NOTES

INTRODUCTION

1. Immordino-Yang M. H., & Damasio A. (2007). We feel, therefore we learn: The relevance of affective and social neuroscience to education. Mind, Brain, and Education 1(1), 2, 4.

2. Ibid., p. 10.

3. The concept of myth is included in Claudia Henrion’s book Women in Mathematics as well as in this one. We use this notion independently of each other. It was only recently that we became aware of our shared use of this concept and some similarities in Henrion’s and our formulations.

4. Bronowski, J. (1965). Science and human values. New York: Harper and Row, pp. 62–63.

CHAPTER 1. MATHEMATICAL BEGINNNINGS

1. A short biographical list is included at the end of the book describing key achievements and facts about each of the mathematicians whose names appear in the text.

2. Ulam, Stan. (1976). Adventures of a mathematician. New York: Scribner, p. 10.

3. Dyson, Freeman J. (2004). Member of the club. In John Brockman (Ed.). Curious minds: How a child becomes a scientist. New York: Pantheon Books, p. 61.

4. Sacks, Oliver (2001). Uncle Tungsten: Memories of a chemical boyhood. New York: Alfred Knopf, p. 26.

5. Murray, Margaret (2000). Women becoming mathematicians. Cambridge, Mass.: Massachusetts Institute of Technology. Includes each of the mathematicians whose names appear in the text. Ulam (1976), pp. 10, 83.

6. Wigner, Eugene (1992). The recollections of Eugene P. Wigner as told to Andrew Szanton. New York: Plenum Press, p. 23.

7. Ibid., p. 45.

8. Ibid., pp. 47–48.

9. Brockman, John (Ed.) (2004). Curious minds: How a child becomes a scientist. New York: Pantheon Books, pp. 193–194.

10. Albers, Don (2007). John Todd—Numerical mathematics pioneer, College Mathematics Journal 38(1), 5.

11. Ibid.

12. Ibid.

13. Reid, Constance (1996). Julia, a life in mathematics. Washington, D.C.: Mathematical Association of America, p. 3.

14. Perl, Teri (1978). Biographies of women mathematicians and related activities. Menlo Park, Calif.: Addison-Wesley, p. 64.

15. Osen (2004).

16. James, I. (2002). Remarkable mathematicians. Cambridge: Cambridge University Press, pp. 231–232.

17. Ibid.

18. Ibid., p. 122.

19. Massera, J. L. (1998). Recuerdos de mi vida academica y politica (Memories of my academic and political life). Lecture delivered at the National Anthropology Museum of Mexico City, March 6, 1998, and published in Jose Luis Massera: The scientist and the man. Montevideo, Uruguay: Faculty of Engineering. Text translated by Frank Wimberly. These quotes are from the unpaginated text of Massera’s lecture.

20. Mordell, L. (1971). Reminiscences of an octogenarian mathematician, American Mathematical Monthly 78, 952–961.

21. Tikhomirov, V. M. (2000). Moscow mathematics 1950–1975. In Jean Paul Pisier (Ed.). Development of mathematics 1950–1975. Boston: Birkhäuser, pp. 1109–1110.

22. Ibid.

23. Singh, Simon (1998). Fermat’s Last Theorem. London: Fourth Estate, pp. 5–6.

24. Ibid., p. 6.

25. Feldman, David H. (1986). Nature’s gambit. New York: Basic Books, p. 16.

26. Wolpert, Stuart (2006). Terence Tao, “Mozart of Math,” is UCLA’s first mathematician awarded the Fields Medal, often called the “Nobel Prize in Mathematics.” UCLA News, August 22, 2006. Retrieved April 10, 2008, from http://newsroom.ucla.edu/portal/ucla/Terence-Tao-of-Math-7252.

27. Chang, Kenneth (2007). Journeys to the distant fields of prime. New York Times, March 13, 2007. Retrieved April 10, 2008 from http://www.nytimes.com/2007/03/13/science/13prof.html?_r=1&sq=The%20Mozart%20of%20MAth&st=nyt&oref=slogin&scp=1&pagewanted=print.

