Kant and Mathematical Knowledge
© Thomas J. McFarlane 1995

Reason, or the ratio of all we have already known, is not the same that it shall be when we know more.

           -William Blake [31]

In the Preface to the Second Edition of the Critique of Pure Reason, Kant proposes to save metaphysics from its random groping by instigating a revolution analogous to those which set mathematics and physics upon the secure path of science. The true method, he claims, rests in the realization that reason can only have certain knowledge of that which is necessarily presupposed a priori by reason itself. In particular, the objective validity of mathematical knowledge according to Kant rests on the fact that it is based on the a priori forms of our sensibility which condition the possibility of experience. Mathematical developments in the past two hundred years, however, have challenged Kant's theory of mathematical knowledge in fundamental respects. In this paper I will critically examine the challenges non-Euclidean geometry poses to Kant's theory and then, in light of this analysis, briefly speculate on how we might understand the foundations of mathematical knowledge. Before we enter into the details of criticism, however, we must first examine Kant's view of mathematical knowledge in some of its details. This discussion will introduce important elements of Kant's theory which will be relevant to the criticism which follows.

To explain the secure method of science which is the secret of mathematical certainty, Kant offers us the example of demonstrating the properties of the isosceles triangle. The procedure is not to empirically examine an imperfect sketch or drawing. Nor is it to just think about the definition. Rather, the true method of discovering its properties is for the mathematician to "bring out what was necessarily implied in the concepts that he had himself formed a priori, and had put into the figure in the construction by which he presented it to himself."[32] Only in this way, by considering the very presentation of the triangle as determined a priori, is certain knowledge of it possible at all. In an appearance, we thus arrive at a distinction between the form, which we necessarily supply a priori, and the content, which is given to us in sensation.

Because Kant is ascribing the a priori conditions to the sensibility, they provide us with certain knowledge that is more than purely logical. In other words, Kant is affirming the existence of a priori synthetic knowledge, which Kant distinguishes from analytic knowledge. Analytic propositions are tautological truths, which rest on definition and logic alone, and thus are all a priori. For example, 'it is either raining or not raining' is an analytic proposition. Although it contains reference to empirical facts, its truth is independent of the empirical situation. Analytic propositions are not really about empirical facts but about logical relations. Thus to deny an analytic proposition is to assert a contradiction. The denial of a synthetic proposition, on the other hand, is not a logical contradiction. But its denial may nonetheless contradict the state of affairs, for synthetic propositions extend our knowledge beyond mere logical relations. Empirical knowledge (e.g.., 'it is raining') is synthetic, for it asserts more than is determined by logic and definitions alone. While most of our synthetic knowledge is empirical, Kant makes the bold claim that there is synthetic knowledge which is nonetheless a priori. In other words, independently of experience we can have knowledge which tells us more than logic alone will. Mathematics, according to Kant, is synthetic a priori knowledge. While it is not based on experience, it still tells us something positive about what is necessarily the case for our world. How is this possible?

How synthetic a priori judgments are possible is the famous question which begins Kant's Critique. In the case of mathematics, Kant answers this question in the Transcendental Aesthetic. There he argues that space and time are conditions of the possibility of appearances, i.e., they must be presupposed for appearances to be represented to us at all. Space and time are therefore the a priori forms of our sensible intuition. And because we have access to these forms through pure intuition (independent of sensation) we can know a priori the properties of space and time. Furthermore, since all appearances necessarily conform to these properties, we have a priori synthetic knowledge of what is the case. As Kant puts it, "the apodeictic certainty of all geometrical propositions, and the possibility of their a priori construction, is grounded in this a priori necessity of space."[33] We should here make particular note of Kant's claim that (Euclidean) geometrical propositions are not at all arbitrary but possess necessary synthetic truth a priori.

There are two points concerning Kant's theory of space which will be important for us later. First, it should be emphasized that while judgments about the properties of space have objective validity (i.e., space is empirically real), Kant maintains that space is not a thing-in- itself with objective existence (i.e., space is transcendentally ideal). We must not forget that space is a subjective condition of sensibility and that space is nothing if considered independent of our sensibility (i.e., in itself).

