Some Different Philosophies of Constructive Mathematics

Abstract

It is possible to distinguish at least five different philosophies of constructive mathematics, plus at least one “semi-constructive” philosophy. By a philosophy of constructive mathematics, we mean a philosophy that subscribes to the two fundamental principles:

• (i) “there exists an x” means we can find x explicitly

• (ii) “truth” has no a priori meaning; a proposition is true just in case we can find a proof of it.

On the basis of these two principles alone, a considerable amount of agreement concerning mathematical practice can be reached. That is to say, any constructive mathematician can read Bishop or Bridges and find the proofs correct. That is not say that every constructive mathematician would agree with Bishop on all philosophical points! In fact, there are some significant differences between the various schools. In this chapter, we shall examine these different views in detail.