SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 15 afford a solid philosophical basis for the theory of computation, and construc- tive formalism a point of departure for the development of a better computer language. Certainly recursive function theory, which has played a central role in the philosophy of computation, is inadequate to the task. The development of a constructive formalism at any given level would seem to be no more difficult than the development of a classical formalism at the same level. See [17], [18], [20], [21], [22], and [a] for examples. For a discussion of constructive formalism as a computer language, see 121 . Let us return to the technical development of constructive mathematics, and ask what is meant constructively by a function f: Z + Z (where Z is the set of integers). We improve the classical treatment right away - instead of talking about ordered pairs, we talk about rules. Our definition takes a func- tion F: Z + Z to be a rule that associates to each (constructively defined) integer n a (constructively defined) integer f(n), equal values being associ- ated to equal arguments. For a given argument n, the requirement that f(n) be constructively defined means that its decimal form can be computed by a finite, purely routine process. That's all there is to it. Functions f : Z + Q, f: Q + Q, f: Zt + Q are defined similarly. (Here Q -is the set of rational t t numbers and Z the set of positive integers.) A function with domain Z is called a sequence, as usual. Now that we know what a sequence of rational numbers is, it is easy to define a real number. A real number is a Cauchy sequence of rational num- bers! Again, I have improved on the classical treatment, by not mentioning equivalence classes. I shall never mention equivalence classes. To be sure we completely understood this definition, let us expand it a bit. Real numbers are not pre-existent entities, waiting to be discovered. They must be con- structed. Thus it is better to describe how to construct a real number, than to say what it is. To construct a real number, one must (a) construct a sequence {x 3 of rational numbers, n (b) construct a sequence {N 3 of integers, n (c) prove that for each positive integer n we have 1 - x . l g 'xi j whenever i r Nn and j 2 Nn . Of course, the proof (c) must be constructive, as well as the constructions (a) and (b).
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