The second point which will be of importance is Kant's distinction between what Allison[34] calls a determinate (conceptualized) intuition of space and an indeterminate (unconceptualized) intuition of space. Space is not only represented a priori as the form of sensible intuition but as itself an intuition containing a manifold in its own in unity: "Space, represented as object (as we are required to do in geometry), contains more than mere form of intuition; it also contains combination of the manifold, given according to the form of sensibility, in an intuitive representation, so that the form of intuition gives only a manifold, the formal intuition gives unity of representation."[35] The pure intuition of space, therefore, is not an immediate grasping of the spatial form of intuition itself, but rather an intuition whose content is this form of intuition. This is Kant's explanation of how the a priori knowledge of pure intuition functions. Thus a priori synthetic judgments of empirical space are possible because all our intuitions, both pure (formal) intuitions and empirical intuitions, rest on this form of intuition; and through the science of geometry as the study of the pure intuition of space we have systematized this knowledge in a science.

The claim that geometry is, in fact, a priori synthetic knowledge of space was challenged when mathematicians in the last century constructed non-Euclidean geometries -- consistent geometrical systems based on axioms which differ slightly from Euclid's. (Before this time Euclidean geometry was the only geometry and no other was considered to be possible.) But although these alternate geometries were demonstrated to be logical possibilities, and thus analytic a priori, most philosophers still maintained that only Euclid's geometry was synthetic a priori.[36] Einstein's theory of general relativity, however, demonstrated that empirical space is non-Euclidean, and this development made it clear, according to Rudolf Carnap, that we must "distinguish between pure or mathematical geometry and physical geometry. . .Mathematical geometry holds indeed a priori, as Kant asserted, but only because it is analytic. Physical geometry is indeed synthetic; but it is based on experience and hence does not hold a priori."[37]

This situation seems to demand that we abandon Kant's claim that geometry is a priori synthetic, for on Kant's theory the Euclidean space which we intuit a priori as the condition of all sensible intuition is the only possible empirical space. But since the actual empirical space (according to Einstein) is non-Euclidean, Kant's theory is thus denied. A reevaluation of both Kant's theory and the basis of geometrical truth is thus called for. Along these lines I will focus on the following questions. First of all, how do we reconcile the empirical facts of general relativity with the inner necessity of Euclidean geometry? If we must forfeit the synthetic a priori with respect to space, what are we to make of Kant's arguments? Does Kant's theory actually constrain him to Euclidean geometry in the first place? What does it mean to say that empirical space has a certain structure?

Euclidean geometry is based on a set of axioms. The axiomatic structure of geometry is the source of the system's power as well as its weakness, for since all the theorems can be logically deduced from the axioms alone, any theorem is only as true as the axioms from which it was derived. But Euclid's axioms are so self-evident that the truth of the entire system of theorems is assured. While the demonstrations from the axioms to the theorems proceeds analytically, the truth of the axioms themselves must rest on something other than mere logic alone, and Kant therefore regards geometry as synthetic.[38] When the axiom of the parallels was altered, however, and non-Euclidean geometries were consistently developed from such a non-intuitive axiom, it became clear that geometrical systems need not have self-evident axioms. At this point mathematics has two options. It could exclude from its domain the study of systems which are not based on self-evident truths, perhaps regarding them as non-intuitive nonsense. Or mathematics could regard the truth of its assumptions as irrelevant to its purposes, defining mathematical truth as merely implicational. Whether or not the axioms of geometry are true in themselves or not is then an empirical (or perhaps philosophical) question. What is truth, and where does it rest? In empirical verification? In intuition or self-evidence? Or in logical relations alone?

