Constructive proof

Summary

A constructive proof of an existence statement $\exists x \, P(x)$ gives a specific object $a$, or a method to find it, and proves $P(a)$. A non-constructive proof shows that an object exists without giving it, for example by proof by contradiction.

Remarks

A classical non-constructive example: there are irrational numbers $a$ and $b$ such that $a^b$ is rational. Consider $\sqrt{2}^{\sqrt{2}}$. It is rational or irrational, and each case gives such $a$ and $b$. Intuitionistic logic accepts only constructive proofs of existence.

Explanations

L1 human

A constructive proof that a solution exists shows you the solution. A non-constructive proof only shows that "no solution" is impossible. Both are correct in classical mathematics, but the constructive proof tells you more.