Navigating the History of Mathematics

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Notes

¹Ivor Grattan-Guinness, ‘History or Heritage? An Important Distinction in Mathematics and for Mathematics Education’, American Mathematical Monthly, 111 (2004), 1–12.

²Izabella Grigor'evna Bashmakova and Ioannis M. Vandoulakis, ‘On the Justification of the Method of Historiographical Interpretation’, in Trends in the Historiography of Science, edited by Kostas Gavroglu, Jean Christianidis and Efthymios Nicolaidis, (Dordrecht/Boston/London: Kluwer Academic Publishers, 1994), 249–64.

⁴Ivor Grattan-Guinness, The Norton History of the Mathematical Sciences: The Rainbow of Mathematics (New York/London: W. W. Norton, 1997; 1st American ed., 1998) (p. 7).

³Jesper Lützen and Walter Purkert, ‘Conflicting Tendencies in the Historiography of Mathematics: M. Cantor and H.G. Zeuthen’, in The History of Modern Mathematics, Vol. III: Images, Ideas, and Communities, edited by Eberhard Knobloch and David E. Rowe (Boston, MA: Academic Press, 1994), 1–42 (p. 1).

⁵George Berkeley, The Analyst: Or, a Discourse Addressed to an Infidel Mathematician. Wherein it is Examined Whether the Object, Principles, and Inferences of the Modern Analysis are More Distinctly Conceived, … Than Religious Mysteries … By the Author of The Minute Philosopher (London: printed for J. Tonson/Dublin: printed by and for S. Fuller, and J. Leathly, 1734; London: printed for J. and R. Tonson and S. Draper, 2nd ed., 1754) (p. 59). For details of Grattan-Guinness's account, see his ‘Berkeley's Criticism of the Calculus as a Study in the Theory of Limits’, Janus, 56 (1970), 215–27; erratum: 57 (1971), 80.

⁶ Annals of Science, 34 (1977), 193–202. An example is the History of Astronomy (London/New York: Putnams, 1909) by George Forbes (1849–1936), mentioning the discovery of Neptune with the aid of computation based on the Bode-Titius Law. Forbes failed to give the mathematical statement of the law itself: that the distance of the n ^th planet from the sun is 0.4 + 0.3 · 2 ⁿ , or an account of its history, but is satisfied to attempt a verbal explanation and provide the corresponding table. The original statement, in the Deutliche Anleitung zur Kenntniß des gestirnten Himmels (Hamburg: Johann Georg Büsch, 2te durchgehends verbesserte und stark vermehrte Auflage, 1772) of Johann Elert Bode (1747–1826) is rendered as: a=(n+4)/10, for a=0, 3, 6, 12, 24, 32, 48, … In fact, the actual distances between the planets show the inaccuracy of the law. It transpires in fact that the distribution of the planets is closer that given by the Fibonacci sequence than to the distribution as represented the law, and that some modification of the Titius-Bode Law based on the Fibonacci sequence provides a better empirical fit; see, e.g. H.W. Gould, ‘Cale's Rule for Planetary Distances’, Proceedings of the West Virginia Academy of Science, 37 (1966), 243–57. Grattan-Guinness mentions the Bode-Titius Law (p. 297) as a secular replacement in the later editions of Principia for Newton's statement about God's placement of orbits of the planets in the first edition.

⁷Thomas S. Kuhn, The Structure of Scientific Revolutions (Chicago/London: University of Chicago Press, 1962; reprinted: 1969; 2nd enlarged ed., 1970).

⁸Errett A. Bishop, ‘The Crisis in Contemporary Mathematics’, Historia Mathematica, 2 (1975), 507–17.

⁹Joong Fang, ‘Is Mathematics an “Anomaly” in the Theory of “Scientific Revolutions”?’, Philosophia Mathematica, 10 (1973), 92–101.

¹⁰Michael J. Crowe ‘Ten “Laws” Concerning the Patterns of Change in the History of Mathematics’, Historia Mathematica, 2 (1975), 161–66; reprinted: Donald A. Gillies (ed.), Revolutions in Mathematics (Oxford: Clarendon Press, 1992; paperback edition, 1995), pp. 15–20.

¹¹Emelie A. Kenney, ‘On the Possibility of Mathematical Revolutions’, Philosophia Mathematica, 5 (1990), 114–23.

¹²Isaac Asimov, ‘Foreword’ to Carl Benjamin Boyer and Uta C. Merzbach, A History of Mathematics (New York/Chi-chester/Brisbane/Toronto/Singapore: John Wiley & Sons, 2nd revised ed., 1991), vii–viii (pp. vii; viii).

¹³Herbert Mehrtens, ‘T. S. Kuhn's Theories and Mathematics: A Discussion Paper on the “New Historiography” of Mathematics’, Historia Mathematica, 3 (1976), 297–320; reprinted in Gillies (note 10), with an appendix, pp. 21–48; Joseph Warren Dauben, ‘Conceptual Revolutions and the History of Mathematics’, in Transformation and Tradition in the Sciences, edited by Elliott Mendelson (Cambridge: Cambridge University Press), 81–103; reprinted in Gillies (note 10), pp. 49–71; Imre Lakatos, in Proofs and Refutations: The Logic of Mathematical Discovery, edited by John Worrall and Elie Zahar (Cambridge/New York/New Rochelle/Melbourne/Sydney: Cambridge University Press, 1976; reprinted with corrections: 1977, 1979, 1981; reprinted: 1983, 1984, 1987); p. 192, Philip Kitcher, The Nature of Mathematical Knowledge (New York/Oxford: Oxford University Press, 1984).

¹⁴See also Mary Tiles, The Philosophy of Set Theory: An Introduction to Cantor's Paradise (Oxford/New York: Basil Blackwell, 1989; 2^nd ed., Mineola, NY: Dover, 2004).

¹⁵Claire L. Parkinson, ‘Paradigm Transitions in Mathematics’, Philosophia Mathematica, 2 (1987), 127–50 (p. 144).

¹⁶Basel: Birkhäuser/Berlin: Deutscher Verlag der Wissenschaft, 1990.