Kant's Failure to Argue for Synthetic a Priori Mathematical Truths In this paper I argue that Immanuel Kant's argument that pure mathematics, both arithmetic and geometry, are a priori synthetic judgments fails to demonstrate that they are not, in fact, a priori analytic judgments. Kant utilizes examples in the B Introduction to his 'Critique of Pure Reason' to demonstrate his point that geometry and arithmetic are both a priori synthetic judgments. These simple examples provide him with a somewhat convincing and strong argument for his conclusions. The crux of his argument focuses on demonstrating their synthetic nature, as the fact that they are a priori is well established and generally agreed upon by his critics and contemporaries at the time. Demonstrating that these sub-disciplines of pure mathematics are a priori synthetic judgments is imperative to supporting the rest of his work - he has to verify that a priori synthetic judgments are actually existing and possible forms of knowledge, something that was not considered as a possibility in Kant's time. Kant argues that arithmetic is made up of a priori synthetic judgments. Kant's argument is entirely by example. He asks the reader to consider the proposition '5 + 7 = 12'. On the face of it, Kant observes, this might be interpreted to be an analytic proposition, which follows from the concept of the law of noncontradiction. However, if one is to consider the statement 'five plus seven' more closely, it is discovered that the concept of the sum of five and seven represents only a unification of these two individual numbers. It does not, in fact, yield the singular number 'twelve' by mere examination of the sum. Kant states that one must "seek assistance in the intuition" (B15) and utilize one's fingers or consider points, and only after unifying those five fingers with seven in addition or gathering five points to seven points can one arrive at the concept of twelve - something must also be done above and beyond the statement 'the sum of five and seven' in order to conclude 'twelve'. In other words, the concept of twelve is not contained within the predicate of the addition of five and seven, and is hence not an analytic judgment. This entails, therefore, that arithmetic is a synthetic judgment. Kant claims that this point is more salient with larger numbers that aren't so readily combined. I take issue with Kant's apparent conception of numbers. He seems to believe that numbers are somehow disconnected from one another. He takes twelve to be a distinct object from five and seven and, it appears, even from the sum of the two. However, this seems misguided. His argument is clearly laid out from the start as '5 + 7 = 12'. This is a statement of identity. Kant has already granted that statements of identity are analytic propositions ''necessarily''. It would be an absolute contradiction for '5 + 7' to not be '12'. This is, of course, what he initially points out in his argument. Kant states that "the concept of twelve is by no means already thought merely by my thinking of that unification of seven and five" (B15), but how isn't it? One can perform all of the same arithmetical operations on 5 + 7 just as if it were 12. Indeed, it seems that 12 is really only a kind of placeholder for '5 + 7'. 12 can be considered to be a name with '5 + 7' as its referent. One can go even further with this idea and claim that 5 (and likewise 7) are names with a similar referent. Giuseppe Peano's axiomatization of the natural numbers yields a very simple system with which we can actually do this. The natural numbers can be constructed with the use of a successor function and a starting point, namely 1. We can represent the object denoted by '2' as 'the successor of 1', or, more simply, s(1). Thus, the claim '5 + 7 = 12' can be written as 's(s(s(s(1)))) + s(s(s(s(s(s(1)))))) = s(s(s(s(s(s(s(s(s(s(s(1)))))))))))'. It may seem that this is simply the same problem with a different, longer name, but it isn't - addition is merely a successor operation. s(1) can be denoted as 1 + 1 if we so choose to translate s(1) in that way - they are isomorphic representations. That is to say, they are absolutely identical. Therefore, our statement is actually 's(s(s(s(s(s(s(s(s(s(s(1))))))))))) = s(s(s(s(s(s(s(s(s(s(s(1)))))))))))', which is clearly a statement of identity, and thus the idea that '5 + 7 = 12' contains twelve within the predicate of a sum, and is an analytic judgment. However, even without considering referents or what the natural numbers actually are, it can be demonstrated that Kant is still misguided in his thinking. He approaches the judgment '5 + 7 = 12' very particularly - namely, he focuses on the sum of five and seven as opposed to the singular twelve. If Kant does not grant that twelve contains many different things within it - twelve is an even number, it is greater than ten, it has six divisors, etc. - it would seem to be an empty kind of object. What exactly are the properties of 'twelve' if we do not have any of these ideas given to us within it? It becomes a vacuous thing, a meaningless inkblot on a page. Kant needs to provide a definition of the number without at all appealing to arithmetical methods. This is not something that I am sure he is able to do, nor has he provided such an understanding. There