Infinity and foundations · Methods and tools
Can paradoxes be a source of mathematical knowledge rather than a threat to it?
A paradox is an argument from acceptable assumptions to an unacceptable conclusion. In mathematics paradoxes have exposed hidden assumptions and forced new foundations. This question asks whether paradoxes damage confidence in mathematics or are one of the ways it grows.
Claims
- Paradoxes show exactly where an assumption fails, which leads to clearer definitions and stronger theories.
- Mathematics' willingness to confront paradoxes openly is a sign of its reliability, not its weakness.
Counterclaims
- A paradox in the foundations shows that mathematics believed to be secure was not, which should make us less confident in today's foundations too.
- Some 'solutions' to paradoxes simply forbid the troublesome constructions, which avoids the problem rather than understanding it.
Real-life situations from mathematics
Russell's paradox
In 1902 Bertrand Russell wrote to Gottlob Frege, whose foundations of arithmetic were going to press, pointing out that the set of all sets that do not contain themselves leads to a contradiction. Frege's system had to be revised, and axiomatic set theory grew partly in response.
The Banach–Tarski paradox
In 1924 Stefan Banach and Alfred Tarski proved that a solid ball can, in theory, be cut into finitely many pieces and reassembled into two identical balls. It is a theorem, not a contradiction, but it shows how strange the consequences of accepted axioms can be.
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 how knowledge develops through problems or disagreement.
Exhibition. A barber's sign reading 'I shave everyone who does not shave themselves' is a playful object for a prompt about language or contradiction.
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 language Methods and tools
How mathematical knowledge is produced and justified.