---
tag: G958EQ8CWW
kind: concept
name: Constructivism
msc: ["03F65", "03F55", "00A30"]
uses: [01XM7JEKHY, 0RYYK8PG4S, 0J0XBKG64S]
sources:
  - url: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)
same_as:
  wikidata: Q1137814
  wikipedia: Constructivism (philosophy of mathematics)
---

## Summary

Constructivism is the view in the philosophy of mathematics that a mathematical object exists only when one can construct it.
To prove that something exists, one must give a method to find it, as in a [[constructive proof|01XM7JEKHY]].
As a result, constructive mathematics does not accept the [[law of excluded middle|0RYYK8PG4S]] as a general principle.
Its main schools are Brouwer's intuitionism, Markov's recursive constructive mathematics, and Bishop's constructive analysis.

## Explanations

### L1 human

A constructivist accepts "there is a number with this property" only if you can, at least in principle, produce it.
A [[proof by contradiction|0J0XBKG64S]] that only shows "it cannot be that no such number exists" is not enough.
For example, the classical proof that there are irrational $a, b$ with $a^b$ rational does not say which pair works.
A constructivist asks for a proof that names the pair.
In return, every constructive existence proof contains an algorithm.