Terence Tao on Nostr: Even in the realm of pure mathematics, where the notion of inerrancy is the most ...

Even in the realm of pure mathematics, where the notion of inerrancy is the most developed (and nowadays even able to be automatically verified by formal proof assistants), it is still true that mathematics *as practiced by mathematicians* still contains important components of accuracy-based thinking rather than inerrancy-based thinking. In line with the "three stages of mathematical development" I discussed in https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ , while it is vitally important to be able to use the inerrant mechanism of formal (or semi-formal) mathematical proof to verify one's arguments, in order to actually *locate* these proofs it is far faster and more efficient to adopt looser, but still reasonably accurate, heuristic models in order to get an early sense of the feasibility or infeasibility of different ways to attack the problem, and to glimpse what the potential outcome of a complex calculation might be. Nowadays one even can employ completely non-rigorous techniques such as numerical simulations or machine learning to map out the landscape of intermediate results (which may be true or false, and whose proof or disproof may be routine, difficult, or hopeless) in order to select the most promising strategy (or to select a more tractable problem to study first). (3/3)