SCIENCE AND SANITY - online book

An Introduction To Non-aristotelian Systems And General Semantics.

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does not apply to the study of the foundations of mathematics. It applies to a large extent to contentless technical mathematics, including so-called 'formal logic' of that system.
Formalism when free from identification becomes a unique comparative tool in search for structure; formalism with identification of different orders of abstractions, a symptom of semantic disturbances, often of a morbid character. It should be realized that we may have one-, two-, three-. , many-, and oo-valued orientations, which with the exception of one-valued, we should utilize when conditions warrant a particular use iri a particular case. Thus in mathematics, for the sake of having mathematics as a standard of evaluation, we select a sharply two-valued orientation by which in the old language 'A is B or not B', to allow sharp statements that for instance, 1 + 1=2. If we would deliberately postulate that 1+1 may sometimes be equal to 2 and sometimes not equal to 2, we would have forms of representation which would apply perhaps more readily to science and life, but mathematics as such would be impossible, and we would be deprived of this sharp tool for evaluation.
It is interesting to notice that mathematicians, by the use of two-valued semantics, (not 'logic', because an el discipline cannot be 'lived through' at all by non-heavily pathological individuals), have produced the most important disciplines. Thus we have, for instance, the theory of 'variance' (the theory of function), the theory of invariance, the differential calculus, the 1, 2, 3, 4, and M-dimensional systems, and a host of other verbal structures similar not only to the world, but to the human nervous system. These results give us means not only to enlarge our mastery of the external world, but when generalized into a non-el, A
-system, give us the means for the mastery of the inner world, leading toward sanity.
It is amusing to discover, in the twentieth century, that the quarrels between two lovers, two mathematicians, two nations, two economic systems ., usually assumed insoluble in a 'finite period' should exhibit one mechanism - the semantic mechanism of identification - the discovery of which makes universal agreement possible, in mathematics and in life.