Discovered or invented? · Methods and tools
Can mathematics give us knowledge about the world before we observe it?
Several famous discoveries began on paper: a calculation said something should exist, and observers then found it. This question asks whether mathematics itself produced that knowledge, or whether the observation was what turned a prediction into knowledge.
Claims
- A successful mathematical prediction shows that reasoning alone can reach truths about the physical world.
- Predictions make knowledge testable: by saying exactly where to look, mathematics guides observation and saves effort.
Counterclaims
- The mathematics only works because it was built on earlier observations and physical theory, so the knowledge was never independent of experience.
- Mathematics also predicts things that do not exist; equations need physical interpretation, and many predicted objects are never found.
Real-life situations from mathematics
The discovery of Neptune
Irregularities in the orbit of Uranus led Urbain Le Verrier (and independently John Couch Adams) to calculate where an unseen planet should be. In September 1846 Johann Galle found Neptune close to Le Verrier's predicted position.
The positron
In 1928 Paul Dirac's equation for the electron had solutions that seemed to describe a particle like the electron but with positive charge. In 1932 Carl Anderson observed such a particle, the positron.
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 prediction, the relationship between reason and observation, or areas of knowledge working together.
Exhibition. A star chart showing where Neptune was found could anchor a prompt about how we gain knowledge of things we cannot yet see.
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.