28. Winner, Ellen (1996). Gifted children: Myths and realities. New York: Basic Books, pp. 36–37.

29. Radford, John (1990). Child prodigies and exceptional achievers. New York: Harvester Wheatsheaf, p. 82.

30. Güstin, William C. (1985). The development of exceptional research mathematicians. In Benjamin S. Bloom (Ed.). Developing talent in young people. New York: Ballantine Books, p. 274.

31. Ibid., p. 279.

32. Winner (1996), p. 187.

33. Rathunde, Kevin, & Csikszentmihalyi, Mihaly. (1993). Undivided interest and the growth of talent: A longitudinal study of adolescents. Journal of Youth and Adolescence 22(4), 385–405.

34. Murray (2000), p. 49.

35. Paulson, Amanda. (2004). Children of immigrants shine in math, science, Santa Fe New Mexican 813 1/04, p. A5. These findings are further supported in a 2008 article in Notices of the American Mathematical Society.

36. Olson, Steve (2004). Count down. Boston: Houghton Mifflin, p. 63.

37. Heims, Steve J. (1982). John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death. Cambridge, Mass.: MIT Press, p. 56.

38. Wiener, Norbert (1953). Ex-prodigy: My childhood and youth. New York: Simon and Schuster, pp. 67–68.

39. Levinson, N. (1966). Wiener’s life, Bulletin of the American Mathematical Society 72(1, II), 3.

40. MacTutor, web site. George Dantzig, Birkhoff, p. 1.

41. Gustin (1985), pp. 279–282.

42. Feldman (1986), p. 12.

43. Howe, Michael J. A. (1990). The origins of exceptional abilities. Cambridge, Mass.: Basil Blackwell, p. 181.

44. Feldman (1986), p. 31.

45. Gustin (1985), p. 277.

46. Feldman (1986), p. 31.

47. Ibid., p. 33.

48. Ibid., p. 33.

49. Winner (1996), p. 19.

50. Ibid., p. 210.

51. Radford (1990), p. 96.

52. Gustin (1985), p. 278.

53. Ibid., p. 287.

54. Wigner (1992), p. 50.

55. Murray (2000), p. 67.

56. Ibid., p. 79.

57. Morrow, Charlene, & Perl, Teri (Eds.). (1998). Notable women in mathematics: A biographical dictionary. Westport, Conn.: Greenwood Press, pp. 190–191.

58. Ibid., p. 192.

59. Ibid., p. 193.

60. Murray (2000), p. 114.

61. Gallian, Joseph A. (2004). A conversation with Melanie Wood. Math Horizons 12 (September 2004), 123.

62. Olson (2004), p. 30.

63. Gallian (2004), p. 13.

64. Hersh, Reuben, & John-Steiner, Vera. (1993). A visit to Hungarian mathematics. Mathematical Intelligencer 15(2), 13–26.

65. Bloom (1985), pp. 308–309.

66. Notes on the theory and application of Fourier transforms, with R.E.A.C. Paley, Transactions of the American Mathematical Society 35.

67. Albers, Donald J., & Alexanderson, G. L. (1985). Mathematical people: Profiles and interviews. Boston: Birkhäuser, p. 287.

68. Ulam (1976), pp. 25–26.

69. Albers & Alexanderson (1985), p. 125.

70. O’Connor, J.J., & Robertson, E. F. (2003). George Dantzig, p. 2.

71. Hersh & John-Steiner (1993), p. 17.

72. Ibid., p. 18.

73. Yau, S. T. (Ed.) (1998). S. S. Chern: A great geometer of the twentieth century. Singapore, International Press.

74. Ibid.

75. Ibid.

76. Murray, Margaret (2000). Women becoming mathematicians, Cambridge, Mass.: Massachusetts Institute of Technology, p. 149.

77. Reid, C. (1976). Courant in Göttingen and New York. New York: Springer-Verlag, p. 255.

78. Hersh & John-Steiner (1993).

79. MacTutor, Birkhoff.

80. “Distant teachers” refers to significant teachers from the past whose work was influential to the novice without their ever meeting. This term comes from the work of Vera John-Steiner.

