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 can be found in the definitions of the concepts used.

A controversy that goes back to the early twentieth century concerns the issue of pure existence theorems. From a constructivist viewpoint, by admitting them mathematics loses its concrete applicability (see nonconstructive proof). The opposing viewpoint is that abstract methods are far-reaching, in a way that numerical analysis cannot be.

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[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.

To a non-constructivist mathematician (that is to say, most mathematicians), this is a problematic concept. This is because it is a tag applied to a theorem, but qualifying its proof. They would say that 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.)

[edit] Constructivist ideas

From the other direction there has been considerable clarification of what constructive mathematics is; 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).