He thought that previous philosophers – including Leibniz and Hume – were guilty of a serious oversight in supposing that mathematical judgments are analytic (and hence that the denial of a true mathematical judgment is self-contradictory).This cannot be right, he argues, since we must have recourse to construction in order to determine the truth or falsity of any mathematical judgment. So, in the case of the geometrical question concerning the sum of the internal angles of a triangle, he holds that we need to draw a triangle, either in imagination or e.g. on paper, and proceed to prove the judgment by showing, through the use of the diagram, how the angles must add up to 180 degrees. The specific procedure is well discussed at A 713–24/B 741–52, from which the following is an extract:
[The geometrician] at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of his triangle and obtains two adjacent angles, which together are equal to two right angles. He then divides the external angle by drawing a line parallel to the opposite side of the triangle, and observes that he has thus obtained an external adjacent angle which is equal to an internal angle – and so on. In this fashion, through a chain of inferences guided throughout by intuition, he arrives at a fully evident and universally valid solution of the problem. (A 716–17/B 744–5)
The essential ingredients in this example are, I think, these. First, the geometrical concepts involved (line, triangle, angle, etc.) are ones that we devise for ourselves, entirely a priori, ‘without having borrowed the pattern from any experience’ (A 713/B 741). Second, although the figure (the triangle) which we draw on paper is, of course, an empirical one, the way that we use the figure in the demonstration is entirely independent of any of the contingencies of this particular drawn figure (its size, the colour of the drawn lines and so on). As a result, the drawn figure can stand for all possible triangles. Third, the proof proceeds by means of the construction of the triangle, together with some further construction (extending the base line beyond one of the sides, etc.), in order to demonstrate – that is, to show – by means of the diagram that certain truths obtain for this particular figure and, therefore, for all possible figures that are constructed in accordance with these same a priori concepts.
In Kantian terminology, the method may be described as producing a proof on the basis of ‘sensible intuition’. More will be said about intuition in the section on space and time. The point to note here about basing a proof on sensible intuition is that it involves recourse to the construction of figures, in a sensuous field, in accordance with our own a priori concepts. This construction can occur either by means of the imagination alone, in what Kant calls ‘pure intuition’, or by means of drawing a figure in reality, e.g. on paper, in what he calls ‘empirical intuition’. But in either case, the construction must employ the a priori geometrical concepts, discounting any contingent features inherent in the drawn figure. And the demonstration itself – for Kant, a ‘demonstration’ can occur only in mathematics – consists in being able to exhibit (A 716/B 744) or observe (A 717/B 745) that certain relations obtain in intuition as a result of the construction. (A more general discussion, which was added in the second edition, is given at B 14–17.)
Without reference to intuition, Kant’s claim is that there is no possibility of proving any of the axioms or first principles of geometry. Thus, the a priori concepts of triangle, angle,etc., however deeply they are analysed, will never entail that the denial of the judgment concerning the internal angles of a triangle is self-contradictory. Hence, the judgment cannot be an analytic a priori truth. I need to engage in a demonstration, a showing: I need, that is, to go outside the a priori concepts involved and exhibit in intuition, by means of the constructing of a diagram, that the corresponding judgment holds. That is why the a priori judgment is synthetic, and not analytic. Moreover, since geometrical demonstrations essentially involve proving synthetic yet necessary relationships about extension and figure (inherently spatial concepts), geometry is conceived as describing the structure of space. As Kant sees it, geometry is a body of synthetic a priori judgments that determines the properties of space (B 40).
A parallel approach is adopted with judgments of arithmetic. The judgment that 7 + 5 = 12 requires that the sum 7 and 5 must be constructed. For the concepts 7, 5 and the sum of – concepts which, as Kant sees it, are not derived from experience – never yield through analysis the concept 12. They merely together tell us how the result is to be constructed, viz. by counting (whether by means of mental arithmetic or e.g. by using one’s fingers). Only when the a priori concepts of 7 and 5 are realized in intuition, and then joined together, can the resulting number be brought into existence, i.e. exhibited by this constructive process of adding the two numbers. Since one must go outside any mere analysis of the concepts involved, and exhibit the result by construction in intuition, the judgment must be synthetic as well as a priori.
Although it is not difficult to see why, given Kant’s claim about the manner of proving geometrical judgments, he should argue for a parallel process for arithmetic, it is not so easy to grasp why he thinks that arithmetic, together with pure mechanics, are possible only by means of our possession of an intuition of time. His argument seems to be that, in the case of arithmetic, the various concepts of number are all realized by successive addition or subtraction (succession being a temporal notion); and in the case of mechanics, its fundamental concepts are all concerned in some way with motion (which itself requires a recognition of how the same thing can be in distinct places, viz. through its existence in these distinct places, one after the other).At any event, in his discussion of the synthetic a priori judgments of arithmetic and pure mechanics, he links these judgments with the necessity of our having a sensible intuition of time.
Criticism of the thesis that mathematical judgments are synthetic a priori
By the end of the first quarter of the twentieth century, a consensus had grown up that Kant’s view of the synthetic a priori nature of mathematics is untenable. The basis for the consensus can be illustrated most easily in the case of geometry. Euclidean geometry – the criticism runs – should be considered as either an a priori or an a posteriori discipline. Let us assume, first, that it is an a priori discipline. As such, the whole Euclidean system is a set of uninterpreted formulae, like a logical calculus. Proofs, on this account, are entirely formal, and there is no need to draw any diagrams, whether in reality or in the imagination, to establish any axiom or theorem. Thus Euclid’s axiom ‘A straight line is the shortest distance between two points’ is taken as an uninterpreted definition of the term ‘straight line’; and so on for all the other axioms of the system.The theorems of the system are established by means of rules of inference from these axioms. So understood, Euclidean judgments can have no application to the world, since the terms in the proofs are all uninterpreted, being either undefined terms or defined by means of the undefined terms. Admittedly, because the theorems result logically from the axioms, i.e. by means of the system’s rules of inference, they do follow necessarily from the axioms. But since the basic terms are merely uninterpreted symbols, there can be no question here of any Euclidean judgment holding for the structure of space.
Second, let us assume that Euclid’s system is a posteriori. As such, the Euclidean axioms can, in principle, be true judgments about the structure of space. Further, since they are based on experience, they must be synthetic (to deny any of them is not self-contradictory). But although the axioms may be true synthetic judgments, they are not, pace Kant, necessarily true. They will be merely true empirical generalizations about the structure of space. Thus the straight line axiom should here be taken as an empirical claim to the effect that since it has always been found in fact that the shortest distance between any two points in space is a straight line – where ‘straight line’ is given some suitable empirical definition, e.g. ‘the path of a light wave in a vacuum’ – it is held, on inductive grounds, that this is always the case. Consequently, the theorems of the system, since they follow logically,