Symbolic Systems Forum - Andrew Arana, Philosophy Department, "Purity in Mathematics"

Andrew Arana, Philosophy Department, "Purity in Mathematics"

ABSTRACT:

A proof of a proposition P is `pure', roughly speaking, if it is comprised of definitions (including axioms) of terms occurring in P and deductive consequences of those definitions. A proof of a proposition regarding prime numbers that uses just arithmetic premises is pure, while a proof of this proposition using premises from geometry, or complex analysis, or mechanics, is impure. There has been an ongoing debate on the value of pure proofs since antiquity, and it continues today. I will try to give a flavor of what is at stake in this debate. I'll do so by contrasting some pure and impure proofs, and then analyzing their differences. What is at stake turns out to be as much epistemological as mathematical.