SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS excluded middle for instance completely convincing. Fortunately the problem is hypothetical, because such proofs do not seem to arise. It does raise the interesting point that a classically acceptable proof of Goldbachls conjecture might not be constructively acceptable, and therefore the classical and the constructive interpretations of Goldbachls conjecture must differ in some fundamental respect. We shall see later that this is indeed the case. Classical mathematics fails to observe meaningful distinctions having to do with integers. This basic failure reflects itself at all levels of the classi- cal development of mathematics. Consider the number n defined to be 0 if 0 ' the Riemann hypothesis is true and 1 if it is false. The constructivist does not wish to prevent the classicist from working with such numbers (although he may personally believe that their interest is limited). He does want the classicist to distinguish such numbers from numbers which can be "com- 10 puted, l1 such as the number n of primes less than lolo . Classical 1 mathematicians do concern themselves sporadically with whether numbers can be "computed, l1 but only on an ad hoc basis. The distinction is not ob- served in the systematic development of classical mathematics, nor would the tools a ~ i l a b l e the classicist permit him to observe the distinction system- atically even if he were so inclined. The constructivists are frequently accused of displaying the same insen- sitivity to shades of meaning of which they accuse the classicist, because they do not distinguish between numbers that can be computed in principle, such as the number n defined above, and numbers that can be computed in fact. 1 Thus they violate their own principle (D). This is a serious accusation, and one that is not easy to refute. Rather than attempting to refute it, I shall give you my personal point of view. First, it may be demanding too much of the constructivists to ask them to lead the way in the development of usable and sy stematic methods for distinguishing computability in principle from compu- tability in fact. If and when such methods are found, the constructivists will gratefully incorporate them into their mathematics. Second, it is by no means clear that such methods are going to be found. There is no fast distinction be- tween computability in principle and in fact, because of the constant progress of the state of the art among other reasons. The most we can hope for is some good systematic measure of the efficiency of a computation. Until such is found, the problem will continue to be treated on an ad hoc basis.

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