Discovered or invented? · Scope
Do mathematical objects such as numbers exist independently of human minds?
You can point to three apples but not to the number three. Yet statements about numbers seem objectively true. This question asks what mathematical objects are: abstract things that exist outside space and time, patterns in the world, or useful fictions.
Claims
- Mathematical statements are true whether or not anyone believes them, and true statements are about something, so numbers must exist in some sense.
- Facts about numbers, such as the digits of π, can be discovered, which suggests they were already fixed independently of us.
Counterclaims
- We never interact with numbers in space and time, so it is hard to explain how we could know about them if they exist independently of us.
- Numbers can be understood as tools we built for counting and measuring; the truths follow from how we defined them.
Real-life situations from mathematics
Computing π
In 2022 a team at Google Cloud led by Emma Haruka Iwao computed π to 100 trillion digits. NASA's Jet Propulsion Laboratory has explained that about 15 decimal places are enough for its interplanetary navigation. The extra digits were fixed before anyone computed them, but did they 'exist'?
Counting without number words
Researchers working with the Pirahã people of the Amazon in the 2000s reported that their language has no words for exact numbers beyond roughly 'one' and 'two'. The findings are debated, but they raise the question of whether exact number is universal or learned.
Check dates and figures in a reliable source before you use them, and cite that source.
Use this in your TOK work
Essay. Use with care in titles about the nature of knowledge; make sure you connect the philosophy to how mathematicians actually work.
Exhibition. A printout of a thousand digits of π invites questions about what we know when we know a number.
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 Knowledge and indigenous societies Scope
What mathematics is about, and where its edges lie.