81. John-Steiner, Vera (1997). Notebooks of the mind: Explorations of thinking. New York: Oxford University Press, p. 54.

82. Gustin (1985), p. 326.

83. Ibid.

84. Ibid., p. 329.

CHAPTER 2. MATHEMATICAL CULTURE

1. Albers & Alexanderson (1985), p. 127.

2. Schwartz, L. (2001). A mathematician grappling with his century. Basel: Birkhäuser, p. 38.

3. Peter, R. (1990). Mathematics is beautiful. Mathematical Intelligencer 12(1), p. 58.

4. Boas, R. (1990). Interview. In D. J. Albers, G. L. Alexanderson, & C. Reid (Eds.). More mathematical people: Contemporary conversations. Boston: Birkhauser, p. 25.

5. Gardner, H. (1993). Creating minds. New York: Basic Books.

6. Everett, D. (2005). Cultural constraints on grammar and cognition in Piraha. Current Anthropology (Aug./Sept.), pp. 622–623.

7. Mandelbrot, B. (1985). Interview. In D. J. Albers & G. L. Alexanderson (Eds.), Mathematical people: Profiles and interviews. Boston: Birkhäuser, p. 209.

8. Ibid. p. 210.

9. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser, p. 391.

10. Hadamard, J. (1945). The psychology of invention in the mathematical field. New York: Dover, p. 140.

11. Ibid., p. 142.

12. Weil, A. (1992). The apprenticeship of a mathematician. Basel: Birkhäuser, p. 91.

13. Csikszentmihalyi, M. (1990). Flow: The psychology of optimal experience. New York: Harper Perennial, p. 53.

14. Parikh, C. (1991). The unreal life of Oscar Zariski. Boston: Academic Press, p. 50.

15. Ibid., p. 51.

16. Atiyah, M. (1984). Interview. Mathematical Intelligencer 6(1), 17.

17. Macrae, Norman. (1992). John von Neumann. New York: Random House, pp. 44-51.

18. Gregory, Graves, & Bangert, N. (2005). Math with heart: Why do mathematicians love math? Unpublished manuscript.

19. Chang (2007), p. D1.

20. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser, p. 310.

21. Ibid., p. 311.

22. Vygotsky, L. S. (1962). Thought and language. Cambridge, Mass.: MIT Press.

23. Rota, G. C. (1997). Indiscrete thoughts. Boston: Birkhäuser, pp. 45–46.

24. Bell, E. T. (1965). Men of mathematics. New York: Simon and Schuster, p. 378.

25. Ruelle, D. (2007). The mathematician’s brain. Princeton, N.J.: Princeton University Press, p. 129.

26. Ibid., p. 96.

27. Ibid.

28. Hardy, G. H. (1991). A mathematician’s apology. Cambridge: Cambridge University Press, p. 85.

29. Halmos, P. R. (1985). I want to be a mathematician. New York: MAA Spectrum, Springer-Verlag, p. 51.

30. Ibid., p. 73.

31. Mozzochi, C. J. (2000). The Fermat diary. Providence, R.I.: American Mathematical Society, pp. 64–65.

32. Singh (1998), p. 304.

33. da C. Andrade, E. N. (1954). Sir Isaac Newton: His life and work. Garden City, N.Y.: Anchor Books, p. 35.

34. Ibid.

35. Krantz, S. G. (2002). Mathematical apocrypha: Stories and anecdotes of mathematicians and the mathematical. Washington, D.C.: Mathematical Association of America, pp. 24–25.

36. Krantz, S. G. (2005). Mathematical apocrypha redux: More Stories and anecdotes of mathematicians and the mathematical. Washington, D.C.: Mathematical Association of America, p. 74.

37. Krantz (2002), p. 38.

38. Nasar, S., & Gruber, D. (2006). Manifold destiny. The New Yorker, August 28, 2006, p. 52.

39. Ibid., p. 45.

40. Collins, M. A., & Amabile, T. M. (1998). Creativity and motivation. In R. J. Sternberg (Ed.). Handbook of creativity. Cambridge: Cambridge University Press, p. 298.

