Existence theorem

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In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all x, y, ... there exist(s) ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not do so explicitly, as usually stated in standard mathematical language. For example, the statement that the sine function is continuous; or any theorem written in big O notation . The quantification is then hidden in definitions.

A controversy that goes back to the early twentieth century concerns the issue of pure existence theorems, and the related accusation that by admitting them mathematics was betraying its responsibilities to concrete applicability (see nonconstructive proof). The point from the mathematical side was always that abstract methods are far-reaching, in a way that numerical analysis cannot be.

[edit] 'Pure' existence results

An existence theorem may be called pure if the proof given of it doesn't also indicate a construction of whatever kind of object the existence of which is asserted. But this is a problematic concept.

The reason is that it is a tag applied to a theorem, but qualifying its proof. Pure is here defined in a way which violates the standard way mathematical theorems are encapsulated. Theorems are statements for which the fact is that a proof exists, without any 'label' depending on the proof. They may be applied without knowledge of the proof; and indeed if that's not the case the statement is faulty.

Such pure existence results are in any case ubiquitous in contemporary mathematics. For example for a linear problem the set of solutions will be a vector space, and some a priori calculation of its dimension may be possible. In any case where the dimension is provably at least 1, an existence assertion has been made, that a non-zero solution exists.

Theoretically, a proof could proceed by way of a meta-existence theorem, stating that a proof of the original theorem exists (for example, that a proof by exhaustion search for a proof would always succeed). It is unclear whether such a theorem would be pure, as the theorem itself (when proved) would be a proof of the original theorem, and thus could be considered constructive.

[edit] Constructivist ideas

From the other direction there has been considerable clarification of what constructive mathematics might be; without the emergence of a 'master theory'. For example according to Errett Bishop's definitions, the continuity of a function (such as sin x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a promise that can always be kept. One could get another explanation from type theory, in which a proof of an existential statement can come only from a term (which we can see as the computational content).