an a priori, and not an empirical, consciousness of space. Whenever we think of outer appearances, we are conscious of them as spatially located (even a single outer appearance has spatial relations). But we can think about space without being conscious of any outer appearances. For instance, whenever we imaginatively engage in geometrical constructions and demonstrations. Consequently, if we are capable of thinking of space as empty of outer appearances, while all outer appearances must be spatially located, it follows that our concept of space is a presupposition of our consciousness of the appearances of outer intuition. Our concept of space must therefore be a priori, since it makes possible our outer empirical consciousness, rather than being dependent upon it. (Again, see the corresponding argument in the Metaphysical Exposition of time.)
Note that Argument 2 does not depend, as some have claimed, on the doubtful psychological generalization that we can think of an entirely void space. When it is said that we can think of space as empty of objects, what is meant is that we can think of space without thinking of outer appearances, of the objects of an empirical intuition. As I have just indicated, Kant believes that this occurs when we imaginatively undertake geometrical constructions. It can also occur if we start with the consciousness of an empirical object, and then think away all features besides the form of the object. Kant gives us an example in his Introduction: ‘Gradually remove from your experiential concept of a body everything that is empirical in it – the colour, the hardness or softness, the weight, even the impenetrability – there still remains the space that was occupied by the body (which has now entirely disappeared), and you cannot leave that out’ (B 5–6; see also A 20–1/B 35).
The second set of arguments, numbered 3 and 4 in the B edition, are designed to show that space is an intuition, indeed an a priori intuition, and not a general concept.
Argument 3: According to the view that space is a general concept, we have built up this concept from observing the relationships between different sets of empirical objects, and then taken the general concept space to stand for any similar relationships between objects. But this view cannot be correct, claims Kant, because ‘we can represent to ourselves only one space; and if we speak of diverse spaces, we mean thereby only parts of one and the same unique space’ (A 25/B 39). In other words, if space were a general concept, we could sensibly talk of genuinely different spaces, each of which individually falls under this general concept. But the truth is that we think of space as essentially one continuum, and not as made up of a number of individual spaces, each instantiating the general concept space. Compare a familiar example of a general concept: namely, the concept horse. Here we can think, indeed must be able to think, of many possible individual horses –in Kantian phraseology, ‘we can represent to ourselves many diverse horses’ – and when we do talk about diverse horses, we plainly do not mean that these are, let alone must be, only parts of one and the same horse. In employing any general concept, we necessarily think of it as capable of having innumerable instances, each capable of existing in its own right, rather than as a mere part of one all-embracing instantiation of the concept in question. But we cannot think of space in this way: all the so-called individual spaces are thought of as essentially parts of one all-embracing space. Space, therefore, cannot be a general concept. We must, rather, acquire our idea of it through acquaintance, through intuition.
Moreover, because we think of space as essentially one, and not made up of separable spaces, the intuition of space must be a priori, and not empirical. Consider a single empirical intuition like the consciousness of a colour expanse. Here, we certainly can have the consciousness of distinct expanses of that colour; and although it is possible that all these diverse expanses may be discovered to be parts of one overall expanse (by finding empirically that there are further expanses of the colour joining all the original ones), there is clearly no requirement that this must be the case. We might, alternatively, be conscious of e.g. empty space between the various colour expanses. On the other hand, we must think of space as forming one continuum. We cannot first have the intuition of two spatial areas, and then discover empirically that there is intervening space joining them up (as can be done with the colour expanses). Our notion of the unity of space is not something that we have merely discovered empirically; it is, rather, something that we think of as essential to it. Our intuition of space is, therefore, an a priori and not an empirical one (since its unity is thought of as necessary).
The question of why we cannot conceive of genuinely distinct spaces will be taken up later by Kant (see Two Problems about Kant’s Account of Space and Time). In the Metaphysical Exposition, he is taking it as given that each of us does think of space as essentially one.
Argument 4: An additional ground is provided, in Argument 4, for holding that space must be an intuition, rather than a general concept. We think of space as presented to us as infinite in extent. Now although a general concept can have an infinite number of instances falling under it, this must always be in virtue of a limited number of common characteristics between the instances. For no concept can be used if an infinite number of characteristics need to be instantiated before it can be applied. Space, however, is conceived as presented to us as infinite in extent, as having an unbounded number of parts. Although no general concept can be thought of as constituted by a boundless number of parts, an a priori intuition can be given to us as boundless in extent. Thus, whenever we draw a line, whether in fact or in the imagination, we can think of extending that line without limit; and it is by the thought of such a limitless progression or construction in a priori intuition that we are enabled to conceive of space as a presentation that is infinite in extent (cf. Prol, Sect. 12). Space must, then, be an intuition, indeed an a priori intuition (since the thought of its boundless nature is acquired through a process of geometrical construction), and not a general concept.
Transcendental Exposition
I turn now to the Transcendental Exposition of space (later referred to as a Deduction, see A 87/B 119). Not only is this the clearest and, given its premisses, the most compelling of the two expositions, it also directly connects with Kant’s attempt to explain the existence of synthetic a priori judgments in mathematics.
The term ‘transcendental’ surfaces many times in the First Critique, as well as in the other works of Kant’s critical period, so it will be useful to quote his own definition of it in relation to knowledge: ‘I entitle transcendental all knowledge which is occupied not so much with objects as with the mode of our knowledge of objects, insofar as this mode of knowledge is meant to be possible a priori’ (A 11/B 25; italics original). So, for example, the transcendental exposition of space is called ‘transcendental’ because it is occupied with explaining how we can be in possession of a body of synthetic a priori knowledge (geometry) holding for the structure of space – and, in consequence (as will emerge later), for the empirical objects that can come in space.
The Transcendental, unlike the Metaphysical, Exposition does not begin with certain very general thoughts that we have about space, and then proceed to draw conclusions from them concerning our concept of space. Instead, it starts with an agreed body of synthetic a priori knowledge, and proceeds to argue that such knowledge is possible if and only if space is the form of our outer intuition.
Kant believes that he has already shown in the Introduction that geometry is a body of synthetic a priori judgments, and that these judgments hold for the structure of space. The point now is to explain how this can be. Since the judgments are synthetic, it cannot be that they are proved merely from analysis of the concepts involved. Such a procedure could only produce analytic a priori judgments. But neither can the judgments proceed by consulting experience. At best, this could only give us synthetic a posteriori truths about how the differing singular figures, presented in empirical intuition, have each, in fact, been discovered to be structured. Rather, the proofs will need to proceed on the basis of our own a priori construction of figures in intuition. That is, we shall need ourselves to produce the specific sense fields (by means