41. Sternberg, J., & Lubart, T. I. (1991). The investment theory of creativity and its development. Human Development 34, 1–31.

42. Diacu F., & Holmes, P. (1996). Celestial encounters. Princeton, N.J.: Princeton University Press, p. 42.

43. Barrow-Green, J. (1997). Poincaré and the three body problem. Providence, R.I.: American Mathematical Society, p. 162.

44. Mira, C. (2000). I. Gumowski and a Toulouse research group in the “prehistoric” times of chaotic dynamics. In R. Abraham and Y. Ueda (Eds.). The chaos avant-garde: Memories of the early days of chaos theory. Singapore: World Scientific, p. 188.

45. Ueda, Y. (2000a). Strange attractors and the origins of chaos. In R. Abraham & Y. Ueda (Eds.). The chaos avant-garde: Memories of the early days of chaos theory. Singapore: World Scientific, p. 34.

46. Ueda, Y. (2000b). My encounter with chaos. In R. Abraham & Y. Ueda (Eds.). The chaos avant-garde: Memories of the early days of chaos theory. Singapore: World Scientific, pp. 48–49.

47. Ueda, Y. (2000c). Reflections on the origin of the broken-egg chaotic attractor. In R. Abraham & Y. Ueda (Eds.). The chaos avant-garde: Memories of the early days of chaos theory. Singapore: World Scientific, p. 65.

48. Abraham (2000), p. 89.

49. See Allyn Jackson’s article in Notices of the American Mathematical Society which appeared in 1994.

50. Jackson, A. (1994). Fighting for tenure: The Jenny Harrison case opens a Pandora’s box of issues about tenure, discrimination, and the law. Notices of the American Mathematical Society 41(3), p. 187.

51. John-Steiner, V. (2006). Harrison interview, December 4, 2006, Berkeley, Calif.

52. Harrison, J. (2007), web site.

53. Ibid.

54. Ibid.

55. Jackson (1994), p. 190.

56. Moore (2007). Mathematics at Berkeley: A history. Wellesley, Mass.: A. K. Peters, p. 288.

CHAPTER 3. MATHEMATICS AS SOLACE

1. Rota, G. C. (1990). “The lost café,” in Indiscrete Thoughts. Boston: Birkhäuser, p.

2. Bollobas, Béla (Ed.). (1986). Littlewood’s miscellany. Cambridge, Cambridge University Press.

3. “Rogers-Ramanujan identities” are identities between sums and products of rational functions, independently discovered by L. J. Rogers and Srinivasa Ramanujan.

4. Dyson, F. J. (1988). A walk through Ramanujan’s garden. In G. E. Andrews et al. Ramanujan revisited. Boston: Academic Press, p. 15.

5. Weil’s autobiography contains a myth about how the Finnish function theorist Rolf Nevanlinna saved his life. Nevanlinna claimed that when Weil was under arrest in Finland, Nevanlinna was at a state dinner also attended by the chief of police, and that when coffee was served the chief of police came to Nevanlinna saying: “Tomorrow we are executing a spy who claims to know you. Ordinarily I wouldn’t have troubled you with such trivia, but since we’re both here anyway I’m glad to have the opportunity to consult you.” “What is his name?” “Andre Weil.” Upon hearing this, Nevanlinna told Weil he was shocked. “I know him,” he told the police chief. “Is it really necessary to execute him?” “Well, what do you want us to do with him?” “Couldn’t you just escort him to the border and deport him?” “Well, there’s an idea; I hadn’t thought of it.” “Thus,” Weil wrote, “was my fate decided.” We now retract this tale, which we innocently repeated in a version of this chapter published in the Mathematical Intelligencer. After that publication we belatedly learned that Nevanlinna’s self-glorifying fairy tale was refuted in 1992 by the Finnish mathematician Osmo Pekonen. Pekonen read Weil’s dossier in the Finnish police archives and found that Weil was never condemned to death, and Nevanlinna was never involved in his case. (In an interview in the United States in 1934, Hermann Weyl called Nevanlinna a “Finnish Nazi.” In World War II Nevanlinna served as chairman of the Committee for the Finnish Volunteer Battalion of the Waffen-SS.) [Osmo Pekonen, L’affaire Weil à Helsinki en 1939. Gazette des mathématiciens 52 (April 1992), pp. 13–20.]

