632 VIII. ON THE STRUCTURE OF MATHEMATICS |
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for the
many-to-one correspondences of causal factors, typified, for instance, by sunshine or cod liver oil producing a similar effect.It should be understood that in what is said here, the numerical values do not matter. In most of the cases we are not advanced enough to be able to deal with such numerical values. What is to be emphasized is the structure of the language we use. The method should enable us, instead of dealing with generalizations in the old language, which somewhere have to be contradicted, to use a language of mathematical structure which shall account for the facts and leave room for the great individual varieties of organisms in structure and function.
After all, we should not be surprised that the theory of functions and language of functions is structurally appropriate in expressing, and so in understanding, the
functioning of the nervous system, or any other system. Personally I have benefitted greatly through this method; and many baffling structural complexities have been much simplified.Structurally, when we use the language of functions, variables. , we automatically introduce
extensional structure, as already explained, and we have at our disposal methods of translation of different orders of abstractions dynamic into static, and vice versawhich is a neurological structural necessity for being rational and sane. And surely science should try to be rational. It should be stressed again that in our problem numerical values matter very little, but structure and method, for the many reasons already explained, are of paramount importance. Perhaps even the value of numbers is due mainly to the structural fact that it has forced upon us extensional and relational methods. It is the only language which is in accordance with the structure and functioning of the nervous system, and so helps to co-ordinate these activities instead of disorganizing them.That these simple structural dependences have been discovered so late is really astonishing. The only explanation I can give of this is that we have been so engrossed in generalizing and generalizations that we lost sight of the fact that in life we deal structurally with
absolute individuals, and that the only language which preserves the extensional structural individuality for its elements is found in mathematicsspecifically, in numbers.It may be that a study of mathematical structure and the psycho-logics of mathematics will give results of unparalleled human values, particularly for our sanity. The problems of sanity are problems of adjustment, and no means of adjustment should be disregarded. It may also be that the main importance of mathematics will be found some day to be more in the mathematical methods and structure which it has originated, methods forced upon the mathematician |
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