"Philosophy of Mathematics, Narrative and Fermat’s Last Theorem"

"Philosophy of Mathematics, Narrative and Fermat’s Last Theorem"http://www.haverford.edu/calendar/details/193402Gest Center 1012012-02-17T14:00:002012-02-17T16:30:00

February 17, 2:00PM

Gest Center 101

The Department of Philosophy in conjunction with the Distinguished Visitors Program presents a talk by Emily Grosholz, Liberal Arts Research Professor, Department of Philosophy, Pennsylvania State University

Description

In a Platonic dialogue, an intelligible unity is apprehended, but imperfectly, and analysis leads to increased understanding via a search for the conditions of intelligibility. The form of dramatic dialogue expresses something essential to the process of coming to understand, for a dialogue may be read both as a set of arguments (in many of which reductio ad absurdum plays a pivotal role) which uncover the logical presuppositions of various claims, but it may also be read as a narrative, a process in time and history. Mathematical analysis is the search for both conditions of the intelligibility of problematic objects, and conditions of solvability of objective problems. It is an ampliative process, that increases knowledge as it proceeds. From an analysis, an argument can be reconstructed, as when Andrew Wiles finally wrote up the results of his seven years-long search in the over one hundred page full dress proof in the May, 1995 issue of Annals of Mathematics. The story of Wiles’ arrival at the insights that drove his proof is dramatic, and indeed must be narrated historically as well as logically presented. (Up till now, logicians have not been satisfied with Wiles’ proof, though it is accepted by the community of number theorists.) The Italian philosopher of mathematics Carlo Cellucci contrasts the analytic method of proof discovery with the axiomatic method of justification, which reasons “downwards,” deductively, from a set of fixed axioms, as well as with algorithmic problem-solving methods. He argues that the primary activity of mathematicians is not theory construction but problem solution, which proceeds by analysis, a family of rational procedures broader than logical deduction. We will take a closer look at the kind of ampliative reasoning that Wiles carried out, and also see what the logicians have to say about it.

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