is no reason to suppose that twelve cannot be the subject of this proposition, and it certainly contains within it a variety of concepts, particularly those of factoring. The sum runs both ways, and as a result each part of the identity must contain something for which the particular predicate appeals to concepts. Thus, arithmetic has a well-grounded foundation in a priori analytic judgments, not synthetic. Kant also argues that the propositions of geometry are a priori synthetic judgments. Kant appeals to an example in his argumentation for this point as he did to demonstrate the a priori synthetic nature of arithmetic. He claims 'the straight line between two points is the shortest' is a synthetic proposition, as "[his] concept of the straight contains nothing of quantity, only of quality" (B16). We must appeal to our intuitions in order to marry to this proposition of 'the straight' to the concept of 'shortest distance', and only through this synthesis of our knowledge is a judgment possible. Kant also offers up another example in a secondary writing: "that the radius can be carried over into the circumference 6 times cannot be derived from the concept of circumference" (Notes, page 718, number 18 refers to another text). These examples are closely related to those that he provided for the a priori synthetic nature of arithmetic. Kant utilizes these two examples to demonstrate that geometry, as a pure conception, is a priori synthetic knowledge. This might very well be true; a conception of straight does not within it readily provide the idea of a shortest distance. However, geometry considered in a pure form has an axiomatic structure. That is to say, pure geometry does not make truth claims. Precisely for this reason 'the straight' _should not_ contain the concept of 'shortest distance', as it is not the case that such a thing is true - it is only the case that this is a consequence of the distance formula, derived from the axiomatic principles outlined by Euclid. Indeed, Kant only considers Euclidean geometry as a case of 'pure geometry', which is rather understandable considering other geometries were completely unheard of in Kant's time. For example, hyperbolic geometry, formally explored in the 19^{th}century, provides a robust axiomatic system demonstrating an entirely different way of thinking about space (in a geometric sense) and objects within this space. It stems from a rejection of the parallel postulate, which is one of the five axioms of Euclidean geometry. The parallel postulate states that through any line R there exists a point P such that exactly one line can be drawn containing P that does not intersect R - that is to say, any line has a uniquely determined parallel line. In hyperbolic geometry, there are at least two such lines. The axiomatic nature of this system and of other pure geometries - a variety exist beyond Euclidean and hyperbolic, such as elliptic geometry - demonstrates that they are not synthetic judgments (Gardner, 58). Even if such geometries were in fact synthetic judgments, it is not necessarily the case that they are a priori. Roughly 100 years after the publishing of Kant's 'Critique', Einstein demonstrated with his theory of relativity that the universe was not even a Euclidean space - it is only locally Euclidean. This means that Euclidean geometry only offers up a close approximation of the nature of space on a small enough scale. This discovery was empirically derived, demonstrating that the conception, which Kant had, of geometry was not an a priori one. Considering these facts, it is clear that Kant has failed to demonstrate that geometry is either a priori or even synthetic - indeed, he is wrong in both cases. From these points it is clear that Kant has failed to establish the a priori synthetic nature of arithmetic or geometry, the two pure mathematical disciplines that he considers in his 'Critique of Pure Reason'. The nature of numbers that Kant considers seems far too limited or inadequate when one explores what precisely his interpretation of the statement '5 + 7 = 12' entails, or rather what it does not entail. Overall, Kant offers an unconvincing argument for synthetic a priori judgments in arithmetic and concedes that his example is more clear when larger numbers are considered, which might show that arithmetic propositions are really only synthetic when they take a prolonged time period of time to verify or evaluate - a rather vague explanation of what it means to be an a priori synthetic judgment to say the least. Likewise, his understanding of geometry is neither necessary nor synthetic depending in the consideration. However, his failing with respect to geometry seems to more so be a limitation of his time period rather than a discrepancy in his argumentation - this is simply a case where Kant is just wrong in his understanding due to an inadequate comprehension of what precisely the universe is like in a rigorously defined physical or empirical verified framework. Overall, Kant simply fails to establish strong enough reasons to believe that he is right to assert that pure mathematical propositions, let alone mathematical propositions in general, are a priori synthetic judgments. ________________________________________________________________________________ Dilyn Corner (C) 2020-2022