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A Detail in Kronecker's Program

Published online by Cambridge University Press:  14 March 2022

E. T. Bell
E. T. Bell*
Affiliation:
California Institute of Technology
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Extract

It was Kronecker who sought to avoid the use in mathematics of all numbers (negatives, fractions, irrationals) other than the positive integers, and he outlined the means for carrying through this program. In the introductory sections of his memoir he briefly indicates the personal philosophy which made such a project appear desirable.

Type
Research Article
Information
Philosophy of Science , Volume 3 , Issue 2 , April 1936 , pp. 197 - 207
Copyright
Copyright © Philosophy of Science Association 1936

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References

1 L. Kronecker, Uber den Zahlbegriff, Journal für die reine und angewandte Mathematik vol. 101 (1887) pp. 251–280; Werke, vol. III 1, pp. 251–280.

2 E. W. Hobson, The Theory of Functions of a Real Variable, etc., (1907) Art. 17.

3 A. C. Black, The Nature of Mathematics (1933), p. 177. Black refers “for a full discussion and criticism of Kronecker's views” to

4 L. Couturat, De l'Infini Mathématique, (1896), pp. 603–616. This book had a wide circulation a generation ago.

5 A. L. Cauchy, Exercises d'Analyse et de Physique Mathématique, Tome 4, (1847), pp. 87–110. One remark of Cauchy's is rather amusing, coming as it does from the inventor of the theory of functions of a complex variable:

“In the theory of algebraic equivalences [congruences] substituted for the theory of imaginaries, the letter i will cease to represent the symbolic sign , which we repudiate completely, and which we can abandon without regret, because nobody knows what this sham sign signifies, nor what meaning is to be attributed to it.”

6 J. H. M. Wedderburn, Algebraic Fields, Annals of Mathematics, Second Series, vol. 24 (1923), pp. 236–264.

7 The sign ≡ of identical equality in the first formula is not in danger of confusion with the same sign in the second formula, for in the latter ≡ is part of the compound sign “≡… mod” of congruence. The “mod” indicates that congruence, not identical equality is meant.

8 The “+” in [f(x)] + [g(x)] refers to addition of classes, as defined; the “+” in [f(x) + g(x)] refers to addition in the semi-field D. Similarly for x.

9 For the postulates of a field see L. E. Dickson, Algebras and their Arithmetics, p. 201, or the papers by Dickson or E. V. Huntington in Transactions of the American Mathematical Society, vol. 4 (1903) pp. 13–20, vol. 6 (1905) pp. 181–204.

10 Since this note was written, a paper by H. S. Vandiver has appeared (Proceedings of the National Academy of Sciences, vol. 20, November 1934, pp. 579–584) bearing partly on Kronecker's program. It would be of considerable historical interest to know the exact date of Schatunovsky's first work on objections to the unrestricted use of the law of excluded middle, as some of his examples are even more illuminating than some of Brouwer's.