example. I can draw four equal lines bound together. When I apply it to an object in the real world, my construction or drawing would be confirmed because I would find a geometric figure which would fit the description of the four equal lines, and this we might call a ‘square’. From these two examples, it follows that from the a priori nature of space, Kant deduces the apodictic certainty of geometry.    

But let me return to Kant’s arguments, particularly his second argument since it is the proof of the first as we have shown, and state my objection. Kant does not provide any proof for his claim that we can conceive or imagine space without objects. In fact, this claim is problematic. Perhaps Kant would say that it can be proven by a kind of thought-experiment or by introspecting. But even if we were to grant this, Kant does not tell us how exactly we are able to tell the difference between representing empty space and not representing space at all. In addition, Kant’s two arguments only explain how space must be represented, or that we must have the representation of space. Put simply: the upshot of Kant’s failure to provide proof is that his arguments only show that space is a necessary representation. They do not provide any clearly stated premise for the conclusion that space is an a priori representation. 

Moreover, we have a higher problem. We contend that Kant’s claim that we can conceive or imagine space without objects, vis-a-vis empty space, weakens his conclusion for the apriority of space. For just as pure or empty space can be conceived, it is not contradictory for it to be conceived as non-existent. Finally, it is difficult to explain how space can still be existent even when all sensible contents (objects) are thought away. 

In conclusion, as we have argued, Kant’s two arguments are inextricably linked. The first argument is Kant’s thesis and the second argument the proof. That is why the condition of ‘necessity’ is sort of implied or embedded in the former until it is stated and proven in the latter. Moreover, Kant nowhere proves his conclusion as we just now mention. Even so, his arguments are sound and strongly made.

Reference/Notes:

(¹) Kant, Immanuel. Critique of Pure Reason. trans. & ed. By Paul Guyer and Allen W. Wood.   The Cambridge Edition of the Works of Immanuel Kant. Cambridge University Press, 2019. It is the first of Kant’s three critiques, the other two being the Critique of Practical Reason and the Critique of Judgement.

(²) First argument: “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 <and next to> 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” (B38, pg. 174-5)

Second Argument: “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” (A24/B39, pg. 175).