Constructivism

Remarks

Example: the proof that for each natural number $n$ there is a prime $p > n$, by taking a prime factor of $n! + 1$, is constructive. It gives a method: factor $n! + 1$ and take any prime factor $p$; then $p > n$, because no number from $2$ to $n$ divides $n! + 1$. Non-example: the classical proof that there are irrational numbers $a, b$ with $a^b$ rational is not constructive. It argues that either $\sqrt{2}^{\sqrt{2}}$ is rational, or it is not and then $(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = 2$; it does not say which case holds. It uses the law of excluded middle, which fails principle 2. In constructive mathematics, a proof by contradiction proves negated statements $\neg P$, but in general does not prove $P$ from $\neg \neg P$; see the double negation law.