nonconstructive argument

In philosophy, a nonconstructive argument is a type of reasoning that proves the existence of something without providing a method to actually construct or find it. This contrasts with constructive arguments, which offer a way to build or identify the object in question.1

A classic example is the proof that irrational numbers exist such that their power is rational. The argument shows that either ( \sqrt{2}^{\sqrt{2}} ) is rational or ( \sqrt{2}^{\sqrt{2}} ) is irrational. If ( \sqrt{2}^{\sqrt{2}} ) is rational, the proof is complete. If it's irrational, then ( (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = 2 ), which is rational. The nonconstructive aspect is that this initial argument doesn't specify which case is true.1

While some nonconstructive arguments are criticized for their lack of explicit construction, they are a fundamental part of logic and mathematics, particularly in areas like set theory where axioms like the axiom of choice can lead to nonconstructive proofs. The term "nonconstructive" can also broadly mean unproductive or not serving to build or create something.12