Proof and certainty · Methods and tools

If mathematics rests on axioms that cannot themselves be proved, in what sense is it certain?

Every proof starts from assumptions. In geometry these were once thought to be self-evident truths about space; today mathematicians usually treat them as chosen starting points. This question asks whether certainty in mathematics is certainty about the world, or only certainty that conclusions follow from assumptions.

Claims

Mathematical certainty is conditional: if the axioms hold, the theorems must hold. That conditional certainty is still far stronger than anything other areas of knowledge offer.
Axioms are not arbitrary: they are chosen because they capture ideas such as number and space that we already understand, so results built on them tell us something real.

Counterclaims

Different choices of axioms give different, equally consistent mathematics, which suggests mathematics describes possibilities rather than one certain truth.
Some axioms are accepted because they are useful even though their consequences seem strange, so the foundations rest partly on convenience and taste.

Real-life situations from mathematics

Euclid's parallel postulate

For about two thousand years mathematicians tried to prove Euclid's fifth postulate from his others. In the 1820s and 1830s Lobachevsky and Bolyai (and, privately, Gauss) showed that replacing it gives a consistent geometry in which a triangle's angles add up to less than 180°.

The axiom of choice

This widely used axiom of set theory implies the Banach–Tarski result of 1924: a solid ball can, in theory, be split into finitely many pieces and reassembled into two balls of the same size. Most mathematicians accept the axiom because so much useful mathematics depends on it.

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 assumptions, certainty or whether knowledge needs foundations. Compare axioms with the paradigms or assumptions of another area of knowledge.

Exhibition. A triangle drawn on a ball (its angles add up to more than 180°) shows that 'obvious' geometry depends on assumptions.

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.