Discovered or invented? · Perspectives
Is mathematics discovered or invented?
Do mathematicians find truths that were already there, like explorers mapping a coastline, or create them, like composers writing music? Your answer affects what you think a proof shows and why mathematics works. The best discussions avoid a simple either-or.
Claims
- Mathematical results are not up to us: once the definitions are fixed, no one can choose whether there are infinitely many primes. That suggests they are discovered.
- Different cultures working independently have found the same results, such as the relationship between the sides of a right-angled triangle, which is what you would expect of discoveries.
Counterclaims
- Notation, definitions and axioms are human choices, and different choices lead to different mathematics, which suggests invention.
- Some objects, such as imaginary numbers, began as tools people were reluctant to believe in, and became accepted because they were useful, not because anyone saw them.
Real-life situations from mathematics
Imaginary numbers
In 1545 Gerolamo Cardano published a method for solving cubic equations that sometimes required square roots of negative numbers. Rafael Bombelli worked out rules for them in 1572. They were long treated with suspicion, yet today they are essential in electrical engineering and quantum physics.
The Mandelbrot set
This intricate shape comes from repeating one simple rule on complex numbers. Benoit Mandelbrot made it famous in 1980 using computer plots, but its detail was fixed by the rule before anyone drew it. Was it invented with the rule, or discovered by the pictures?
Check dates and figures in a reliable source before you use them, and cite that source.
Use this in your TOK work
Essay. A classic, so make it fresh: use one specific example in depth, and argue that the answer may differ for definitions and for results.
Exhibition. A printed zoom of the Mandelbrot set works well for prompts about whether knowledge is created or found.
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 Perspectives
Different views of what mathematics is and whose it is.