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Finite or Infinite?

Published online by Cambridge University Press:  14 March 2022

E. T. Bell
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Extract

When I undertook to write an article for mathematical laymen on the mathematical infinite. I did not realize the depths of my own layness, I do now. Having refreshed my memory of the classics of infinity by re-reading among other things the famous papers of Cantor and Zermelo, and having struggled like a boa constrictor to swallow the latest papal bull on the human significance of the infinite, I am completely reduced to what Professor E. W. Hobson aptly and somewhat indignantly calls Mathematical Nihilism.

Type
Research Article
Information
Philosophy of Science , Volume 1 , Issue 1 , January 1934 , pp. 30 - 49
Copyright
Copyright © Philosophy of Science Association 1934

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References

1 An old proof of this may be outlined thus. If √2 the ratio of two whole numbers, a, b, then √2 = a/b. If a, b have a common factor greater than I, it can be cancelled from both a and b without altering the value of a/b, so we shall assume that a, b have no such factor. Square both sides of √2 = a/b. Then 2 = a 2/b 2, and hence, multiplying throughout by b 2, we have 2b 2 = a 2, where a, b are whole numbers with the greatest common divisor I. Since 2b 2 is even, and 2b 2 = a 2, a 2 must be even. Hence a is even; say a = 2c, where c is a whole number. From 2b 2 = a 2 we now get 2b 2 = (2c)2, or 2b 2 = 4c 2, or b 2 = 2c 2. Repeating the argument we get b even; say b = 2d. So, in the original 2b 2 = a 2, we have b = 2d, a = 2c. That is, b, a have the common divisor a, contrary to the assumption that I is their greatest common divisor.

In this sketch many subtle properties of arithmetical divisibility are presupposed. These were all proved by the Greeks and expounded by Euclid. Unless the reader is experienced in this kind of thing, some of the presuppositions will perhaps have passed unnoticed. But they are all justifiable by strict reasoning; nothing has been put over.