Discovered or invented? · Perspectives
How do new mathematical ideas come to be accepted as knowledge?
Negative numbers, irrational numbers, imaginary numbers and infinite sets were all resisted at first, sometimes for centuries. This question asks what changed people's minds: proof, usefulness, familiarity or the authority of respected mathematicians.
Claims
- Ideas are accepted when they are shown to be consistent and fruitful: once rules for a new kind of number are clear and lead to correct results, the case for it is strong.
- Resistance is part of how knowledge is checked; it forces new ideas to be justified carefully.
Counterclaims
- Acceptance often follows familiarity and education rather than argument: each generation takes for granted what the previous one doubted.
- Respected figures can delay or speed acceptance, so the history of mathematics is partly a social history, not only a logical one.
Real-life situations from mathematics
Negative numbers
Brahmagupta stated rules for calculating with zero and negative numbers in 628 CE. In Europe, many mathematicians treated negative solutions as false or absurd well into the seventeenth and eighteenth centuries.
Cantor's infinite sets
Georg Cantor's work on different sizes of infinity, from the 1870s, was strongly opposed by some leading mathematicians, notably Leopold Kronecker. A generation later David Hilbert defended Cantor's set theory as something mathematics should not give up.
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 changes over time, or the role of authority. Compare with paradigm shifts in the natural sciences.
Exhibition. A thermometer reading below zero is a simple object for a prompt about how ideas once rejected become everyday 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.