SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 2 1 constructive terminology (at least mentally) and then translating that into the language of whatever computer was being used. Third, no gain in precision is actually achieved. One of the procedures for defining the value of a recursive function is to search a sequence of integers one by one, and choose the first that is non-zero, having first proved that one of them is non-zero. Thus the notion of a recursive function is at least as imprecise as the notion of a cor- rect proof. The latter notion is certainly no more precise than the naive notion of a (constructive) sequence of integers. The desire to achieve complete precision, whatever that is, is doomed to frustration. What is really being sought is a way to guarantee that no dis- agreements will arise. Mathematics is such a complicated activity that dis- agreements are bound to arise. Moreover, mathematicians will always be tempted to try out new ideas that are so complicated or innovative that their meaning is questionable. What is important is not to develop some framework, such as recursive function theory, in the vain hope of forestalling questionable innovations, but rather to subject every development to intense scrutiny (in terms of the meaning, not on formal grounds). Recursive functions come into their own as the source of certain counter-examples in constructive mathematics, the most famous being the word-problem in the theory of groups. Since the concept of a (constructively) recursive sequence is narrower than the concept of a (constructive) sequence, it is easier to demonstrate that there exist no recursive sequences satisfying a given condition G. Such a demonstration makes it extremely unlikely that a (constructive sequence satisfying G will be found without some radically new method for defining sequences being discovered, a discovery that many view as almost out of the question. Although some very beautiful counter-examples have been given by means of recursive functions, they have also been used as a source of counter- examples in many situations in which a prior technique due to Brouwer [20] would have been both simpler and more convincing. Brouwer's idea is to counter-example a theorem A by proving A + LPO. Since nobody seriously thinks LPO will ever be proved, such a counter-example affords a good indi- cation that A will never be proved. As an instance, Brouwer has shown that the statement that every bounded monotone sequence of real numbers con- verges implies LPO.
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