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An Introduction To Non-aristotelian Systems And General Semantics.

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566                   ADDITIONAL STRUCTURAL DATA
It becomes obvious that we should not identify the manipulation of mathematical symbols with the semantic aspects of mathematics. History and investigations show that both aspects are necessary and important, although of the two, the semantic discoveries are strictly connected with the revolutionary advances in science, and have invariably marked a new period of human development. In Chapter XXXIX, the reader will find a very impressive example of this general fact. Thus, what is known as the 'Lorentz transformation', looks like the 'Einstein transformation'. When manipulated numerically both give equal numerical results, yet the meanings, and the semantic aspects, are different. Although Lorentz produced the 'Lorentz transformation' he did not, and could not have produced the revolutionary Einstein theory.
It is well known that when it comes to the manipulation of symbols mathematicians agree, but when it comes to the semantic aspects or meanings . , they are admittedly hopelessly at variance. In a prevailingly A world we have had no satisfactory theory of 'infinity', or a A definition of numbers and mathematics. This necessarily resulted in the fact that the semantic aspects of practically all important mathematical works by different authors often involve individual semantic presuppositions, or orientations concerning fundamentals. My presentation intends to be primarily semantic and elementary, and is only remotely concerned with the manipulation of symbols. A A -system, which rejects 'identity', differs very widely from A attitudes, and introduces distinct A requirements. I had, therefore, to select from many works, with their individual presuppositions, those which were less in conflict with A principles than the others.
A survey of important mathematical treatises shows that although the majority of modern mathematicians explicitly abjure the 'infinitesimal', yet, in some presentations, this notion persists. In my presentation I reject the 'infinitesimal' explicitly and implicitly, although the formulae are not altered. 'Modern' calculus is based officially on the theory of limits, but as the theory of limits involves the unclarified theory of 'infinity'. , nothing would be gained semantically and for my purpose, had I stressed these formal possibilities of the calculus. Quite the opposite, if I had done so, I would have failed to stress the most fundamental A principle and task of establishing the similarity of structure between languages and the unspeakable levels and happenings as the first and crucial consequence of the elimination of identity. For these weighty reasons, in my presentation, I followed some older textbooks, particularly Osgood's, which, from a point of view, are sounder than the newer, largely A rationalizations and apologetics.
However, it should be realized that practically all outstanding and creative mathematicians have had, and still have, A attitudes, yet, these private beneficial attitudes, not being formulated in a -system, could not become conscious, simple, workable, public, and educational assets. We can be simple about this point. With the elimination of identity, structure becomes the only possible content of 'knowledge'and structure of the un-speakable levels has to be discovered. Discovery depends on the finding of new, and therefore different