Infinity and foundations · Scope
How can finite minds have knowledge of the infinite?
No one has ever counted infinitely many things, yet mathematicians prove precise facts about infinity, including that some infinities are bigger than others. This question asks how such knowledge is possible and whether it is the same kind of knowledge as knowing that 7 is prime.
Claims
- Proof lets us reason about infinitely many cases at once, using a finite argument, so knowledge of the infinite is possible without counting.
- Results about infinity are tested by consistency and by their usefulness, for example in calculus, so they are as reliable as other mathematics.
Counterclaims
- Some mathematicians, the finitists and intuitionists, have argued that we only know about infinity as a process that never finishes, and that treating infinite sets as completed objects goes beyond what we can know.
- Our intuitions about size break down with infinity, so knowledge here depends heavily on definitions we accept rather than on understanding.
Real-life situations from mathematics
Cantor's diagonal argument
In 1891 Georg Cantor gave a short argument that the real numbers cannot be listed, even with an infinite list, so there are 'more' of them than whole numbers. Cantor himself linked the idea of an absolute infinite with God.
Hilbert's hotel
David Hilbert's thought experiment of a full hotel with infinitely many rooms that can still take new guests, by moving everyone along one room, is often used to show how infinite sets break everyday rules.
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 the limits of knowledge, imagination or intuition. Keep the maths accurate and simple.
Exhibition. A hotel key card labelled 'Room ∞' could anchor a prompt about the limits of what we can know.
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 religion Scope
What mathematics is about, and where its edges lie.