Beauty, intuition and the knower · Perspectives
Is beauty a reliable guide to mathematical truth?
Mathematicians often call proofs beautiful or ugly, and some physicists have trusted beautiful equations as signs of truth. This question asks whether a sense of beauty is evidence, a useful guide, or only a feeling.
Claims
- Beautiful results usually reveal deep connections and simple structure, which are signs of having understood something true.
- Many mathematicians report that aesthetic judgement guides their research successfully, choosing which problems and approaches to pursue.
Counterclaims
- Some true results have long, ugly proofs, and some elegant ideas are false; beauty cannot replace proof.
- Judgements of beauty vary with training and culture, so they are a perspective rather than evidence.
Real-life situations from mathematics
Euler's identity
The equation e^(iπ) + 1 = 0 has repeatedly topped polls of mathematicians' most beautiful equations, including a reader poll in The Mathematical Intelligencer published in 1990. In a 2014 brain-imaging study led by Semir Zeki, mathematicians' ratings of beautiful equations were linked to activity in the same brain region as other experiences of beauty.
Hardy on ugly mathematics
In A Mathematician's Apology (1940) G. H. Hardy argued that beauty is the first test of mathematics and that ugly mathematics has no lasting place.
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 linking the arts and mathematics, or about the role of emotion in knowledge.
Exhibition. A poster of Euler's identity, or a page with a proof you find beautiful, is a personal object for a prompt about the knower.
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 Perspectives
Different views of what mathematics is and whose it is.