Rigor in Math

Sometimes progress requires rigor, and sometimes progress can’t wait for rigor—at least in math. 

In a recent interview at 3:16AM, Richard Marshall asks philosopher John Burgess (Princeton) “What’s the importance of rigor in maths?”

Professor Burgess answers:

As a member in good standing of the circle of sciences, mathematics has the right to conduct its own internal affairs as seems to mathematicians—historically in some cases only after long-continued and quite sharp disagreement—appropriate. It has seemed to mathematicians appropriate to uphold standards of rigor, which does not mean ignoring discoveries that have not been rigorously established, but rather sharply distinguishing them, as conjectures, from results that rank as theorems. 

As I have said elsewhere, on the one hand, rapidly developing seventeenth and eighteenth century physical science could not afford to wait for calculus and the like to be established even with Euclid’s kind of rigor, let alone Frege’s; on the other hand, the mathematics used in twentieth-century physics (Riemannian geometries, Hilbert space) could scarcely have been arrived at if mathematicians had remained satisfied with the relaxed standards prevailing circa 1800.

I would say mathematicians went in for rigorization at just about the time it was coming to be needed for imminent future scientific applications. 

Despite differences in the types of relationships philosophy and math bear to the sciences, one might wonder whether there lessons here for how to think about rigor and its place in philosophy. Discussion welcome.

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