SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS In discussing the principles of'constructivism, I shall try to separate those aspects of constructivism that are basic to the philosophy from those that are merely convenient (or inconvenient, as the case may be). Four prin- ciples stand out as basis: (A) Mathematics is common sense. (B) Do not ask whether a statement is true until you know what it means. (C) A proof is any completely convincing argument. (D) Meaningful distinctions deserve to be maintained. Surprisingly many brilliant people refuse to apply common sense to mathematics. A frequent attitude is that the formalization of mathematics has been of great value, because the formalism constitutes a court of last resort to settle any disputes that might arise concerning the correctness of a proof. Common sense tells us, on the contrary, that if a proof is SO involved that we are unable to determine its correctness by informal methods, then we shall not be able to test it by formal means either. Moreover the formalism cannot be used to settle philosophical disputes, because the formalism merely re- flects the basic philosophy, and consequently philosophical disagreements are bound to result in disagreements about the validity of the formalism. Principle (B) resolves the problem of constructive truth. For that mat- ter, it would resolve the problem of classical truth if the classical mathe- maticians would accept it. We might say that truth is a matter of convention. This simply means that all arguments concerning the truth or falsity of any given statement about which both parties possess the same relevant facts occur because they have not reached a clear agreement as to what the state- ment means. For instance in response to the inquiry "Is it true the construc- tivists believe that not every bounded monotone sequence of real numbers converges?, " if I am tired I answer "yes. " Otherwise I tell the questioner that my answer will depend on what meaning he wishes to assign to the state- ment (*), that every bounded monotone sequence or real numbers converges. Moreover I tell him that once he has assigned a precise meaning to statement (*I, then my answer to his question will probably be clear to him before I give it. The Cwo meanings commonly assigned to (*) are the classical and the con- structive. It seems to me that the classical mathematician is not as precise as he might be about the meaning he assigns to such a statement. I shall show you later one simple and attractive approach to the problem of meaning in classical mathematics. However in the case before us the intuition at least is
Previous Page Next Page