---
tag: RR10R8JN02
kind: definition
name: Intuitionism
msc: [03F55, 03A05]
defines: XQHF63BBYF
uses: [EFFRJFW67Y, 01XM7JEKHY]
sources:
  - url: https://en.wikipedia.org/wiki/Intuitionism
---

## Definition

*Intuitionism* is the position in the [[foundations of mathematics|EFFRJFW67Y]] that consists of these theses:

1. mathematical objects are mental constructions, and an object exists only if it can be constructed;
2. a mathematical statement is true only if it has a [[constructive proof|01XM7JEKHY]], so truth means provability by construction;
3. as a consequence, the law of excluded middle $P \lor \neg P$ is not accepted for arbitrary statements $P$.

## Remarks

The statement "there exist irrational numbers $a$ and $b$ with $a^b$ rational" has an intuitionistic proof.
Take $a = \sqrt 2$ and $b = 2 \log_2 3$; then $a^b = 2^{\log_2 3} = 3$.
Both numbers are shown to be irrational directly.
The classical proof considers the two cases "$\sqrt 2^{\sqrt 2}$ is rational" and "it is irrational".
This proof is not intuitionistic: it uses excluded middle and does not say which case holds.
The logic of intuitionism was formalized by Arend Heyting as [[intuitionistic logic|Q11N8X99FK]], and its meaning of the connectives is the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].