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Modern Mathematics and Some Problems of Quantity, Quality, and Motion in Economic Analysis

Published online by Cambridge University Press:  14 March 2022

Karl H. Niebyl
Karl H. Niebyl*
Affiliation:
Carleton College
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Abstract

It can not be our purpose to give here a complete account of the phenomenological history of mathematical doctrine. It will be enough to refer to the battle of opinions in mathematical theory which was waged within the last eighty-five years, since Riemann's inaugural lecture on Non-Euclidean Geometry. Furthermore, the revolution which Einstein's theory of general relativity created is indicative of the complete absence of any general awareness that mathematics as a science has any relation to social reality. If we are going to establish such relations, we are conscious of the fact that they would be hotly contended by many mathematicians.

Type
Research Article
Information
Philosophy of Science , Volume 7 , Issue 1 , January 1940 , pp. 103 - 120
Copyright
Copyright © The Philosophy of Science Association 1940

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References

Notes

1 Cf., Hans Reichenbach, Relativitätstheorie und Erkenntnis Apriori, Verlag von Julius Springer, Berlin, 1920.

2 l.c., p. 4.

3 Ibid.

4 Cf., Albert Einstein, Relativity, The Special and General Theory, London: Methuen, 1920; The Meaning of Relativity, Four Lectures delivered at Princeton University, May, 1921, London: Methuen, 1922; Sidelights on Relativity, London: Methucn, 1921, especially, Part II; “Zur Einheitlichen Feldtheorie,” Sitzungen der Preussischen Akademie der Wissenschafften, Berlin, 1929.

5 Cf., Science at the Crossroads, Papers presented to the International Congress of the History of Science and Technology, held in London, June 29-July 3, 1931, Kniga Ltd., London, no date.

6 Reichenbach, l.c., pp. 6-7.

7 The development of Markets with the concomitant separation of producer and final consumer made necessarily for a tendency towards an abstract picture of human relations, as reflected, e.g., in the transition from arithmetic to algebra. A similar development had once before taken place in Greece, between 700 and 400 B.C.

8 Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen (Habilitationsvortrag Göttingen 1854; Sonderausgabe mit Erläuterungen von Weyl, 1923).

9 Herrn. Weyl, Philosophie der Mathematik und Naturwissenschaft (Handbuch der Philosophie), München und Berlin: R. Oldenbourg, 1927, p. 63.

10 Professor Paul Laberenne writes in “Mathematik und Technik,” (Die Wissenschaft … Jean-Christophe—Verlag, Zürich, 1937, translated into German by Dr. Hans Mühlestein, p. 24), that the real importance of the non-Euclidean geometry and of the absolute differential calculus appeared only with the development of Einstein's theory of relativity.

11 Cf., Max Black, The Nature of Mathematics, New York: Harcourt, Brace, & Co., 1934. p. 2.

12 Cf., L. E. J. Brouwer, “Intuitionism and Formalism,” The Bulletin of the American Mathematical Society, Vol. 20, 1914, pp. 81-82.

13 Cf., Bertrand Russell, Principles of Mathematics, New York: W. W. Norton, Second Edition, 1938, p. xvi ff.

14 Ibid., pp. xvi-xvii.

15 l.c., p. xvii.

16 Russell, l.c., p. 3.

17 Cf., V. J. McGill, “An Evaluation of Logical Positivism,” Science and Society, Vol. I, No. 1, pp. 56-57.

18 Struik, l.c., p. 97.

19 Russell, l.c., p. V. For modifications, cf., pp. IX ff.

20 There exists considerable confusion concerning the meaning of the term dynamic. Russell, and most economists mean by it a linear change, while we want it to infer a qualitative, i.e., dialectic change.

21 The difficulty involved in this reduction is indicated by Struik when he refers to the fact that “when capitalism began its triumphant conquest of Central and Western Europe … it commenced … in the universities and other institutions of learning … with an awakened interest in Euclid, followed by a similar interest in Archimedes. … In philosophy, the Aristotelian school lost its supremacy. It is typical of this period that the worship of Aristotle was replaced, for awhile, by a revival of interest in Plato. Plato, in whose world the deity is always geometrizing, and whose school stimulated mathematics as the key to understanding, stressed the value of measurement and quantitative research much more than Aristotle. Quantity had to be stressed at the expense of quality. This explains why Plato, despite his idealism, was preferred to a philosopher like Aristotle, who was much closer to a materialist viewpoint” (all italics mine). (D. J. Struik, “Concerning Mathematics,” Science and Society, Vol. I, No. 1, 1936, p. 136.) We may add that it was not “despite” Plato's idealism, but because of it that his philosophy became the correlate to Euclidean geometry and quantitative mathematics, at a time when this Euclidean geometry, originally developed for definite material purposes, began to be used as a technique in the process of production, the latter being characterized by its alienating from itself the wholeness of human activity. I.e., Euclidean geometry was being used abstractly.

22 Russell, op. cit., p. 4.

23 l.c., p. 4.

24 Wittgenstein, Tractatus Logico-Philosophicus, p. 31.

25 l.c., p. 109.

