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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 19th
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. 


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