ERRETT A. BISHOP Common sense is a quality that is constantly under attack. It tends to be supplanted by methodology, shading into dogma. The codification of insight is commendable only to the extent that the resulting methodology is not elevated to dogma and thereby allowed to impede the formation of new insight. Con- temporary mathematics has witnessed the triumph of formalist dogma, which had its inception in the important insight that most arguments of modern math- ematics can be broken down and presented as successive applications of a few basic schemes. The experts now routinely equate the panorama of mathe- matics with the productions of this or that formal system. Proofs are thought of as manipulations of strings of symbols. Mathematical philosophy consists of the creation, comparison, and investigation of formal systems. Consis- tency is the goal. In consequence meaning is debased, and even ceases to exist at a primary level. The debasement of meaning has yet another source, the wilful refusal of the contemporary mathematician to examine the content of certain of his terms, such as the phrase "there exists. " He refuses to distinguish among the differ- ent meanings that might be ascribed to this phrase. Moreover he is vague about what meaning it has for him. When pressed he is apt to take refuge in formalistics, declaring that the meaning of the phrase and the statements of which it forms a part can only be understood in the context of the entire set of assumptions and techniques at his command. Thus he inverts the natural order, which would be to develop meaning first, and then to base his assump- tions and techniques on the rock of meaning. Concern about this debasement of meaning is a principal force behind constructivism. Since meaning is debased and common sense is rejected, it is not sur- prising to find that the means are inappropriate to the ends. Applied mathe- matics makes much of the concept of a model, as a tool for dealing with reality by mathematical means. When the model is not an adequate representation of reality, as happens only too often, the means are inappropriate. One gets the impression that some of the model-builders are no longer interested in reality. Their models have become autonomous. This has clearly happened in mathe- matical philosophy: the models (formal systems) are accepted as the pre- ferred tools for investigation the nature of mathematics, and even as the font of meaning.
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