Intuitionism
Branch · 1900s-1930s · Mathematics, stage 2: Map the branches
Intuitionism, founded by L. E. J. Brouwer in the early twentieth century, holds that mathematical objects are constructions of the mind rather than things discovered, so a statement is true only when it has been constructed and proved. It follows that a proof of existence must produce the object, and that the law of excluded middle, which says every statement is either true or false, cannot be assumed when infinite collections are involved. Poincaré and Kronecker had raised related objections to non-constructive methods earlier, and are usually treated as predecessors. The position rules out proofs most mathematicians accept, and it survives today mainly through constructive mathematics and its links to computing.
What it claims
- A mathematical object exists only when a construction of it has been given.
- The law of excluded middle cannot be applied to statements about infinite collections.
- A proof by contradiction that establishes existence without producing the object proves nothing.
Key ideas
People
Sources
- Intuitionism in the Philosophy of Mathematics Stanford Encyclopedia of PhilosophyBrouwer's position stated carefully, along with what accepting it costs.
- Intuitionism Wikipedia
