20 ERRETT A. BISHOP every function anyone has ever been able to construct from R to R has turned out to be continuous, in fact uniformly continuous on bounded subsets. (The function f that is 1 for x 2 0 and 0 for x 0 does not count, because for those real numbers x for which we have no proof of the statement "x 2 0, or x 0" we are unable to compute fk). ) Brouwer had hopes of proving that every function from IR to R is continuous, using arguments involving free choice sequences. He even presented such a proof [7] . It is fair to say that almost nobody finds his proof intelligible. It can be made intelligible by re- placing Brouwer's arguments at two critical points by axioms, that Kleene and Vesle y [2 11 call t'Brouwer's principlew and "the bar theorem. I ' My objection to this is, that by introducing such a theorem as "all f : R + R are contin- uous" in the guise of axioms, we have lost contact with numerical meaning. Paradoxically this terrible price buys little or nothing of real mathematical value. The entire theory of free-choice sequences seems to me to be made of very tenuous mathematical substance. If it is fair to say that the intuitionists find the constructive concept of a sequence generated by an algorithm too precise to adequately describe the real number system, the recursive function theorists on the other hand find it too vague. They would like to specify more precisely what is meant by an algor- ithm, and they have a candidate in the notion of a recursive function. They admit only sequence of integers or rational numbers that are recursive (a con- cept we shall not define here: see 1201. for details). Their reasons are, that the concept is more precise than the naive concept of an algorithm, that every naively defined algorithm has turned out to be recursive, and it seems unlikely we shall ever discover an algorithm that is not recursive. This requirement that every sequence of integers must be recursive is wrong on three funda- mental grounds. First and most important, there is no doubt that the naive concept is basic, and the recursive concept derives whatever importance it has from some presumption that every algorithm will turn out to be recursive. Second, the mathematics is complicated rather than simplified by the restric- tion to recursive sequences. If there is any doubt as to this, it can be re- solved by comparing some of the recursivist developments of elementary anal- ysis with the constructivist treatment of the same material. Even if one is oriented to running material on a computer, the recur sivist formulation would constitute an obstacle, because very likely the recursive presentation would be translated into computer language by fir st translating into common
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