SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 13 terminology just introduced affords as good a framework as is presently avail- able for expressing the content of classical mathematics. If you wish to do classical mathematics, first decide what non-finite attributes you are willing to grant to God. You may wish to grant him LPO and no others. Or you may wish to be more generous and grant him EM, the principle of the excluded middle, possibly augmented by some version of the axiom of choice. When you have made your decision, avail yourself of all the apparatus of the constructivist, and augment it by those additional powers (LPO or EM or whatever) that you have granted to God. Although you will be able to prove more theorems than the constructivist will, because your being is more powerful than his, his theorems will be more meaningful than yours. Moreover to each of your theorems he will be able to associate one of his, having exactly the same meaning. For example, if LPO is the only non-finite attribute of your God, then each of your theorems "A" he will restate and prove as "LPO implies A. " Clearly the meaning will be preserved. On the other hand, if he proves a theorem "B. " you will also be able to prove "B. " but your "B" will be less meaningful than his. The classical interpretation of even such simple results as Goldbach's conjecture is weaker than the con- structive interpretation. In both cases the same phenomena - - the results of certain finitely performable computations -- are predicted, but the degree of conviction that the predicted phenomena will actually be observed is greater in the constructive case, because to trust the classical predictions one must be- lieve in the theoretical validity of the concept of a God having the specified attributes, whereas to trust the constructive predictions one must only believe in the theoretical validity of the concept of a being who is able to perform arbitrarily involved finite operations. It would thus appear that even a constructive proof of such a result as "the number of zeros in the first n digits of the decimal expansion of n does not exceed twice the number of ones" would leave us in some doubt as to whether the prediction is correct for any particular value of n, say for n = 1000. We have brought mathematics down to the gut level. My gut tells me to trust the constructive prediction and be wary of the classical prediction. I see no reason that yours should not tell you to trust both, or to trust neither. In common with other constructivists, I also have gut feelings about the relative merits of the classical and constructive versions of those results which, unlike Goldbach's conjecture, assert the existence of certain cpantit'ies.

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