Numbers: Their Tales, Types, and Treasures.

Notes

CHAPTER 1: NUMBERS AND COUNTING

1. Marvin Minsky, The Society of Mind (New York: Simon & Schuster, 1988), p. 192.

2. Paul Auster, The Music of Chance (New York: Viking, 1990), p. 73.

3. Bertrand Russell, Introduction to Mathematical Philosophy (New York: Macmillan, 1920), 2nd ed., chapter 2. Retrieved from http://www.gutenberg.org/ebooks/41654.

4. It is much more difficult to “count” infinite sets. A definition of the cardinality of infinite sets is beyond the scope of this book.

5. This notion comes from a lengthy discussion in chapter 1 of Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (New York: John Wiley & Sons, 2000).

CHAPTER 2: NUMBERS AND PSYCHOLOGY

1. K. C. Fuson, “Research on Learning and Teaching Addition and Subtraction of Whole Numbers,” in Analysis of Arithmetic for Mathematics Teaching, ed. G. Leinhardt, R. Putnam, and R. A. Hattrup (Hillsdale, NJ: Lawrence Erlbaum Associates, 1992), p. 63.

CHAPTER 3: NUMBERS IN HISTORY

1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (New York: John Wiley & Sons, 2000).

2. Ibid., p. 538.

3. Ibid., p. 414.

CHAPTER 4: DISCOVERING PROPERTIES OF NUMBERS

1. Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics, rev. and updated ed. (New York: Oxford University, 2011), p. 104.

2. Aristotle, Metaphysics, trans. W. D. Ross (The Internet Classics Archive), book 1, part 5, http://classics.mit.edu/Aristotle/metaphysics.1.i.html.

3. Euclid, Euclid's Elements, trans. and ed. Thomas L. Heath (New York: Dover, 1956), book 7, http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0086.

4. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (New York: John Wiley & Sons, 2000), pp. 5–6.

5. Aristotle, Metaphysics.

6. Giovanni Reale, A History of Ancient Philosophy, trans. and ed. John R. Catan (Albany: State University of New York, 1987), pp. 63–64.

CHAPTER 5: COUNTING FOR POETS

1. James D. McCawley, The Phonological Component of a Grammar of Japanese (The Hague: Mouton, 1968).

CHAPTER 9: NUMBER RELATIONSHIPS

1. Leonardo Fibonacci, The Book of Squares (Liber Quadratorum), trans. L. E. Sigler (Orlando, FL: Academic Press, 1987).

CHAPTER 10: NUMBERS AND PROPORTIONS

1. Aristotle, Metaphysics, trans. W. D. Ross (The Internet Classics Archive), book 1, part 5, http://classics.mit.edu/Aristotle/metaphysics.1.i.html.

2. Herodotus, The Histories, trans. and ed. A. D. Godley (Cambridge, MA: Harvard University Press, 1920), book 2, chapter 124, http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0126.

CHAPTER 11: NUMBERS AND PHILOSOPHY

1. Charles Hermite, Correspondance d'Hermite et de Stieltjes, vol. 2, ed. B. Baillaud and H. Bourget (Paris: Gauthier-Villars, 1905), p. 398.

2. Charles Hermite, quoted in “Notice Historique sur Charles Hermite,” in Eloges académiques et discours, by G. Darboux (Paris: Hermann, 1912), p. 142.

3. G. H. Hardy, A Mathematician's Apology, 19th ed. (Cambridge, UK: Cambridge University Press, 2012), pp. 123–24.

4. E. B. Davies, “Let Platonism Die,” Newsletter of the European Mathematical Society, June 2007, p. 24.

5. Ibid., p. 25.

6. Barry Mazur, “Mathematical Platonism and Its Opposites,” Newsletter of the European Mathematical Society, June 2008, p. 19. (Italics in original.)

7. Reuben Hersh, “On Platonism,” Newsletter of the European Mathematical Society, June 2008, p. 17. (Italics in original.)

8. Richard Dedekind, “The Nature and Meaning of Numbers: Preface to the First Edition, 1887,” in Essays on the Theory of Numbers, trans. Wooster Woodruff Beman (Chicago: Open Court, 1901), p. 14, http://www.gutenberg.org/ebooks/21016. (Italics in original.)

9. Bertrand Russell, The Principles of Mathematics, vol. 1 (Cambridge, UK: Cambridge University Press, 1903), p. xliii.

10. Ibid., p. 111.

11. Eric Temple Bell, Mathematics: Queen and Servant of Science (New York: McGraw-Hill, 1951), p. 21.

12. Bertrand Russell, Introduction to Mathematical Philosophy (London: George Allen & Unwin, 1919), p. 22, http://www.gutenberg.org/ebooks/41654.

13. Ibid., p. 11.

14. Hermann Weyl, “Mathematics and the Laws of Nature,” in The Armchair Science Reader, ed. Isabel Gordon and Sophie Sorkin (New York: Simon and Schuster, 1959), p. 300.

15. Paul Benacerraf, “What Numbers Could Not Be,” in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam (Cambridge, UK: Cambridge University Press, 1964), p. 290.

16. Ibid., p. 291.

17. Heike Wiese, Numbers, Language, and the Human Mind (Cambridge, UK: Cambridge University Press, 2003), p. 79.

18. Albert Einstein, “Geometry and Experience,” in The Collected Papers of Albert Einstein, vol. 7, The Berlin Years, ed. M. Janssen et al. (Princeton, NJ: Princeton University Press, 2002), p. 385. See also The Digital Einstein Papershttp://einsteinpapers.press.princeton.edu.

19. Eugene P. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13, no. 1 (February 1960), New York: John Wiley & Sons, 1960.

20. Stewart Shapiro, “Philosophy of Mathematics and Its Logic: Introduction,” in The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro (Oxford, UK: Oxford University Press, 2007), p. 5.

21. Einstein, “Geometry and Experience,” p. 385.

22. Steven Weinberg, Dreams of a Final Theory (New York: Random House, 1993), p. 167.

23. Thomas Forster, “Does Mathematics Need a Philosophy?” (notes for paper presented to the Trinity Mathematical Society on October 21, 2013, https://www.srcf.ucam.org/tms/talks-archive/).

24. Mazur, “Mathematical Platonism and Its Opposites,” p. 20.