Mathematical Revolutions

Catalog description

Mathematics is a central part of our intellectual experience; some view it simply as rigorous common sense. It is connected to sophisticated philosophical perspectives and also to fundamental views in the sciences. Those perspectives and views have changed in revolutionary ways; so, has mathematics. The common view that mathematics - if not directly "static" - is evolving in a linear fashion, does not withstand historical scrutiny. Indeed, there are many dramatic conceptual changes concerning the very nature and object of mathematics. We examine three episodes in the recent past that reflect radical transformations of the subject. The episodes fall within the period from 1854 to 1954, but are rooted in the past and impact the present. They are framed by a discussion, at the beginning, of the axiomatic method and, at the end, of contemporary computational models of mathematical thinking. The three episodes are deeply connected and paved the way to traditional and generative AI. The first episode deals with the shift from geometry through arithmetic to set theory as the foundational discipline for mathematics. The accompanying change in the methodology of mathematics - when joined with contemporaneous logical developments - underlies the formalization of mathematics and is at the center of the second episode. Given Turing's analysis of "formal or mechanical methods", the question "How much of mathematical reasoning can be accomplished by computing machines?" is addressed in the third part. Completing a full circle, we incorporate central features of the axiomatic method into computational models of mathematical thinking.