6. Weil (1992).

7. Massera (1998).

8. Each day, for some hours, each prisoner could have tools in his cell, that were delivered by an inmate, with the objective of making handicrafts. The sale of these, during the vexing and frustrating work toward release, helped with family sustenance.

9. The Isla (island) was a place of punishment where the prisoner stayed in a state of complete isolation; the only thing within the walls was his body. At night they delivered a blanket and quilt. And three times a day a soldier, with whom he was prohibited communication, delivered a meager food ration. It was the coldest, least hospitable place in the prison.

10. The entry of books written in languages besides Spanish was always prohibited. And in the epoch of which we write, every mathematics book was prohibited from entering: they might have codes that the censor couldn’t interpret.

11. Abramowitz, I. (1946). The great prisoners. New York: E. P. Dutton, p. 142.

12. Turán, P. (1997). Note of welcome. Journal of Graph Theory 1(1), 1.

CHAPTER 4. MATHEMATICS AS AN ADDICTION: FOLLOWING LOGIC TO THE END

1. Schneps, L. (2008). Grothendieck-Serre correspondence, book review. Mathematical Intelligencer 30(1), 66–68.

2. Roy Lisker translated the first 100 pages of Récoltes et Semailles (Reaping and Sowing) and published it in his magazine Ferment.

3. Ibid.

4. Ibid.

5. Ibid.

6. Schwartz (2001).

7. Ibid.

8. Ibid.

9. Grothendieck, A. (1986). Recoltes et Semailles, unpublished manuscript, promenade #9.

10. Grothendieck, A., Colmez, P. (Ed.), & Serre, J. P. (2001). Grothendieck-Serre correspondence. Paris: Societe Mathematique Francaise. Bilingual edition, Providence, R.I.: American Mathematical Society, 2004.

11. Grothendieck (1986), promenade #10.

12. Ibid., promenade #11.

13. Ibid., promenade #13.

14. Ibid., promenade, #7.

15. Ruelle, D. (2007). The mathematician’s brain. Princeton, N.J.: Princeton University Press, p. 40.

16. Ibid.

17. Grothendieck, A. (1989). Letter refusing the Crafoord Prize, Le Monde, May 4, 1998. Mathematical Intelligencer 11(1), 34–35.

18. Grothendieck (1986), promenade #17.

19. The unsettling story was reported in the Ukrainian Weekly on May 12, 1996, and August 31, 1997, and is available on the World Wide Web.

20. Baruk, H. (1978). Patients are people like us. New York: William Morrow, pp. 184–187.

21. Ibid.

22. Ibid.

CHAPTER 5. FRIENDSHIPS AND PARTNERSHIPS

1. Bollobás (1986), p. 8.

2. Koblitz, A. (1983). A convergence of lives. Boston: Birkhäuser, pp. 99–101.

3. Ibid., p. 101.

4. Ibid., p. 113.

5. James (2002), pp. 233–234.

6. Ibid.

7. Reid, C. (2004). Hilbert. New York: Springer, pp. 12–13.

8. Ibid., p. 14.

9. Ibid., p. 91.

10. Ibid., p. 95.

11. Ibid.

12. Ibid.

13. Hardy, G. H. (1967). A mathematician’s apology. New York: Cambridge University Press, pp. 147, 148.

14. Snow, C. P. (1967). Foreword to Hardy (1967). A mathematician’s apology. New York: Cambridge University Press, p. 29.

15. Bollobás (1986), p. 2.

16. Ibid., p. 11.

17. Tattersall, J., & McMurran, S. (2001). An interview with Dame Mary L. Cartwright, D. B. E., F. R. S. College Mathematics Journal 32(4), 249.

