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An Introduction To Non-aristotelian Systems And General Semantics.

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by Alfred Korzybski
We are here dealing with a concrete mathematical problem which is not trivial, but at the same time is solvable, and I cannot imagine that any mathematician can find the courage to elude its honest solution by means of a metaphysical dogma. (549)                                           Hermann weyl
I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a fagon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restrictions.
(74)                                                                                                                     K. F. GAUSS
A very extensive literature shows that the problems of 'infinity' pervade human psycho-logical reactions, starting from the lowest stage of human development up to the present and that without some theory of 'infinity', modern mathematics would be impossible. Up to date, no satisfactory theory of infinity, on which all mathematicians could agree, has been produced. The results are rather bewildering because what appears to some prominent mathematicians as perfectly sound mathematics is evaluated by other equally prominent scientists as a 'mental' disease (Poincare); or we find opinions that a large portion of mathematics is devoid of proof and has to be accepted on faith; or that some parts of mathematics must be treated as non-sense (Kronecker, Brouwer, Weyl . ,). 'There are eminent scholars on both sides and the chance of reaching an agreement within a finite period is practically excluded', says Brouwer, and certainly such a state of affairs does not allow us to have any satisfactory modern standards of proof and rigour; the last thing we should expect in mathematics.
The majority of those mathematicians who take interest in the soundness of their science seem to believe that the main difficulty centres around the validity of the 'law of excluded third' ('A is B, or not B') of the accepted, sharply two-valued, chrisippian form of A 'logic'. They disregard the fact that we are born, bred, educated, speak a language, live under conditions, institutions . , which still remain desperately A or even pre-aristotelian. If we attempt to reject one of the two-valued 'laws of thought' or postulates of the A -system, but retain A or pre-aristotelian elementalistic 'psychologies', 'logic', and s.r, no agreement in 'a finite period' can be expected, and the present mathematical chaos would continue.
♦Paper presented before the American Mathematical Society at the New Orleans, Louisiana, Meeting of the A.A.A.S. December 28, 1931. I continue to use the abbreviations introduced in this book.