10 - Frege’s mathematical setting

Summary

I have not yet any clear view as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties.

John Graves, 'Letter to William Hamilton'

INTRODUCTION

Although Gottlob Frege was a professional mathematician, trained at one of the world's greatest centres for mathematical research, it has been common for modern commentators to assume that his interests in the foundations of arithmetic were almost entirely 'philosophical' in nature, unlike the more 'mathematical' motivations of a Karl Weierstrass or Richard Dedekind. As Philip Kitcher expresses the thesis:

The mathematicians did not listen [to Frege because] . . . none of the techniques of elementary arithmetic cause any trouble akin to the problems generated by the theory of series or results about the existence of limits.

Indeed, Frege's own presentation of his work easily encourages such a reading. Nonetheless, recent research into his professional background reveals ties to a rich mathematical problematic that, pace Kitcher, was as central to the 1870s as any narrow questions about series and limits per se. An appreciation of the basic facts involved (which this essay will attempt to describe in non-technical terms) can only heighten our appreciation of the depths of Frege’s thought and of the persistent difficulties that any adequate philosophy of mathematics must confront. Although it may be possible to appreciate Frege’s approach to language on its own terms, some awareness of the rather unusual examples that he encountered in the course of his mathematical work can enhance our understanding of his motivations within linguistic philosophy as well.