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An Introduction To Non-aristotelian Systems And General Semantics.

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culties because of the unfamiliarity of a new terminology which embodies the new structural assumptions, and because of the necessity of a re-canalization of our neuro-linguistic habits, etc. Yet after the new orientations are acquired, the new issues become much simpler than the older, because they are better understood (see p. 97).
In at least one historical case, it was the omission of an unnecessary artificial assumption that brought about a transformation of the whole system. I speak here about euclidean geometry, which assumes the equal distance of parallels, and the non-euclidean geometries, which eliminated this equal distance postulate as unnecessary. The results were very striking. Thus, in the euclidean system we build curves out of little bits of 'straight lines'. We do the opposite in the newer geometries - we start with curves, shortest distances, etc., not 'straight lines' (as no one knows what that means), and build up 'straight lines' as the limit of an arc of a circle with an 'infinite radius' (see p. 590).
Further explanations are given in the text, but I hope that I have conveyed to the reader the fundamental character of these problems and some of the difficulties encountered at first when new structural factors are introduced. Even the elimination of a postulate may be translated into an introduction of a new negative factor. This translation is important in life, although it may be unimportant in technical mathematics. In science as well as in life we deal all the time with this kind of problems, and when they are not understood structurally, we are only plunged into paradoxes and bewilderment, and potential maladjustment.
Section F. Non-aristotelian methods.
There is an especially broad generalization, already referred to, which empirically indicates a fundamental difference between the traditional, aristotelian, intensional orientations, and the new non-aristotelian extensional orientations, and in many ways summarizes the radical differences between the two systems. This is the problem of intension (spelled with an s) and extension. Aristotle, and his followers even today, recognized the difference between intension and extension. However, they considered the problem in the abstract, never applying it to human living reactions as living reactions, which can be predominantly intensional or predominantly extensional. The interested reader is advised to consult any textbook on 'logic' concerning 'intension' and 'extension', as well as the material given in this text (see index).
The difference can be illustrated briefly by giving examples of 'defini-