---
tag: XQHF63BBYF
kind: concept
name: Intuitionism
msc: [03F55, 03A05]
sources:
  - url: https://en.wikipedia.org/wiki/Intuitionism
same_as:
  wikidata: Q10879018
  wikipedia: Intuitionism
---

## Summary

Intuitionism is the view, founded by L. E. J. Brouwer, that mathematics is a construction of the human mind.
A mathematical object exists only when it can be constructed, and a statement is true only when it has a proof.
As a result, intuitionists do not accept the law of excluded middle in general.
Its logic, [[intuitionistic logic|Q11N8X99FK]], is a central example of [[constructivism|G958EQ8CWW]].

## Explanations

### L1 human

For an intuitionist, saying "there is a number with this property" means "I can show you such a number".
Proving that "no such number" leads to a contradiction is not enough.
Take a real number $x$ given by an infinite list of digits.
The statement "either $x = 0$ or $x \ne 0$" cannot always be settled by a finite check, so an intuitionist does not assert it.
This makes intuitionistic mathematics more careful, and every intuitionistic proof contains a method of construction.