Fermat’s Last Theorem and the Logicians

Abstract

I re-examine Enderton’s exposition of number theory in his logic textbook, and look at Gödel’s incompleteness theorems, to show that there too definitional extension is both too strong and two weak; the disparity and the resultant ambiguity in both the textbook and the proofs testify to the disparity between the referential discourse of arithmetic and number theory and the analytic discourse of logic. Then I examine parts of Andrew Wiles’ proof of Fermat’s Last Theorem in more detail, since the proof combines disparate discourses in a strikingly ampliative and inspiring way. I also discuss attempts by the logicians McLarty, Friedman, and Macintyre to rewrite the proof and try to show that the aims of logicians are different from the aims of number theorists. This disparity can however contribute to the growth of knowledge as long as both sides tolerate each other and remain open to novel kinds of interaction, with neither claiming to have the Ultimate Discourse.