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Some Remarks on the Foundations of Mathematics: an Expository Article

Published online by Cambridge University Press:  03 November 2016

G. Kreisel
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Extract

It is tempting to state the problems of the foundations of mathematics in general logical. terms, with words like “truth”, “proposition”, “necessary conclusions” In this way a sometimes useful, and always at any rate verbal, connection is made with old problems of logic; and for this reason also mathematical investigations into the foundations of mathematics have found a good number of popular exponents. But it seems that at times these expositions suffer considerably from rather ill-chosen slogans.

Type
Research Article
Information
The Mathematical Gazette , Volume 35 , Issue 311 , February 1951 , pp. 23 - 28
Copyright
Copyright © The Mathematical Association 1951

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References

Page 23 of note * The criticism of indirect proofs of theorems without existence symbols is particularly unconvincing; see, for example, Gazette, No. 300, pp. 202–3, where it is said that the ordinary proof of the irrationality of √2 (p 2 ≠ 2q 2 for integers p and q) does not “tell” you how much p 2/q 2 differs from 2; yet it tells you that p 2 ≠ 2q 2 and, since p and q are integers, that ∣p 2 – 2q 2 ∣ ≥ 1. No attempt is made in this article to describe precisely which proofs (of analysis) are direct and which are not. For, we believe, the logical difficulties which are often put down to indirect proofs arise from the very form, of the theorem which is proved, and our discussion concentrates on this.

Page 24 of note * In school mathematics such definitions simply do not occur, and the reader might wonder what the fuss is about: hence we give an example.

Page 26 of note * At this point, the false impression mentioned in the introduction may be created: the proofs of a system, that is, the classification of formulae into proved, disproved (and indeterminate) ones, may well be useful without an interpretation. However, if one is determined to have an interpretation, one is led to some such formulation as the one below, I think.

Page 26 of note † It is here clear why one needs a formal system. To give precise translation rules, one must at least say what formulae are to be translated, and from below, how they are to be proved. Giving a formalization of proofs is just that.

Page 28 of note * The crux of the matter is: the experimental proposition “λ is a value of the observable” is associated with the formulae , where is a bound variable formula.