Proof and certainty · Methods and tools

Is a mathematical proof a stronger kind of justification than the evidence used in the natural sciences?

Mathematicians say a result is known once it is proved, however many examples already support it. Scientists, by contrast, accept claims on the strength of evidence that could, in principle, be overturned. This question asks whether proof really is a different and better kind of justification, or only a different one.

Claims

A valid proof shows a result must hold in every case, including the infinitely many no one will ever check, so it gives a certainty that no amount of observation can.
Once proved, mathematical results tend to stay proved: Euclid's theorem that there are infinitely many primes is as secure today as it was over two thousand years ago, while many accepted scientific theories have been replaced.

Counterclaims

A proof is only as certain as its axioms and the people checking it. Published proofs have contained errors that went unnoticed for years, so in practice mathematical knowledge also rests on trust and testing.
Mathematicians use evidence too: they believe conjectures because of numerical checks long before anyone proves them, and much research is guided by that kind of experimental confidence.

Real-life situations from mathematics

Goldbach's conjecture

The claim that every even number greater than 2 is the sum of two primes has been checked by computer for every even number up to about 4 × 10¹⁸, yet it is still called a conjecture. To a scientist that much evidence would settle the matter; to a mathematician it does not.

The Pólya conjecture

Proposed in 1919, this claim about how many whole numbers have an odd number of prime factors held for every number anyone checked. In 1958 Haselgrove showed it was false, and the smallest counterexample, found later, is 906,150,257. Evidence from the first few hundred million cases had been misleading.

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

Use this in your TOK work

Essay. Useful for titles that compare how areas of knowledge justify claims, or that ask whether certainty is possible. Contrast mathematics with a natural science, and make sure your counterclaim shows that proof also depends on people.

Exhibition. A printout of a long computer check of a conjecture, beside a short handwritten proof, makes a strong object for a prompt about certainty or about what counts as evidence.

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.