Constructivism (mathematics) - Constructive Mathematics

Constructive Mathematics

Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle which states that for any proposition, either that proposition is true, or its negation is. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable. It is just that the general law is not assumed as an axiom. The law of non-contradiction (which states that contradictory statements cannot both at the same time be true) is still valid.

For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem (where x, y, z ... are the free variables in the proposition p). In this sense, propositions restricted to the finite are still regarded as being either true or false, as they are in classical mathematics, but this bivalence does not extend to propositions which refer to infinite collections.

In fact, L.E.J. Brouwer, founder of the intuitionist school, viewed the law of the excluded middle as abstracted from finite experience, and then applied to the infinite without justification. For instance, Goldbach's conjecture is the assertion that every even number (greater than 2) is the sum of two prime numbers. It is possible to test for any particular even number whether or not it is the sum of two primes (for instance by exhaustive search), so any one of them is either the sum of two primes or it is not. And so far, every one thus tested has in fact been the sum of two primes.

But there is no known proof that all of them are so, nor any known proof that not all of them are so. Thus to Brouwer, we are not justified in asserting "either Goldbach's conjecture is true, or it is not." And while the conjecture may one day be solved, the argument applies to similar unsolved problems; to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution.

With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property which classical logic does not: whenever is proven constructively, then in fact is proven constructively for (at least) one particular, often called a witness. Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.