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The Locus of Mathematical Reality: An Anthropological Footnote

Published online by Cambridge University Press:  14 March 2022

Leslie A. White
Leslie A. White*
Affiliation:
University of Michigan, Ann Arbor, Michigan
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Extract

“He's [the Red King's] dreaming now,” said Tweedledee: “and what do you think he's dreaming about?”

Alice said, “Nobody can guess that.”

“Why, about you!” Tweedledee exclaimed, clapping his hands triumphantly. “And if he left off dreaming about you, where do you suppose you'd be?”

“Where I am now, of course,” said Alice.

“Not you!” Tweedledee retorted contemptuously. “You'd be nowhere. Why, you're only a sort of thing in his dream!”

“If that there King was to wake,” added Tweedledum, “you'd go out—bang!—just like a candle.”

“I shouldn't!” Alice exclaimed indignantly. “Besides, if I'm only a sort of thing in his dream, what are you, I should like to know?”

“Ditto,” said Tweedledum.

“Ditto, ditto!” cried Tweedledee.

He shouted this so loud that Alice couldn't help saying “Hush! You'll be waking him, I'm afraid, if you make so much noise.”

“Well, it's no use your talking about waking him,” said Tweedledum, “when you're only one of the things in his dream. You know very well you're not real.”

“I am real!” said Alice, and began to cry.

“You won't make yourself a bit realler by crying,” Tweedledee remarked: “there's nothing to cry about.”

“If I wasn't real,” Alice said—half laughing through her tears, it all seemed so ridiculous —“I shouldn't be able to cry.”

“I hope you don't suppose those are real tears?” Tweedledum interrupted in a tone of great contempt.

—Through the Looking Glass

Type
Research Article
Information
Philosophy of Science , Volume 14 , Issue 4 , October 1947 , pp. 289 - 303
Copyright
Copyright © Philosophy of Science Association 1947

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Footnotes

∗

I am greatly obliged to Professor R. L. Wilder of the Department of Mathematics, University of Michigan, for reading this paper in manuscript and for acquainting me with Jacques Hadamard's The Psychology of Invention in the Mathematical Field. It goes without saying, however, that Professor Wilder is not responsible for any errors or shortcomings that this essay may contain.

References

1 She wrote the following works, some of which went into several editions: The Mechanism of the Heavens, 1831 (which was, it seems, a popularization of the Mécanique Céleste of Laplace); The Connection of the Physical Sciences, 1858; Molecular and Microscopic Science, 1869; Physical Geography, 1870.

2 Personal Recollections of Mary Somerville, edited by her daughter, Martha Somerville, pp. 140–141 (Boston, 1874).

3 ibid., p. 375. See, also, A.D. White, The History of the Warfare of Science with Theology &c, Vol. I, p. 225, ftn.∗ (New York, 1930 printing).

4 Quoted by E. T. Bell in The Queen of the Sciences, p. 20 (Baltimore, 1931).

5 G. H. Hardy, A Mathematician's Apology, pp. 63–64 (Cambridge, England; 1941).

6 The mathematician is not, of course, the only one who is inclined to believe that his creations are discoveries of things in the external world. The theoretical physicist, too, entertains this belief. “To him who is a discoverer in this field,” Einstein observes, “the products of his imagination appear so necessary and natural that he regards them, and would like to have them regarded by others, not as creations of thought but as given realities,” (“On the Method of Theoretical Physics,” in The World as I See It, p. 30; New York, 1934).

7 P. W. Bridgman, The Logic of Modern Physics, p. 60 (New York, 1927).

8 Edward Kasner and James Newman, Mathematics and the Imagination, p. 359 (New York, 1940).

9 Principles of Philosophy, Pt. I, See. XVIII, p. 308, edited by J. Veitch (New York, 1901).

10 Les Formes Élémentaires de la Vie Religieuse (Paris, 1912) translated by J. W. Swain (London, 1915). Nathan Altshiller-Court refers to Durkheim's treatment of this point in “Geometry and Experience,” (Scientific Monthly, Vol. LX, No. 1, pp. 63–66, Jan., 1945).

11 Les Règles de la Methode Sociologique (Paris, 1895; translated by Sarah A. Solovay and John H. Mueller, edited by George E. G. Catlin; Chicago, 1938).

12 Durkheim did not use the term culture. Instead he spoke of the “collective consciousness,” “collective representations,” etc. Because of his unfortunate phraseology Durkheim has been misunderstood and even branded mystical. But it is obvious to one who understands both Durkheim and such anthropoligists as R. H. Lowie, A. L. Kroeber and Clark Wissler that they are all talking about the same thing: culture.

13 See, e.g., E. B. Tylor, Anthropology (London, 1881); R. H. Lowie, Culture and Ethnology, New York, 1917; A. L. Kroeber, “The Superorganic,” (American Anthropologist, Vol. 19, pp. 163–213; 1917); Clark Wissler, Man and Culture, (New York, 1923).

