Is Conceptualism a Sufficient Reason for the Rejection of Non-Constructive Existence Proofs in Mathematics?

Abstract

Non-constructive existence proofs (which prove the existence of mathematical objects of a certain kind without giving any particular examples of such objects) are rejected by constructivists, who hold a conceptualist view that mathematical objects exist only if they are constructed. In the paper it is argued that this conceptualist argument against non-constructive proofs is fallacious, because those proofs establish the existence of objects belonging to certain kinds rather than the existence of those objects per se. Moreover, to engage in proving existence theorems in a given mathematical theory one has to define all of the objects of this theory at the very beginning, which can be interpreted as establishing the existence of these objects before any theorem about them is proven. It is also argued that the constructivist may escape these objections by adopting the actualistic view, according to which a mathematical sentence is true if and only if it is established as true, but this view is very implausible, as it seems unable to explain the strictness and objectiveness of mathematics and the fact that it differs so fundamentally from, for example, fictional discourse.