Proof and certainty · Methods and tools

Why can one counterexample overturn a mathematical claim that thousands of examples could not establish?

In mathematics a general claim is overturned by a single case that breaks it, while no number of agreeing cases is enough to prove it. This asymmetry between confirming and refuting is also discussed in the philosophy of science. This question asks why it is so sharp in mathematics.

Claims

A mathematical claim usually says 'for every case', so one verified failure makes it false outright; there is no room for 'mostly true'.
A counterexample in mathematics can be checked completely by anyone, so it gives decisive knowledge in a way that a surprising experiment, which might be faulty, does not.

Counterclaims

When a counterexample appears, mathematicians often repair the claim by changing definitions or adding conditions rather than abandoning it, so refutation is less final than it looks.
Checking an enormous counterexample may itself need a computer, so even refutation can depend on trust in tools.

Real-life situations from mathematics

Euler's sum of powers conjecture

Euler suggested that you need at least n nth powers to add up to another nth power. In 1966 Lander and Parkin used a computer search to find that 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, four fifth powers adding to a fifth power, and the conjecture fell.

Lakatos and Euler's polyhedron formula

In Proofs and Refutations (1976) the philosopher Imre Lakatos traced how mathematicians responded to shapes that broke the formula V − E + F = 2: some called them 'monsters' and excluded them, others refined the definition of a polyhedron. Refutation reshaped the concept.

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

Use this in your TOK work

Essay. Strong for titles comparing how areas of knowledge deal with contrary evidence. Link to falsification in the natural sciences.

Exhibition. A card showing the equation 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ is a small, striking object about how knowledge is overturned.

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 Methods and tools

How mathematical knowledge is produced and justified.