Mathematics — Methods & Tools
Mathematics
Methods and Tools
How mathematicians work: proof, conjecture, theorem. Fermat’s Last Theorem as a case study.
Notes
Case study: Fermat’s Last Theorem
\[\forall\, n \in \mathbb{Z},\ n>2,\ \nexists\, a,b,c \in \mathbb{Z}_{>0} \text{ such that } a^n + b^n = c^n\]
Stated by Fermat in 1637. Proved by Andrew Wiles in 1995. The key TOK questions:
- What is the difference between a and a ?
- How do mathematicians establish ?
- Does mathematics use empirical testing? (No — you cannot prove a theorem by checking examples.)
- Is there creativity in mathematics? Does emotion play a role? (Wiles wept when he found the gap in his proof.)
- Is a that is “mostly right” still a proof? (No — one error invalidates the whole argument.)