Frege ' s Theorem : An Introduction

1. Opening w HAT IS T H E EPISTEMOLOGICAL STATUS O F O U R KNOWledge of the truths of arithmetic? Are they analytic, the products of pure reason, as Leibniz held? O r are they high-level empirical truths that we know only a posteviori, as some empiricists, particularly Mill, have held? O r was Kant right to say that, although our lulowledge that '5+7=12' depends essentially upon intuition, it is nonetheless a pviori? I t was with this problem that Gottlob Frege was chiefly concerned throughout his philosophical career. His goal was to establish a version of Leibniz's vie\\ (and so to demonstrate the independence of arithmetical and geometrical reasoning), to show that the truths of arithmetic follow logically from premises which are themselves truths of logic. I t is this view that we now call Lotjicism. Frege's approach to this problem had a number of strands, but it is simplest to divide it into two: a negative part and a positive part. The negative part consists of criticisms of the views of Mill and Kant, and others who share them. These criticisms, although they are found in a variety of places, mostly occur in the first three chapters of Die Grundlatjen der Avithmetik.' The positive part consists of an attempt to show (not just that but) how arithmetical truths can be established by pure reason, by actually giving proofs of them from premises which are (or are supposed to be) truths of pure logic. There is thus a purely mathematical aspect of Frege's project: it is this on which I intend to focus here. Frege was not the first t o attempt to show how arithmetical truths can be proven from more fundamental assumptions. His approach, however, was more rigorous and encompassing, by far, than anything that had come before. Leibniz, for example, had attempted to prove such arithmetical truths as "2+2=4." His proofs, however, like those of Euclid before him, rest upon assumptions that he does not make explicit: for esample, Leibniz make free appeal to the associative law of addition, which says that (a+b)+c = a+(b+c), i.e., allows himself freely to "re-arrange" parentheses. But it is essential to any attempt to determine the epistemological status of the laws of arithmetic that we be able precisely to determine upon what