Should we believe in nonstandard analysis?

Abstract: Nonstandard analysis has been the one of the focal points for debate about the role of the axiom of choice in mathematics. I'll argue that this discussion often conflates two distinct issues - the question of whether mathematical arguments are valid, and the question of whether all mathematical objects should be understood to "exist" in the same way. I'll discuss various ways of showing that most uses of nonstandard analysis in mathematics don't actually use the axiom of choice, and how this perspective can be used to obtain new mathematical results (including applications, joint with William Simmons, to finding new bounds for primality testing in polynomial rings). On the other hand, I'll argue (based on joint work with Kenny Easwaran) that the same perspective argues against interpreting nonstandard values too literally when considering applications with real-world interpretations.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic