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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 of the construction of geometrical figures in accordance with our own a priori concepts), and to base the proofs on these constructions. Not only will this explain how we can be in possession of synthetic a priori judgments holding for the structure of outer intuition; it should also make it possible to understand how the necessity of these judgments applies equally to the structural relationships among the appearances presented in outer intuition.

By way of analogy, compare the case of a mathematician constructing geometrical figures on a television screen, and thereby (with a knowledge of the curvature of the screen) producing demonstrations – at least as Kant would have it – which hold for these figures. In theory, prior to any transmitted images appearing on the screen, the mathematician could determine the rules governing the possible structural relations of these images.The geometry of the screen would lay down, in advance of the appearance of any transmitted images, the rules concerning how they could be internally structured and related to one another. Of course, the images on the screen will be physical images; and as such, they will be taken, at the common-sense level, to exist whether or not we are, or could be, aware of them. Equally, at the common-sense level, the screen exists independently of our possible awareness of it. Such independence does not apply to what Kant understands by appearances and by a priori intuition. In particular, we do not first apprehend a unified intuition and then construct a geometrical figure upon it: rather, it is the construction of the figure, in accordance with our a priori geometrical concepts, that brings into existence a unified intuition. None the less, the analogy does bring out a crucial point in the transcendental exposition: namely, that it is because we possess the capacity to construct figures a priori in outer intuition, and thereby to demonstrate synthetic a priori judgments about these geometrical figures, that even prior to any experience, we can be in possession of synthetic, yet necessary, rules governing the structure of the appearances in outer intuition.

Still, it may be objected, all that the argument explains so far is how geometry, as a body of synthetic a priori judgments, can describe the structure of outer intuition (and thereby how we can grasp its application to what is given in outer intuition, viz. appearances). It has not explained how geometry is able to describe the structure of space (and the possible structural forms of objects in space). That, of course, is so if space is not identified with the form of our outer sense, i.e. with pure outer intuition. But unless we do identify the two, there can be no explanation of how geometry is a body of synthetic a priori truths holding for the structure of space. For if space referred not to our form of outer sense, but to what exists in itself (absolutely or relationally), then, since the mathematician produces his geometrical demonstrations by recourse to outer intuition, there can be no guarantee that what necessarily holds for any construction in intuition must hold for space (and, a fortiori, for empirical objects in space). In other words, if space is not a property of the mind but exists independently of it, any demonstration that rests on what is a property of the mind (outer intuition) cannot be acknowledged to hold with certainty for what exists independently of the mind. Since, however, geometrical demonstrations are acknowledged to hold with certainty for the structure of space, then space and the form of our outer sense, pure outer intuition, must be one and the same. Such an identification can alone account for the status of geometry.

There is a good summary of the upshot of the Transcendental Exposition at A 48–9/B 66:

If, therefore, space (and the same is true for time) were not merely a form of your intuition, containing conditions a priori, under which alone things can be outer objects to you, and without which subjective condition outer objects are in themselves nothing, you could not in regard to outer objects determine anything whatsoever in an a priori and synthetic manner. It is, therefore, not merely possible or probable, but indubitably certain, that space and time, as the necessary conditions of all outer and inner experience, are merely subjective conditions of all our intuition, and that in relation to these conditions all objects are therefore mere appearances, and not given to us as things in themselves which exist in this manner.