Proof and certainty · Perspectives

How far does mathematical knowledge depend on the agreement of a community of experts?

We often imagine a proof as something anyone could check alone. In reality, new results are accepted when the experts in a field have read them, questioned them and agreed. This question asks what happens when that community is small, disagrees or makes mistakes.

Claims

Peer review spreads the work of checking across many experts, so the community's acceptance is good evidence that a proof is correct.
The history of famous proofs shows that the community does catch mistakes: errors are found, discussed openly and either repaired or the claim is withdrawn.

Counterclaims

When a proof is so specialised that only a handful of people understand it, 'the community agrees' may mean only that a few people are persuaded, which is a weaker kind of knowledge.
Reputation can influence whether a proof is read carefully at all, so acceptance may depend partly on who the author is rather than only on the mathematics.

Real-life situations from mathematics

Fermat's Last Theorem

Andrew Wiles announced a proof in June 1993. During checking, a gap was found in one part of the argument. Working with Richard Taylor, Wiles repaired it in 1994 and the proof was published in 1995. The community's checking process found the problem and then accepted the fix.

The abc conjecture dispute

In 2012 Shinichi Mochizuki released a long proof of the abc conjecture built on new theory of his own. In 2018 Peter Scholze and Jakob Stix said one key step did not work. The proof was published in 2021 in a journal Mochizuki helps to edit, yet most number theorists still do not regard the conjecture as proved.

Check dates and figures in a reliable source before you use them, and cite that source.

Use this in your TOK work

Essay. Good for titles about the role of experts, consensus or disagreement. Contrast mathematics with a human science where disagreement is expected.

Exhibition. A journal article with a published correction, or a referee's report form, could anchor a prompt about who decides what counts as knowledge.

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 politics Perspectives

Different views of what mathematics is and whose it is.