The curious case of broken theorems
8 days ago
- #proof-errors
- #formal-logic
- #mathematics
- Andrew Wiles announced a proof of Fermat’s Last Theorem in 1993, but a gap was later found, requiring a fix.
- Mathematics often contains errors in proofs and theorems, yet the field continues to progress despite these issues.
- Amnon Neeman disproved a 1961 'theorem' in homological algebra, highlighting how errors can persist unnoticed for decades.
- Formal logic suggests that a single contradiction should invalidate a mathematical theory, but in practice, mathematics remains robust.
- Mathematicians like Raphaël Rouquier and Terry Tao have published errata, showing that errors are common but manageable.
- Efforts to formalize mathematics, such as using proof assistants like Lean, aim to reduce errors but reveal existing gaps in proofs.
- Kevin Buzzard's team encountered a broken lemma in crystalline cohomology, but the issue was fixable due to the theory's extensive use.
- Mathematicians often rely on intuitive meaning attached to symbols, blending methodological Platonism and formalism in their work.
- Conceptualism views mathematics as a human activity combining formal logic with intuitive concept-building.
- Mathematics breaks down like pottery (kintsugi) rather than shattering, with errors being fixable and contained.