Existence Theorem - 'Pure' Existence Results

'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.

From a more rigorous point of view, this is a problematic concept. This is because it is a tag applied to a theorem, but qualifying its proof; hence, pure is here defined in a way which violates the standard proof irrelevance of mathematical theorems. That is, 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. Thus, many constructivist mathematicians work in extended logics (such as intuitionistic logic) where pure existence statements are intrinsically weaker than their constructivist counterparts.

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 probably at least 1, an existence assertion has been made (that a non-zero solution exists.)

Theoretically, a proof could also proceed by way of a metatheorem, stating that a proof of the original theorem exists (for example, that a proof by exhaustion search for a proof would always succeed). Such theorems are relatively unproblematic when all of the proofs involved are constructive; however, the status of "pure existence metatheorems" is extremely unclear,