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LTT: Simon DeDeo
November 13, 2020 @ 11:00 am - 12:30 pm EST
Simon DeDeo, CMU & the Santa Fe Institute
Explosive Proofs of Mathematical Truths
This will be on online lecture via Zoom, and pre-registration is required. Register here: https://pitt.zoom.us/webinar/register/WN_l0jdzxEDTSupgpaXzkJwZw
ABSTRACT: Justifications for believing a mathematical proof are traditionally based in the validity of its underlying deductive steps. However, in a skeptical argument going back to Hume, this should make even weak belief in a theorem unjustified because errors compound exponentially. To understand how and why mathematical arguments appear, by contrast, to be paradigms of certainty, we undertake a data science study of the epistemic structure of actual proofs, ranging from Euclid’s Geometry and Apollonius’ Conics to fifty computer-assisted contemporary proofs including Godel Incompleteness and the Four Color Theorem. Our analysis shows that these proofs share an underlying network structure. This structure enables the emergence of certainty even in the presence of skepticism of the correctness of any particular step. This emergence is explosive, and has an isomorphism to phase transitions in material objects. I finish with some remarks from practicing mathematicians that serve to validate the model of mathematical belief formation we propose.