18. Bollobás (1986), p. 13.

19. URL in references.

20. Tattersall & McMurran (2001), p. 249.

21. Hardy, G. H. (1978). Ramanujan. New York: Cambridge University Press, pp. 2–3.

22. Kanigel, R. (1991). The man who knew infinity. New York: Simon and Schuster, p. 294.

23. Hardy (1967), p. 148.

24. Snow, C. P. (1967), p. 51.

25. Ibid.

26. Ibid., p. 57.

27. Aleksandrov, P. S. (2000). “A few words on A. N. Kolmogorov,” Russian Mathematical Surveys 39(4), 5–7, in Kolmogorov in perspective. Providence, R.I.: American Mathematical Society, p. 142.

28. Kolmogorov, A. N. (2000). “Memories of P. S. Aleksandrov” (Russian Mathematical Surveys 41, 255–246), in Kolmogorov in perspective. Providence, R.I.: American Mathematical Society, p. 145.

29. Ibid., p. 150.

30. Ibid., p. 156.

31. Ibid., p. 152.

32. Ibid., p. 159.

33. Gessen, Masha (2005). Perfect rigor. New York: Houghton Mifflin Harcourt, p. 43.

34. Ulam (1976), pp. 33–34.

35. Heims (1982), p. 42.

36. Abelson, P. (1965). Relation of group activity to creativity in science. Daedalus (summer 1965), p. 607.

37. Rota, G. C. (1987). The lost café, Los Alamos Science, Special Issue, 26, p. 26.

38. Schechter, B. (1998). My brain is open: The mathematical journeys of Paul Erdős, New York: Simon and Schuster, p. 196.

39. Goldstein, R. (2006). Incompleteness: The proof and paradox of Kurt Gödel (Great discoveries). New York: W. W. Norton, p. 29.

40. Ibid., p. 33.

41. Ibid.

42. Grattan-Guinness, Ivor. (1972). A mathematical union Annals of Science 29(2), 105–186.

43. Ibid., p. 115.

44. Ibid., p. 118.

45. Ibid., p. 121.

46. Ibid., p. 123.

47. Ibid., p. 131.

48. Ibid., pp. 131–132.

49. Ibid., p. 140.

50. Ibid., p. 141–142.

51. Ibid., p. 148.

52. Ibid., p. 149.

53. Ibid., p. 151.

54. Ibid., p. 156.

55. Ibid.

56. Ibid., p. 163.

57. Ibid., p. 166.

58. Ibid., p. 177.

59. Ibid., p. 181.

60. Reid (1996), p. 35.

61. Ibid., p. 38.

62. Ibid., p. 37.

63. Ibid., p. 43.

64. Ibid., p. 45.

65. Ibid., p. 79.

66. Ibid.

67. Ibid., 39, 79.

68. Tausky-Todd, O. (1985). Autobiographical Essay. In D. J. Albers and G. L. Alexanderson (Eds.). Mathematical people: Profiles and interviews. Boston: Birkhauser, p. 321.

69. Ibid., p. 325.

70. Birman, J. et al. (1991).

71. Helson, R. (2005). Personal communication.

72. Case, B. A. & Leggett, A. M. (Eds.) (2005). Complexities: Women in mathematics. Princeton, N.J.: Princeton University Press, p. 189.

73. Birman, J. et al. (1991). p. 702.

74. Case, B. A., et al., p. 190.

75. Gardner (1993), p. 385.

CHAPTER 6. MATHEMATICAL COMMUNITIES

1. Ulam (1976), p. 38.

2. Rowe, D. E. (1989). Klein, Hilbert, and the Göttingen mathematical tradition. In K. M. Oleska (Ed.). Science in Germany: The intersection of institutional and intellectual issues. Osiris 5, 189–213.

3. Gowers, T., & Nielson, M. (2009). Massively collaborative mathematics. Nature 461, 879–881.

4. Albers & Alexanderson (Eds.) (1985).

5. Beaulieu (1993). A Parisian café and ten proto-Bourbaki meetings (1934–1935). Mathematical Intelligencer 15(1), 27–35.

