Contemporary Mathematics Volume 39, 1985 SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS Errett A. Bishop During the past ten years I have given a number of lectures on the sub- ject of constructive mathematics. My general impression is that I have failed to communicate a real feeling for the philosophical issues involved. Since I am here today, I still have hopes of being able to do so. Part of the diffi- culty is the fear of seeming to be too negativistic and generating too much hostility. Constructivism is a reaction to certain alleged abuses of classical mathematics. Unpalatable as it may be to have those abuses examined, there is no other way to understand the motivations of the constructivists. Brouwer's criticisms of classical mathematics were concerned with what I shall refer to as "the debasement of meaning. " His incisive criticisms were one of his two main contributions to constructivism. (His other was to establish a new terminology, involving a re-interpretation of the usual con- nective s and quantifier s, which permits the expression of certain important distinctions of meaning which the classical terminology does not. ) The debasement of meaning is just one of the trouble spots in contem- porary mathematics. Taken all together, these trouble spots indicate that something is lacking, that there is a philosophical deficit of major propor- tions. What it is that is lacking is perhaps not clear, but the lack in all of its aspects constitutes a syndrome I shall tentatively describe as "schizophrenia." One could probably make a long list of schizophrenic attributes of con- temporary mathematics, but I think the following short list covers most of the ground: rejection of common sense in favor of formalism debasement of meaning by wilful refusal to accommodate certain aspects of reality inappro- priateness of means to ends the esoteric quality of the communication and fragmentation.
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