*The complexity of simplicity*

Curtis Franks, University of Notre Dame

10:00-10:30 Thursday April 4

[slides, video]

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.

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