6. Dieudonné, J. A. (1970). The work of Nicholas Bourbaki. American Mathematical Monthly 77, 134–145.

7. Ibid.

8. Grothendieck (1986).

9. Cartan, H. M. (1980). Nicolas Bourbaki and contemporary mathematics. Mathematical Intelligencer 2(4), 175–187.

10. Klein, F. (1979). Development of mathematics in the 19th century. Translated by M. Ackerman. In R. Hermann (Ed.). Lie groups, history, frontiers and applications, vol. IX. Brookline, Mass.: Math Science Press.

11. Courant, R. (1980). Reminiscences from Hilbert’s Göttingen. Mathematical Intelligencer 3(3), 159.

12. Honda, K. (1975). Teiji Takagi: A biography. Commentary. Mathematica Universitatis Sancti Pauli XXIV-2, 141–167.

13. Alexandrov (2000).

14. Lewy, Hans (1992). Quoted by F. John.

15. John, F. (1992). Memories of student days in Göttingen. Miscellanea Mathematica, New York: Springer, pp. 213–220.

16. Courant, R. (1980). Reminiscences from Hilbert’s Göttingen. Mathematical Intelligencer 3(3), 163–164.

17. Courant’s eulogy for Friedrichs.

18. Moser, J. (1995). Obituary for Fritz John, 1910–1994. Notices of the American Mathematical Society 42(2), 256–257.

19. Lui, S. H. (1997). An interview with Vladimir Arnol’d. Notices of the American Mathematical Society 42(2), 432–438.

20. Sossinsky, A. B. (1993). In the other direction. In S. Zdravkovska & P. L. Duren (Eds.). Golden years of Moscow mathematics, History of mathematics, vol. 6. Providence, R.I.: American Mathematical Society, pp. 223–243.

21. Ibid.

22. Fuchs, D. B. (1993). On Soviet mathematics of the 1950s and 1960s. In S. Zdravkovska & P. L. Duren (Eds.). Golden years of Moscow mathematics, History of mathematics, vol. 6. Providence, R.I.: American Mathematical Society, pp. 220–222.

23. Ibid.

24. Sossinsky (1993), pp. 223–243.

25. Ibid.

26. Ibid.

27. Ibid.

28. Zelevinsky, A. (2005). Remembering Bella Abramovna. In M. Shifman (Ed.). You failed your math test, Comrade Einstein. Hackensack, N.J.: World Scientific.

29. Henrion, C. (1997). Women in mathematics. Bloomington, Ind.: Indiana University Press, p. 152.

30. Notices of the American Mathematical Society (1997), p. 107.

31. Lewis, D. J. (1991). Mathematics and women: The undergraduate school and pipeline. Notices of the American Mathematical Society 38(7), 721–723.

32. Roitman, J. (2005). In B. A. Case and A. M. Leggett (Eds.). Complexities: Women in Mathematics. Princeton, N.J.: Princeton University Press, p. 251.

