## Formal proofs in geometry

- Date: 10/06/2006

Thomas C. Hales (University of Pittsburgh)

University of Calgary

Traditional mathematical proofs are written in a way to make them

easily understood by mathematicians. Routine logical steps are omitted.

An enormous amount of context is assumed on the part of the reader.

Proofs, especially in topology and geometry, rely on intuitive

arguments in situations where a trained mathematician would be capable

of translating those intuitive arguments into a more rigorous argument.

In a formal proof, all the intermediate logical steps are supplied. No

appeal is made to intuition, even if the translation from intuition to

logic is routine. Thus, a formal proof is less intuitive, and yet less

susceptible to logical errors. It is generally considered a major

undertaking to transcribe a traditional proof into a formal proof.

In recent years, a number of fundamental theorems in mathematics have

been formally verified, including the Prime Number Theorem, the Four

Color Theorem, and the Jordan Curve Theorem.

PIMS Distinguished Chair Lectures 2006