Hans Reichenbach, in The Philosophy of Space and Time claims that mathematics is analytic a priori truth and that the synthetic truth of a geometry is an empirical question. But it is not this simple, he warns: "Geometry is concerned solely with the simplicity of definition. . .We can no more say that Einstein's geometry is 'truer' than Euclidean geometry, than we can say that the meter is a 'truer' unit of length than a yard."[39] This comment stems from the fact that we can maintain Euclidean space even in general relativity so long as we adopt a warped definition of measurement. Once we define our coordinates, however, the objective structure of space becomes an empirical question. Thus the question is not which geometry is empirically real, but which definition of space gives the empirical laws their simplest form. So the truth of the geometrical axioms does not rest in self-evidence or a priori intuition, nor does it rest exclusively in empirical verification. We are left with various axiomatic systems, and the only question is, Which definition of space is most valuable?

The present situation in mathematics and physics seems to indicate that Kant was certainly mistaken when he claimed that Euclidean geometry was synthetic a priori truth. (Or, indeed, that any geometry is synthetic a priori truth.) But is Euclidean geometry a necessary consequence of Kant's theory in the first place? He certainly does not anywhere himself derive Euclid's axioms from the a priori intuition of space. But, as we noted earlier, Kant claims that the a priori synthetic truth of Euclidean geometry does rest on our pure intuition. And since Euclidean geometry is not, in fact, synthetic a priori truth, either (1) it is not entirely based on our pure intuition of space or (2) our pure intuition of space is wrong. The latter alternative (2), I submit, is nonsense. For our pure intuition of space is the necessary precondition for all outer appearances and we can have no appearances which are not in conformity with this a priori condition of our sensibility. Thus it is, by definition, impossible for anything in experience to contradict it and so it is meaningless to say that it is somehow 'wrong.' We are left, then, with the former alternative (1), that Euclidean geometry is not completely determined by our pure intuition of space alone.

This conclusion, however, leads to a further problem. What are we to make of the conviction of necessity which, Kant writes[40], accompanies our judgments in mathematics, and, in particular, Euclidean geometry? In attempt to make sense of this situation, Reichenbach has suggested a theory of what he calls the visual a priori. While, as we noted earlier, he has argued that the notion of a unique a priori empirical space is meaningless, he nonetheless offers the idea of an a priori visual space. In our act of visualization, Reichenbach distinguishes between the mere image- producing function and the normative function that tacitly places logical constraints on the images: "Kant's synthetic a priori of pure intuition springs from the normative function of visualization. It is this function that tends to single out Euclidean geometry from all the others; it seems to compel us to regard Euclid's axiom of the parallels as unquestionably true."[41] For example, if I am asked, "what is the sum of the interior angles of a triangle?" I will answer, in accord with Euclid, "180 degrees." But I have tacitly assumed that the triangle exists in the plane. On a sphere, the angles of a triangle add up to more than 180 degrees. Furthermore, I can easily visualize this and recognize it to be the case without proof. Take, for example, a triangle constructed from three lines on the surface of the earth as follows: At the north pole draw two lines south at right angles to each other. Each of these lines will intersect the equator at right angles. Take the portion of the equator between these lines as the third side of the triangle. Now it should be evident that the angles of this triangle add to 270 degrees, and not 180.

While Reichenbach's idea of the visual a priori seems to undermine Kant's whole effort in the Transcendental Aesthetic, I believe that we can understand his distinction between the image-producing function and the normative function of visualization in terms of Kant's distinction between the form of intuition and formal intuitions. In other words, the mere capacity to produce images is the indeterminate pure form of space itself which is a necessary condition of all appearances, just as Kant maintained. But the normative function is the collection of habitual preconceptions which tacitly underlie our formal intuitions of space as a determinate representation. On this interpretation, a priori space itself is more primitive than any formal intuition of it, Euclidean or non-Euclidean. It still maintains its a priori synthetic truth, but only insofar as Kant demonstrated in the Aesthetic. Formal intuitions of determinate objects, on the other hand, introduce assumptions regarding the character of space and do not have a priori synthetic truth. Rather, they are merely habitual and, consequently, malleable. In fact, Reichenbach maintains that "whoever has successfully adjusted himself to a different congruence is able to visualize non-Euclidean structures as easily as Euclidean structures and to make inferences concerning them."[42] It takes much effort, he adds, but after extended experience, one reaches the point, as in the study of a foreign language, where suddenly it makes sense without being translated into our mother tongue, Euclidean space.

