The complexity of simplicity
Curtis Franks, University of Notre Dame
10:00-10:30 Thursday April 4
Simplicity in reasoning served as a focal point in mathematical research from Aristotle until the height of the foundations movement in the early twentieth century. The idea always was that topically pure demonstrations, by unearthing the simple truths on which a fact depends, do more than convince us of its truth: They provide its grounds. Modern mathematics betrays this ideal in several ways. Most obviously, impure proofs are often more explanatory than their counterparts precisely because they reveal hidden connections across topics. More crucially, our judgements of a single proof's simplicity or complexity often change in light of adjustments in the broader mathematical landscape (adjustments that we make in efforts to contextualize and foster an understanding of an initially complex proof). And in both cases, it is most natural to see relatively high level phenomena explaining more basic facts. Mathematical discovery rarely respects our preconceived notions of a problem's topic. What we deem simple thus changes in the course of our efforts to cope with innovation and does not reflect any criterion isolable independent of those very efforts. Reflecting on this, we can begin to appreciate that our judgements of simplicity are often underwritten by highly complex practices and prior understanding. We can trace this reversal of the foundational attitude—familiar more generally in contemporary social science and aesthetics—to some early remarks of David Hilbert himself. Hilbert asked that we shift our attention from the "objective" fetish of topical purity and towards its "subjective" counterpart.