SCHIZOPHRENLA IN CONTEMPORARY MATKEMATICS Everyone who has taught undergraduate mathematics must have been impressed by the esoteric quality of the communication. It is not natural for ''A implies B" to mean 'hot A or B, " and students will tell you so if you give them the chance. Of course, this is not a fatal objection. The question is, whether the standard definition of implication is useful, not whether it is natu- ral. The constructivist, following Brouwer, contends that a more natural definition of implication would be more useful. This point will be developed later. One of the hardest concepts to communicate to the undergraduate is the concept of a proof. With good reason -- the concept is esoteric. Most mathe- maticians, when pressed to say what they mean by a proof, will have recourse to formal criteria. The constructive notion of proof by contrast is very simple, as we shall see in due course. Equally esoteric, and perhaps more troublesome, is the concept of existence. Some of the problems associated with this concept have already been mentioned, and we shall return to the sub- ject again. Finally, I wish to point out the esoteric nature of the classical concept of truth. As we shall see later, truth is not a source of trouble to the constructivist, because of his emphasis on meaning. The fragmentation of mathematics is due in part to the vastness of the subject, but it is aggravated by our educational system. A graduate student in pure mathematics may or may not be required to broaden himself by passing examinations in various branches of pure mathematics, but he will almost cer- tainly not be required or even encouraged to acquaint himself with the philos- ophy of mathematics, its history, or its applications. We have geared our- selves to producing research mathematicians who will begin to write papers as soon as possible. This anti-social and anti-intellectual process defeats even . its own narrow ends. The situation is not likely to change until we modify our conception of what mathematics is. Before important changes will come about in our methods of education and our professional values, we shall have to dis- cover the significance of theorem and proof. If we continue to focus attention on the process of producing theorems, and continue to devalue their content, fragmentation is inevitable. By devaluation of content I mean the following. To some pure mathe- maticians the only reason for attaching any interpretation whatever to theorem and proof is that the process of producing theorems and proofs is thereby facilitated. For them content is a means rather than the end. Others feel
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