SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 9 there are cases in which it would be naive to run the program and wait around for the result. Everything else about integers follows from the above thesis plus the rules of decimal arithmetic that we learned in elementary school. Two integers are equal if their decimal representations are equal in the usual sense. The order relations and the arithmetic of integers are defined in terms of their decimal representations. With the constructive definition of the integers, we have begun our study of the technical implementation of the constructivist philosophy. Our point of view is to describe the mathematical operations that can be carried out by finite beings, man's mathematics for short. In contrast, classical mathe- matics concerns itself with operations that can be carried out by God. For instance, the above number n is classically a well-defined integer because 0 God can perform the infinite search that will determine whether the Riemann hypothesis is true. As another example, the smart sequences previously dis- cussed may be able to outwit you and me (or any other finite being), but they will not be able to outwit God. That is why statement (*) is true classically but not constructively. You may think that I am making a joke, or attempting to put down classi- cal mathematics, by bringing God into the discussion. This is not true. I am doing my best to develop a secure philosophical foundation, based on meaning rather than formalistics, for current classical practice. The most solid foundation available at present seems to me to involve the consideration of a being with non-finite-powers -- call him God or whatever you will -- in addi- tion to the powers possessed by finite beings. What powers should we ascribe to God? At the very least, we should credit him with limited omniscence, as described in the following limited principle of omniscence (LPO for short): If 1% 1 is any sequence of integers, then either nk = 0 for all k or there exists a k with n # 0. By accepting k LPO as valid, we are saying that the being whose capabilities our mathematics describes is able to search through a sequence of integers to determine wheth- er they all vanish or not. Let us return to the technical development of constructive mathematics, since it is simpler, and postpone the further consideration of classical mathe- matics until later. Our fir st task is to develop an appropriate language to describe the mathematics of finite beings. For this we are indebted to
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