The knowledge that we are claimed to be able to have a priori (without appealing to experience) is divided into two kinds: analytic and synthetic. The analytic a priori truths are those that can be determined to be true on the basis of the meanings of the words (or concepts) involved, while the synthetic a priori truths are not known in that way, but can be determined to be true in some other fashion (the exact details being left to specific theories). The reason this distinction is made (and why Kant made the distinction) is so that we can focus on the epistemological justification of the synthetic a priori truths. The synthetic a priori truths can, at least in principle, include any claim whatsoever, and thus investigating them seems worth doing because of the possibility for such investigations to provide a foundation for disciplines such as metaphysics. Of course there are also those who, upon considering the synthetic a priori, conclude that there are no such statements, which motivates attempts to translate everything believed to be a priori and true into some kind of analytic statement.

The distinction between the two categories, and why we care which statements fall under, thus ultimately comes down to the epistemic distinctions between the two. And it is easy to interpret the motivations behind Frege’s project of proving the truths of arithmetic solely from logic in these terms. Frege at many points says that mere intuition cannot be a suitable basis for arithmetic, which we might interpret as indicating that arithmetic cannot be allowed to be synthetic a priori, since intuition is often considered to be the justification behind that category of truths. And so, by reducing arithmetic to logic, he can demonstrate that arithmetic is actually a collection of analytic truths, and thus free of intuition.

The problem with this understanding of Frege’s project is that it completely misinterprets his ideas on the foundation of mathematics and the distinction between the analytic and synthetic. Let’s first consider how we can come to know logical truths according to Frege. Frege seems to be a logical realist, in the sense that he believes that logical objects, such as functions, have an objective existance “out there” in some sense. He also seems to believe that we have a kind of intellectual access to these logical objects, and thus that we can determine the truth of a logical axiom simply by consulting the logical objects themselves. And Frege doesn’t believe this just about the logical objects. When it comes to language he believes that we have access to the senses of words in basically the same manner. And in his discussion of Hilbert’s proof of the independence of certain geometrical axioms he implies that the geometrical objects are also available in this way. (Actually he asserts that Hilbert cannot meaningfully consider the negation of geometrical axioms because, Frege insists, that those axioms are true given the meaning of geometrical terms. Given that Frege often uses meaning where we would use reference this strongly indicates that Frege thinks that the geometrical axioms are true because of the basis of the geometrical objects, graspable by our intellect, involved. And thus to assert the negation of the axiom would necessitate dealing with different geometrical objects and hence could show nothing about the original geometric system.)

Frege also redefines the analytic-synthetic distinction. For Frege the analytic truths are not those which are true because of the concepts involved (as Kant defined them), rather they are those that are true because they are entailed by logic. Logic is thus analytic by definition, and any truth that is reducible to logic is analytic as well. Thus reducing arithmetic to logic makes it analytic. If it couldn’t be reduced arithmetic would have to be synthetic, as geometry is claimed to be by Frege. But note that Frege is not opposed to the existence of synthetic truths, he has no problem with the idea that we might know the axioms of geometry to be true in virtue of our ability to grasp the geometrical objects.

But now we have a problem, which is that reducing arithmetic to logic no longer seems to serve a purpose. In our original conception of the analytic-synthetic distinction whether a truth was one or the other determined how we could know it to be true, and thus a proper investigation of analytic and synthetic truths would proceed in fundamentally different ways. But for Frege there is no epistemological difference between the analytic and synthetic, we can know statements of both classes to be true by our intellectual grasp of the objects involved, its just that analytic truths involve only our ability to grasp the logical objects. And it isn’t the case that our grasp of logical objects works in a special way or is more certain than our grasp of others, such as the geometrical objects, or at least Frege gives no indications that he believes this to be the case. Thus it simply doesn’t seem to matter whether the arithmetic truths can be known on the basis of our grasping the logical objects or whether they require us to appeal to the numerical objects.

Of course there are some possible answers to this dilemma in Frege’s work, but none of them are significantly elaborated on. One possibility is that by reducing the truths of arithmetic to those of logic that Frege wants to illuminate a connection between the “laws of number” and the “laws of thought”. While tantalizing it is hard to make much sense out of this suggestion since Frege strongly rejects psychologism. At another point Frege also suggests that somehow our intellectual grasp of the numbers is deficient, and that we actually have a better grasp of the geometrical and logical objects. Thus reducing arithmetic to logic would remove the possibility of error when dealing with it. Although this suggestion is more consistent with Frege’s work in general it brings with it its own problems, specifically in the possibility of inaccurately deriving arithmetic from logic, or from deriving something very similar to arithmetic from logic that we falsely believe to be arithmetic. And finally there is the possibility that Frege was primarily concerned with demonstrating that arithmetic was a priori and not deduced from experience, either of the world or of thinking, and since all analytic truths are a priori showing arithmetic to be analytic rules out such possibilities. But while this is certainly a side effect of Frege’s reduction it is hard to understand it as his primary motivation because he spends no time arguing that the logical laws or the geometrical laws are also a priori.