Thus far, I have argued that while the developments in mathematics and physics since Kant falsify Kant's claim that Euclidean geometry is a priori synthetic, the fundamental claim of the Transcendental Aesthetic can nonetheless still be maintained, namely, that space is an a priori form of outer intuition and a condition for the possibility of outer appearances. Beyond this primitive epistemological property, however, it is doubtful that much more can be said without making the arbitrary assumptions of some particular geometry.

With geometry uprooted from its supposed a priori synthetic grounding it is perhaps not at all surprising that the a priori synthetic foundations of other areas of mathematics were also questioned. Indeed, as a result of the revolution in geometry, arithmetic, algebra, calculus, and other branches of mathematics were distilled down to the axioms of set theory. But, as Quine[43] has noted, there is no unique set of axioms for set theory any more than there is a unique set of axioms for geometry. In all of mathematics, therefore, we are free to choose our axioms arbitrarily and develop their logical consequences regardless of whether our system is useful and self-evident or not. As Ayer has argued in Language, Truth, and Logic mathematical truths are all a priori analytic, or tautological: "the axioms of a geometry are simply definitions, and. . .the theorems of a geometry are simply the logical consequences of these definitions. . .All that the geometry itself tells us is that if anything can be brought under the definitions, it will also satisfy the theorems."[44] But this also extends to arithmetic. Whereas Kant argues that the proposition '7+5=12' is synthetic[45], Ayer maintains that its truth "depends simply on the fact that the symbolic expression '7+5=12' is synonymous with '12'."[46]

Reducing all of mathematics to a priori analytic propositions may solve many of the difficulties faced by a view of mathematics akin to Kant's, but it also raises questions of its own. As Poincaré put it, "Are we then to admit that the enunciations of all the theorems with which so many volumes are filled are only indirect ways of saying that A is A?"[47] Ayer's answer, in short, is 'yes': The tautologies are remarkable and surprising simply because they are not immediately evident to our limited minds.[48] But as the foregoing discussion has made clear, these empty tautologies based on arbitrary definition alone were, at least in the case of Euclidean geometry, taken for two thousand years to be self-evident and necessary truths about the world. How is it that we can mistake an a priori analytic truth for an a priori synthetic truth?

The developments in mathematics in the past two hundred years have taught us some profound lessons concerning the nature of mathematical knowledge and the analytic/synthetic distinction in general. We have seen in our most precise area of knowledge how an apodeictic truth can become mere convention, pushing the unknown foundations of unquestioned, self-evident truth even deeper. In a very general sense, the analytic/synthetic distinction can be seen to represent the line between knowledge which we have made fully conscious and knowledge whose foundations are still partially veiled. As the understanding matures the line between the known and the unknown gets pushed deeper down, and what was previously taken as self-evident (synthetic a priori) is revealed as simply one logical (analytic a priori) possibility. Consequently, the line between Kant's real possibility and logical possibility also shifts. What we once consider as governing real possibility and synthetic a priori truth (such as Euclid's axioms), is actually a limitation on logical possibility, or analytic truth, and is only contingent. The ground of synthetic a priori truth, therefore, seems to be conditioned rather than necessary. As the mathematician G. Spencer-Brown says, "the validity of a proof [based on synthetic a priori knowledge] rests not in our common motivation by a set of instructions, but in our common experience of a state of affairs."[49]

The consequences of this view of the synthetic/analytic distinction clearly implies that Kant's views on where the distinction is to be drawn merely reflects the knowledge of his day, and its limitations. But if the a priori synthetic truths condition the possibility of experience, and if the synthetic/analytic line can be shifted, this implies that experience is, in fact, malleable. Like learning to visualize non-Euclidean space, once we recognize that what was previously taken for granted is actually an arbitrary structure, we are free to change that conditioning. In this manner, the very definition of experience could be altered, for we have changed the rules which constitute it. As Kant said, "we can know a priori of things only what we ourselves put into them."[50] And who's to say what we put into things can not be changed? While it is one thing to recognize the possibility of altering our preconditions of experience, it is quite another matter to actually transform those conditions and experience a different world.