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The Interpretation of Imaginary Mathematical Time

Published online by Cambridge University Press:  15 September 2017

G. Windred
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Extract

The conception of a mathematical time, as employed in the theory of relativity and other branches of mathematical physics, is frequently used and of great importance in our development of the theory of the universe. In most problems of mathematical physics it is customary implicitly to assume linearity of the temporal continuum, so that instants and periods are respectively regarded as points and sections of a homogeneous number-system. In this way the temporal and numerical continua are identified, and times become analogous to ordinary numbers. In the same way that the conception of imaginary quantities is necessary to the completion of the ordinary number continuum, so also it becomes necessary in certain problems of applied mathematics to consider imaginary values of the time variable. The present paper represents a survey of the mathematical theory of time, and an attempt to interpret mathematically, by means of a time-system admitting imaginary values, certain optical phenomena of relativistic mechanics.

Type
Research Article
Information
The Mathematical Gazette , Volume 19 , Issue 235 , October 1935 , pp. 280 - 290
Copyright
Copyright © Mathematical Association 1935

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References

page 280 note * For a review of existing literature, see the author’s paper “The History of Mathematical Time”, Isis, vol. 19 (1), p. 121 ; vol. 20 (1), p. 192.

page 281 note * We are using the first two sections of the “Transcendental Aesthetic” from the first edition of the Kritik. There is an English translation by F. Max Müller, London, 1896.

page 281 note † Vide N. K. Smith : A Commentary to Kant’s “Critique of Pure Reason”, p. 130.

page 281 note ‡ The World as Will and Idea. English trans, by Haldane and Kemp, vol. i, p. 9.

page 282 note * Op. cit. vol. ii, p. 205.

page 282 note † Vol. xii, pp. 306-318.

page 282 note ‡ P. xxvii of Preface.

page 282 note § P. 174 : Vide, etiam, J. L. Coolidge, The Geometry of the Complex Domain, 1924, pp. 24-25.

page 283 note * The recently published Source Booh in Mathematics, edited by D. E. Smith, New York, 1929, contains some reference to the paper, which is also briefly mentioned by Coolidge, J. L., Geometry of the Complex Domain, Oxford, 1924 Google Scholar.

page 283 note † Expressed in a lecture on Hamilton delivered at Lehigh University in 1901.

page 284 note * Jahresb. d. d. Math. Verein., vol. 19, 1910, p. 287.

page 284 note † Vide Bergson, Henri Louis, Time and Free Will; English trans, by Pogson, F. L., London, 1912, pp. 91 Google Scholar et seqq. For a discussion of Bergson’s mathematical theory of time, see also Bosanquet, B., Science and Philosophy, London, 1927, pp. 223 Google Scholar et seqq. Cp. etiam Hobhouse, L. T., The Theory of Knowledge, London, 1896, pp. 43 Google Scholar et seqq, pp. 599 et seqq.

page 285 note * Robb’s first contribution is a tract entitled Optical Geometry of Motion, a New View of the Theory of Relativity, Cambridge, 1911. The next, containing the idea of “;conical order”, is A Theory of Time and Space, Cambridge, 1913, which forms an introduction to the final theory contained in a larger volume of the same title, Cambridge, 1914. There is also a short treatise on The Absolute Relations of Time and Space, Cambridge, 1921, explanatory to the larger work.

page 285 note † An Enquiry Concerning the Principles of Natural Knowledge, Cambridge, 1919, praec. ch. ix, pp. 110-120.

page 286 note * Op cit. § 13.

page 286 note † Ibid. § 14 (5).

page 286 note ‡ X Ibid. § 14 (1). Kant here attacks contemporary ideas. On this subject, cf. The Critical Philosophy of Immanuel Kant, Edward Caird, Glasgow, 1889, ch. v of introduction, pp. 179 et seqq.

page 286 note § Op cit. § 14 (5), nota. Vide etiam Smith, N. K., A Commentary to Kant’s Critique of Pure Reason, 1918, pp. 128 Google Scholar et seqq. On the relation between the theories of Kant and Hamilton, from a mathematical viewpoint, see the author’s paper, “History of the Theory of Imaginary and Complex Quantities”, Mathematical Gazette, Vol. XIV, 1929, p. 533.

page 286 note ║ One of the most outstanding critics of the relativity theory is Melchior Palágyi, whose papers, afterwards published collectively in the Ausgewahlte Werke, Leipzig, 1925, contain many attacks on this theory. An excellent appreciation of Palágyi’s criticism is given by Gunn, J. A., The Problem of Time, London, 1925, pp. 208–214 Google Scholar.

page 287 note * A Theory of Time and Space, Cambridge, 1914, p. 2.

page 287 note † Op. cit. p. 7.

page 287 note ‡ Comptes Rendus, xxix (1849), p. 90.

page 287 note § Robb, loc. cit. p. 8 ; also The Absolute Relations of Time and Space, Cambridge, 1921, pp. 11, 12.

page 288 note * Die Welt als Wille und Vorstellung. No. 4 of the propositions concerning Time.

page 288 note ‡ Dissertatio De Mundi “ § 14 (5).

page 289 note * At this point, compare with Cayley’s views on Sir William Rowan Hamilton’s deduction of the form $$a + b-1 -1 in connection with the theory of time noted in § 2 above.

page 289 note † The Absolute Relations of Time and Space, p. 12 ; A Theory of Tim,e and Space, p. 6.

page 290 note * Vide, e.g. Eddington, The Mathematical Theory of Relativity, Cambridge, 1923, p. 17 ; The Lorentz Transformation.

page 290 note † Cp. Eddington, The Mathematical Theory of Relativity, pp. 13-29.