26 This statement is made in conscious contradiction to the one by Brouwer (op. cit., p. 83), that formalism is “diametrically opposed” to Kantian a priorism. The point in question is not that Kant's “judgments” were “independent of experience and not capable of analytical demonstration”, but that Kant's judgments have an idealistic character. Compare to this, Weyl's statement that “Hilbert agrees with Kant who, by the way, emphasized also the symbolic construction in algebra with sign symbols (Critique of Pure Reason, 2nd Ed., German, p. 745), ‘in that mathematics has a secure content independent of all logic and that it therefore never can be proven by logic alone’ (Über das Unendliche, p. 171)“. (Weyl, Philosophie der Mathematik, etc., op. cit., p. 51.)

27 Herrn. Weyl, Philosophie der Mathematik, etc., op. cit., p. 44.

28 Cf., Brouwer, op. cit., p. 83.

29 Cf., also, Weyl, op. cit., p. 49.

30 Bertrand Russell, Principles, op. cit., p. vi.

31 Cf., Lionel Robbins, The Nature and Significance of Economic Science, London, Macmillan Ltd., 1932, p. 75.

32 Cf., Dr. Lilly Hecht, A. Cournot und L. Walras, ein formaler und materialer Vergleich wirtschaftstheoretischer Ableitungen, Band I, Heft 6, Heidelberg: Verlag der Weiss' sehen Universitätsbuchhandlung, 1930, p. 11; also, D. Hilbert, Neubegründung der Mathematik (Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, I. Bd., S. 157 ff., Hamburg, 1922), S. 160/61.

33 Ibid.

34 Black, op. cit., p. 10. Compare to this the statement by F. Engels, (Anti-Dühring) “… mathematics arose out of the needs of men.“

35 Black, l.c., p. 10.

36 Cf., Brouwer, op. cit., p. 85; see also, Poincaré, Henri, Le Science el l'Hypothese, Paris, Flammarion, 1902, p. 104.

37 Brouwer, l.c., p. 86.

38 Cf., Weyl, Philosophie der Mathematik, etc., op. cit., pp. 42-43.

39 Cf., Black, op. cit., pp. 196 ff.

40 Cf., Kurt Lewin, Principles of Topological Psychology, New York, MacGraw-Hill, 1935, p. 8.

41 Cf., Kurt Lewin, A Dynamic Theory of Personality, MacGraw-Hill, New York, 1935, p. 1.

42 L.c., Principles of Topological Psychology, op. cit., p. 15.

43 L.c., Topological Psychology, pp. 15-16.

44 We have to keep in mind, however, that Brouwer had conceived of his continuum as “linear.“

45 Topological Psychology, op. cit., p. 16.

46 Lewin, Topological, op. cit., p. 17.

47 The same objection could be raised if we were to analyze the concrete examples which Lewin gives in amplification of his term “life space” or “life situation.” What actually is described in the example of the life situation of the moment is the “single instant” of a total situation, i.e., a situation not valid to be used as a point of departure.

48 Lewin, Topological, op. cit., p. 19.

49 See also, E. Colman, “The Present Crisis in the Mathematical Sciences and General Outline for their Reconstruction,” Science at the Cross-Roads, Kniga, London, pp. 220-231.

50 In the Preface to “The Conceptual Representation and the Measurement of Psychological Forces,” Durham, North Carolina: Duke University Press, 1938, Lewin points out that the development of theoretical psychology which has the same relation to experimental psychology “as theoretical physics has to experimental physics is extremely important today.” We hope that this paper indicates that the vast amount of uncritically used concepts in economics and the urgency of solving adequately some of the problems with which we arc confronted, shows the necessity of overhauling our conceptual apparatus in economics and of penetrating the semantic veil in which we are enclosed.

51 Lewin follows here the suggestion made by Leibniz and recently repeated by Colman. Leibniz had written in a letter to Hugens van Zulichen: “Et je croy … avoir le moyen, et qu'on pourroit réprésenter des figures et mesme des machines et mouvements (my italics) en caractères, comme l'Alegebre réprésente les nombres ou grandeurs” (Leibnizens mathematische Schriften, hersg. von C. I. Gerhardt, 1850, Berlin, A. Asher & Co., Erste Abteilung, Band II, p. 19). Colman had written: “Why should we not proceed further along this road and attempt to create a new qualitative calculus …?” (E. Colman, “The Present Crisis in the Mathematical Sciences and General Outline for their Reconstruction,” Science at the Crossroads, London: Kniga, no date, p. 222.)

52 Cf., for this and the following, Lewin, “The Conceptual Representation,” etc., pp. 22. ff.

53 Compare above, our discussion of Brouwer and Weyl.

54 Lewin, op. cit., p. 25.

55 By “construct”, Lewin means “a dynamic fact which is determined indirectly as an ‘intervening concept’ by way of ‘operational definition.‘ A construct expresses a dynamic interrelation and permits, in connection with Laws, the making of statements about what is possible and what is not possible.” (Cf., Lewin, Topology, op. cit., p. 213.)

56 This ‘whole’ refers, for Lewin, to life space; for us, it refers to ‘social space.‘

57 Cf., Lewin, “Conceptual Representation,” etc., op. cit., p. 110.

68 Cf., Kurt Lewin, “The Conflict Between Aristotelian and Galileian Modes of Thought in Contemporary Psychology,” reprinted in, A Dynamic Theory of Personality, New York, MacGraw-Hill, 1935, pp. 1-42.

69 Cf., Lewin, “Conceptual Representation,” etc., op. cit., p. 112.

60 Lewin quotes here Helmholtz, Carnap, Blumberg and Feigl, l.c., p. 113.