Beauty, intuition and the knower · Methods and tools
What role does intuition play in producing mathematical knowledge?
Mathematics is famous for proof, yet many results are first guessed, sometimes long before anyone can prove them. This question asks whether intuition is part of mathematical knowledge or only a step before it.
Claims
- Intuition guides mathematicians to true results and promising methods; without it, proof would have nothing to aim at.
- Experienced mathematicians' intuitions are trained by years of practice, so they are a reliable form of expertise.
Counterclaims
- Intuition is often wrong, especially with probability and infinity, so it can never count as knowledge on its own.
- Insisting on proof is exactly how mathematics protects itself against the errors of intuition.
Real-life situations from mathematics
Ramanujan's notebooks
In 1913 Srinivasa Ramanujan, largely self-taught, sent G. H. Hardy pages of remarkable formulas, many without proofs. Most turned out to be correct and some were new to the experts; a few were wrong. Mathematicians have spent decades proving results from his notebooks.
Conjectures before proofs
Fermat's Last Theorem was stated in the 1630s and proved in the 1990s. For over 350 years many mathematicians believed it on the strength of intuition and partial evidence.
Check dates and figures in a reliable source before you use them, and cite that source.
Use this in your TOK work
Essay. Fits titles about intuition, imagination or the role of the individual knower.
Exhibition. A page from your own rough working, with a guess you later proved or disproved, is a genuinely personal object.
Link it to the prescribed title or the exhibition prompt you are working on, in your own words. See using maths examples in your essay and choosing exhibition objects.
Themes and study heading
Knowledge and the knower Knowledge and indigenous societies Methods and tools
How mathematical knowledge is produced and justified.