constructive

A constructive proof is one that enables one to give an example, or give a rule for finding an example, of a mathematical object with some property. A nonconstructive proof might result in us knowing that an example exists, but having no idea how to define it. The axiom of choice in set theory is the classical non-constructive existence axiom: it tells us that a certain set exists, whether or not there is any prospect of finding a condition defining membership in it. Similarly the definition of a function to take the value 0 if every even number is the sum of two primes, and 1 if this is not so, is classically a definition enabling us to assert that there is a number that is the value of the function, although we cannot identify which. The view that such a definition is inadmissible and that mathematics should confine itself to constructive proofs and definitions is known as constructivism. Constructivism will be suspicious of indirect existence proofs, because showing that a contradiction follows from denying that some object exists need not of itself show us how to identify the object. Constructivism frequently involves suspicion of the idea of a completed infinite set, thought of as a self-standing object of investigation, as a finite set would be.