Nonconstructive proof

In mathematics, a nonconstructive proof, as opposed to a constructive proof, is a mathematical proof that purports to demonstrate the existence of something, but does not reveal how to construct it.

Many nonconstructive proofs assume the non-existence of the thing whose existence is required to be proven, and deduce a contradiction. The non-existence of the thing has therefore been shown to be logically impossible, and yet an actual example of the thing has not been found.

The term "pure existence proof" is often used as a synonym for "nonconstructive proof", where "pure" means that the proof just shows existence and yields nothing else. See existence theorem.

According to the philosophical viewpoint of constructivism, nonconstructive proofs constitute a different kind of proof from constructive proofs. Supporters of this view consider nonconstructive existence to be a weaker form of existence than its constructive counterpart. Some constructivists deny the validity of nonconstructive proof altogether.

Some examples of nonconstructive proofs

An example is the following proof of the theorem "There exist irrational numbers $a$ and $b$ such that $a^{b}$ is rational."

• Recall that ${\sqrt {2}}$ is irrational, and 2 is rational. Consider the number $q={\sqrt {2}}^{\sqrt {2}}$. Either it is rational or it is irrational.

• If it is rational, then the theorem is true, with $a$ and $b$ both being ${\sqrt {2}}$.

• If it is irrational, then the theorem is true, with $a$ being ${\sqrt {2}}^{\sqrt {2}}$ and $b$ being ${\sqrt {2}}$, since

$\left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{({\sqrt {2}}\cdot {\sqrt {2}})}={\sqrt {2}}^{2}=2$

A constructive proof of this theorem would leave us knowing values for $a$ and $b$.

Since we do not know this (because we do not know whether q is irrational), this proof is nonconstructive. (The statement "Either q is rational or it is irrational", from the above proof, is an instance of the law of excluded middle, which is not valid within a constructive proof.) (Side note: As it happens, one can prove that q is irrational using the Gelfond-Schneider theorem, proving the above theorem in a different manner and giving an actual example; however, as this is not done in the above proof, the above proof remains nonconstructive. Another constructive proof: ${\sqrt {2}}^{\log _{2}9}=3$, the irrationality of ${\sqrt {2}}$ and $\log _{2}{9}$ being easy consequences of unique factorization. Another non-constructive proof: $a^{b}=2$ defines a bijection from $a\in [2,4]$ onto $b\in [1/2,1]$, under which the uncountable set of irrational $a\in [2,4]$ maps to an uncountable set containing an irrational $b\in [1/2,1]$.)

An even simpler example: ${\sqrt {x}}$ is less than its argument $x$ when $x>1$. To see this, assume that ${\sqrt {x}}>x$. But then, since by definition $({\sqrt {x}})^{2}=x$, this would imply $x>x^{2}$ when $x>1$. This is not true for any such $x$. Now, without knowing anything about how to compute a square root, you've bounded it from above.

Another example of a nonconstructive theorem is John Nash's proof, using the strategy-stealing argument, that the game of Hex is a first-player win.

Nearly every proof which invokes the axiom of choice is nonconstructive in nature because this axiom is fundamentally nonconstructive. The same can be said for proofs invoking König's lemma.