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An Introduction To Non-aristotelian Systems And General Semantics.

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ceeded to coin some very doubtful hypotheses; and such works as produced by Peano, Whitehead, Russell, and others, in which not only all assumptions are stated explicitly, but even the assumptions, given in single undefined terms, are listed. It is not assumed here that even Peano, Whitehead, Russell, and the others have fulfilled this program entirely. It is quite probable that not all of their assumptions are stated explicitly. However, a very serious and revolutionary beginning has been made in this direction. We have still far to go, for at present even mathematicians disregard the threefold relational character of mathematics, and, by a semantic confusion of orders of abstraction, make structurally el assumptions false to the facts of 1933; namely, that mathematics exists 'by itself, detached from the producers, Smith and others. This procedure reminds one of the old N 'I do not make hypotheses', proffered just at the moment he begins to legislate about the structure of the universe and to postulate his 'absolute space' and 'absolute time' 'without reference to any external object whatsoever'. This, of course, was structurally unascertainable, and so was a mere figment of his imagination, inside of his skin, and may become a pathological semantic projection when externalized by affective pressure.
That we must all start with undefined terms, representing blind creeds which cannot be elucidated further at a given moment, may fill the hearts of some metaphysicians with joy. 'Here', they might say, 'we have the goods on the scientists; they criticize us and reject our theories, and yet they admit that they also must start with blind creeds. Now we have full justification for assuming whatever we want to.' But this joy would be short-lived for any reasonably sane individual. In mathematics, we deliberately assume the minimum, and not the maximum, as in metaphysics. The undefined terms selected for use are the simplest of our experience; for instance, 'order' (betweenness). Also, in experimental sciences, we assume the least possible. We demand from a scientific theory, according to the standards of 1933, that it should account for all relevant facts known in 1933 and should serve as a basis for the prediction of new facts, which can be checked by new experiments. If metaphysicians and 'philosophers' would comply with such scientific standards, their theories would be scientific. But their old theories would have to be abandoned and their new theories would become branches of science. Under such structural circumstances, there is no possibility of going outside of science, as we can enlarge the bounds of science without known limits, in search for structure.
It must be pointed out that no set of undefined terms is ultimate. A set remains undefined only until some genius points out simpler and