Existence theorem

A controversy that goes back to the early twentieth century concerns the issue of purely theoretic existence theorems, that is, theorems which depend on non-constructive foundational material such as the axiom of infinity, the axiom of choice or the law of excluded middle.

Such theorems provide no indication as to how to construct (or exhibit) the object whose existence is being claimed.

From a constructivist viewpoint, such approaches are not viable as it leads to mathematics losing its concrete applicability,[2] while the opposing viewpoint is that abstract methods are far-reaching,[further explanation needed] in a way that numerical analysis cannot be.

Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics.

[7] One could get another explanation of existence theorem from type theory, in which a proof of an existential statement can come only from a term (which one can see as the computational content).

Geometrical proof that an irrational number exists: If the isosceles right triangle ABC had integer side lengths, so had the strictly smaller triangle A'B'C. Repeating this construction would obtain an infinitely descending sequence of integer side lengths.