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What is modern about “modern” mathematics? (English) Zbl 0840.01024

Ausgehend von einem bekannten Bonmot von B. Russell (1901) verfolgt der Verf. die Frage “welcher Wahrheitsgehalt kommt der Mathematik zu”, von der Antike bis zur Gegenwart. 1. Die klassische Tradition: Die Ansicht von der Wirklichkeit der Außenwelt und deren Erkenntnis wird anhand klassischer Belege verfolgt. 2. Die romantische Revolution. Um 1800 erfolgt in der Kunst, der Musik sowie in der Mathematik ein Umbruch. Dem absoluten Aspekt wird ein subjektiver entgegengestellt. 3. “Mathematics Too?” Bezeichnend für die Neuzeit ist ein Ausspruch von J. Bolyai: “Aus Nichts habe ich eine mir eigene Welt geschaffen”. Die Erkenntnisse von K. Gödel und R. Thom weisen in dieselbe Geisteshaltung, während D. Hilbert eine prästabilisierte Harmonie zwischen Natur und Denken vermutete. 29 Literaturangaben belegen die Ausführungen.

MSC:

01A60 History of mathematics in the 20th century
00A30 Philosophy of mathematics
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[1] Bertrand Russell, ”Recent Work on the Principles of Mathematics,”International Monthly 4 (1901), 84.
[2] A. E. Housman, ”The Name and Nature of Poetry,” inCollected Poems and Selected Prose, London: Allen Lane, The Penguin Press (1988), p. 364.
[3] Clive Bell,Art, London: Chatto & Windus (1949), p. 25.
[4] Isaiah Berlin,The Crooked Timber of Humanity, New York: Random House (Vintage Books) (1990), pp. 5–6.
[5] David Knowles,The Evolution of Medieval Thought, New York: Random House (Vintage Books) (1962), p. 55.
[6] Plato,Timaeus, 29 ff.
[7] Romans 1:20. · Zbl 1269.81126
[8] Spinoza,Ethics, Part II, prop. vii. · Zbl 0948.01041
[9] Boethius,De Institutione Arithmetica (translated by Michael Masi asBoethian Number Theory), Amsterdam: Editions RodopiB.V, 1983, I, 1.
[10] John Dewey,The Quest for Certainty. New York: Minton, Balch, 1929, p. 23.
[11] Plato,Thaeatetus, 191c ff, 193b-196a, 200c; Aristotle,De anima 424a.
[12] Shakespeare,Measure for Measure, II, ii, 117–122.
[13] The most learned and fascinating expositor and defender of the traditional philosophy of art was Ananda K. Coomaraswamy; see, for example,Traditional Art and Symbolism, Princeton NJ: Princeton University Press (1986).
[14] Dante,Inferno, xxxii, 10-12.
[15] Richard Paul, ”Jean-Philippe Rameau (1683-1764), the Musician as Philosophe,”Proc. Am. Phil. Soc. 114 (1970), 140–154.
[16] Johann Winckelmann, ”History of Ancient Art,” inWinck- elmann: Writings on Art, David Irwin (ed), London: Phaidon (1972), pp. 117ff.
[17] de Honoré Balzac,Le père Goriot, trans. M. A. Crawford, London: Penguin (1951), p. 134.
[18] Isaiah Berlin, (ref. 4)passim.
[19] M. L. Abrams,The Mirror and the Lamp, New York: Oxford University Press (1953).
[20] S. T. Coleridge,Table Talk and Omniana of Samuel Taylor Coleridge, quoted in Ref. 19, p. 58.
[21] Howard E. Hugo (ed.),The Portable Romantic Reader, New York: Viking Press (1957), p. 61.
[22] Gabriel Josipovici,The World and the Book: A Study of Modern Fiction, London: Macmillan (1971), pp. xiii-xiv.
[23] M. Kline,Mathematics in Western Culture, New York: Oxford University Press (1953), p. 9. · Zbl 0053.19301
[24] Ivor Grattan-Guinness,Br. J. History Sci. 7 (1974), 186.
[25] János Bolyai, letter of 3 November 1823, in Roberto Bonola,Non-Euclidean Geometry. New York: Dover (1955), ”Translator’s Introduction” toThe Science of Absolute Space, p. xxviii.
[26] William Wordsworth,The Prelude, III, 142.
[27] On both Godei and Thorn, cf. Philip J. Davis and Reuben Hersch,The Mathematical Experience, New York: Penguin Books (1983), pp. 318–319.
[28] Constance Reid,Hilbert-Courant, New York: Springer- Verlag (1986), p. 194.
[29] G. Frege,The Foundations of Arithmetic, 2nd rev. ed., trans. J.L. Austin, Evanston IL: Northwestern University Press (1980), p. 108.
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