14 See, White, Leslie A., “The Symbol: the Origin and Basis of Human Behavior,” (Philosophy of Science, Vol. 7, pp. 451–463; 1940; reprinted in ETC., a Review of General Semantics, Vol. I, pp. 229–237, 1944).

15 Individuals vary, of course, in their constitutions and consequently may vary in their responses to cultural stimuli.

16 Cf. “Keresan Indian Color Terms,” by Leslie A. White, Papers of the Michigan Academy of Science, Arts, and Letters, Vol. XXVIII, pp. 559–563; 1942 (1943).

17 Article “Space-Time.” Encyclopedia Britannica, 14th edition.

18 op. cit., p. 359.

19 “On the Method of Theoretical Physics,” in The World As I See It, p. 33 (New York, 1934).

20 Science and the Human Temperament, p. 115 (London, 1935).

21 “On the Nature of Axioms,” in Science and Hypothesis, published in The Foundations of Science (The Science Press, New York, 1913).

22 Quoted by E. T. Bell, Men of Mathematics, p. 16 (New York, 1937).

23 The Rules of Sociological Method, Preface to 2nd edition, p. lvi.

24 The Elementary Forms of the Religious Life, p. 424. See also The Rules of Sociological Method, Preface to 2nd edition, p. li, in which he says “we need to investigate … the manner in which social representations [i.e., culture traits] adhere to and repel one another, how they fuse or separate from one another.”

25 Einstein, “The Mechanics of Newton and their Influence on the Development of Theoretical Physics,” in The World as I See It, p. 58.

26 “On the Theory of Relativity,” in The World as I See It, p. 69.

27 Einstein, “The Mechanics of Newton &c,” p. 57.

28 Jacques Hadamard, The Psychology of Invention in the Mathematical Field, p. 50 (Princeton, 1945).

29 ibid., p. 51.

30 The following data are taken from a long and varied list published in Social Change, by Wm. F. Ogburn (New York, 1923), pp. 90–102, in which simultaneous inventions and discoveries in the fields of chemistry, physics, biology, mechanical invention, etc., as well as in mathematics, are listed.

Law of inverse squares: Newton, 1666; Halley, 1684.

Introduction of decimal point: Pitiscus, 1608–12; Kepler, 1616; Napier, 1616–17.

Logarithms: Burgi, 1620; Napier-Briggs, 1614.

Calculus: Newton, 1671; Leibnitz, 1676.

Principle of least squares: Gauss, 1809; Legendre, 1806.

A treatment of vectors without the use of co-ordinate systems: Hamilton, 1843; Grassman, 1843; and others, 1843.

Contraction hypothesis: H. A. Lorentz, 1895; Fitzgerald, 1895.

The double theta functions: Gopel, 1847; Rosenhain, 1847.

Geometry with axiom contradictory to Euclid's parallel axiom: Lobatchevsky, 1836–40; Bolyai, 1826–33; Gauss, 1829.

The rectification of the semi-cubal parabola: Van Heuraet, 1659; Neil, 1657; Format, 1657–59.

The geometric law of duality: Oncelet, 1838; Gergone, 1838.

As examples of simultaneity in other fields we might cite:

Discovery of oxygen: Scheele, 1774; Priestley, 1774.

Liquefaction of oxygen: Cailletet, 1877; Pictet, 1877.

Periodic law: De Chancourtois, 1864; Newlands, 1864; Lothar Meyer, 1864.

Law of periodicity of atomic elements: Lothar Meyer, 1869; Mendeleff, 1869.

Law of conservation of energy: Mayer, 1843; Joule, 1847; Helmholz, 1847; Colding, 1847; Thomsom, 1847.

A host of others could be cited. Ogburn's list, cited above, does not pretend to be complete.

31 Hadamard entitles one chapter of his book “Discovery as a Synthesis.”

32 We use “culture” here in its bacteriological sense: a culture of bacilli growing in a gelatinous medium.

33 The distinguished anthropologist. A. L. Kroeber, defines geniuses as “the indicators of the realization of coherent, patterns of cultural value,” Configurations of Culture Growth, p. 839 (Berkeley, 1944).

34 “Mathematical Creation,” in Science and Method, published in The Foundations of Science, p. 387 (The Science Press; New York and Garrison, 1913).

35 ibid., p. 393.

36 See, W. Köhler's The Mentality of Apes (New York, 1931).

37 See Leslie A. White, “On the Use of Tools by Primates” (Journ. of Comparative Psychology, Vol. 34, pp. 369–374, Dec. 1942). This essay attempts to show that the human species has a highly developed and progressive material culture while apes do not, although they can use tools with skill and versatility and even invent them, because man, and not apes, can use symbols.

38 A Mathematician's Apology, p. 63.

39 “Mathematical Proof,” p. 4 (Mind, Vol. 38, pp. 1–25, 1929).

40 A Mathematician's Apology, pp. 62–63, 65.

41 ibid., p. 63.