MATHEMATICS AND THE NERVOUS SYSTEMS 275 |
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because only then shall we have semantic disturbances eliminated, so that all problems can be analysed properly, and, therefore, agreement
must be reached.The future generations, of course, will have no difficulties whatsoever in establishing the healthy
s.r; neither at present have very young children. These do not need such treatises as the present work. But, before the grown-up parents or teachers can train their children, they must first unlearn a great deal and train themselves to new habits involving thestandards of evaluation. So, for them, such a book, in order to be convincing, must deal with the foundations of their difficulties. The last task is difficult for the writer as well for the reader.What has been said here does not apply, I am sorry to say, to
professional 'philosophers', 'logicians', 'psychologists', psychiatrists, and teachers. These, to be adequate at all for their responsible and difficult professional duties, must become thoroughly acquainted with structure in general, and with the structure of mathematics in particular, as factors in s.r, and must work out the present outline much further.Section B. General.Mathematics in the twentieth century is characterized by an enormous productiveness, by the revision of its foundations, and the quest for rigour, all of which implies material of great and unexplored psychological value, a result of the activity of the human nervous system. Branches of mathematics, as, for instance, mathematical 'logic', or the analytical theory of numbers, have been created in this period; others, like the theory of function, have been revised and reshaped. The theory of Einstein and the newer quantum mechanics have also suggested further needs and developments.
Any branch of mathematics consists of prepositional functions which state certain structural relations. The mathematician tries to discover new characteristics and to reduce the known characteristics to a dependence on the smallest possible set of constantly revised and simplest structural assumptions. Of late, we have found that no assumption is ever 'self-evident' or ultimate.
To those structural assumptions, we give at present the more polite name of postulates. These involve undefined terms, not always stated explicitly, but always present implicitly. A postulate system gives us the structure of the linguistic scheme. The older mathematicians were less particular in their methods. Their primitive propositional functions or postulates were less well investigated. They did not start explicitly with undefined terms. The twentieth century has witnessed in this field |
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