CHAPTER 7. GENDER AND AGE IN MATHEMATICS

1. Marjorie Senechal (2007). Hardy as mentor. Mathematical Intelligencer 29(1), 16–23.

2. Germain-Gauss correspondence <http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Gauss.html>

3. LaGrange <http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Lagrange.html>

4. James (2002), pp. 57–58.

5. Kovalevskaya, S., Kochina, P. Y., & Stillman, B. (1978). A Russian childhood. New York: Springer, p. 35.

6. Crelle’s Journal <http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Crelle.html>

7. Weierstrass <http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Weierstrass.html>

8. Ibid.

9. Kovalevskaya et al. (1978), p. 35.

10. James (2002), p. 235.

11. Smolin, L. (2006). The trouble with physics. London: Houghton Mifflin Penguin.

12. Reid, C. (1986). Hilbert-Courant. New York: Springer-Verlag, p. 143.

13. Weyl, H. (1935). Emmy Noether. Scripta Mathematica 3(3), 201–220.

14. James, I. (2009). Driven to innovate: A century of Jewish mathematicians and physicists. Oxford: Peter Lang.

15. Notices of the American Mathematical Society of America, 2005.

16. Henrion (1997), p. xvii.

17. Ibid., p. 66.

18. Ibid., p. 18.

19. Ibid., p. 44.

20. Ibid., p. 73.

21. Ibid., p. 134.

22. Albers, D., & Alexanderson, G. (1991). A conversation with Ivan Niven. College Mathematics Journal 22(5), 371–402.

23. Ibid., p. 393.

24. Ibid.

25. Hyde, J. (2005). The gender similarities hypothesis. American Psychologist 60, 581–592.

26. Henrion (1997), p. 208.

27. Ibid., p. 134.

28. Bollobás (1986), pp. 15–16.

29. Personal communication.

30. Mordell’s name is usually mentioned today in connection with his conjecture of 1922, finally proved by Gerd Faltings of Germany in 1983: A smooth rational plane curve of genus greater than 1 has finitely many points with rational coefficients.

31. Mordell, L. J. (1970). Hardy’s A mathematician’s apology. American Mathematical Monthy 77, 836.

32. Ibid.

33. Ibid.

34. Ibid.

35. Einstein (1942), p. 150.

36. Weil, A. (1950). The future of mathematics. American Mathematical Monthly 57, 296.

37. Wiegand, S. (1977). Grace Chisholm Young. Association for Women in Mathematics Newsletter 7, 6.

38. Adler, A. (1972). Mathematics and creativity. New Yorker, February 19, 1972. Reprinted in Timothy Ferris (Ed.) (1993). The world treasury of physics, astronomy, and mathematics. Back Bay Books, p. 435.

39. van Stigt, W. P. (1990). Brouwer’s intuitionism. Amsterdam: North-Holland.

40. Dauben, J. (1995). Abraham Robinson: The creation of nonstandard analysis: A personal and mathematical odyssey. Princeton, N.J.: Princeton University Press, p. 491.

41. Schneider, Ivo, e-mail communication.

42. Bell (1937), p. 405.

43. An earlier version of this survey appeared as Hersh, R. (2001). Mathematical menopause, or, a young man’s game? Mathematical Intelligencer 23(3), 52–60.

44. Taylor, S. S. (1999). Research dialogues of the TIM-CREF, no. 62.

45. Henrion, p. 113.

46. Henrion, C. (1997).

47. Henrion (1997), p. 112.

48. Ibid.

49. Bollobás (1986), p. 14.

50. Personal communication.

51. Bollobás (1986), p. 14.

CHAPTER 8. THE TEACHING OF MATHEMATICS: FIERCE OR FRIENDLY

1. Albert Einstein. Quoted in Holton, G. (1973). Thematic origins of scientific thought: Kepler to Einstein. Cambridge, Mass.: Harvard University Press, p. 377.

2. Parker, J. (2004). R. L. Moore: Mathematician and teacher. Washington, D.C.: Mathematical Association of America, p. 3.

3. Megginson, R. E. (2003). Yueh-Gin Gung and Dr. Charles Y. Hu award to Clarence F. Stephens for distinguished service to mathematics. American Mathematical Monthly 110(3), 177–180.

4. Mathematicians of the African Diaspora web site.

5. Megginson (2003), p. 177.

6. Donaldson (1989), p. 450.

7. Dictionary of American Biography, p. 163.

8. Parker (2004), pp. vii, viii.

9. Moore specialized in what became known as “Moore spaces.” These are topological spaces satisfying a technical additional condition that was specified in part 4 of Moore’s Axiom 1. [Wilder, R. L. (1982). The mathematical work of R. L. Moore: Its background, nature and influence. Archive for the History of Exact Sciences 26, 73–97]. A “complete Moore space” satisfied all four parts of Axiom 1. According to Wilder, over 300 papers had then been published on the question, When is a Moore space metrizable? In 1951 Moore’s student R H Bing proved that a Moore space is metrizable if it is “collectionwise normal.”

10. Parker (2004), p. 244.

11. Albers, D. J. (1990). More mathematical people. Boston: Harcourt Brace Jovanovich, p. 293.

12. Parker (2004), p. 271.

13. Ibid.

14. Ibid., p. 226.

15. Mathematicians of the African Diaspora web Site.

16. Ibid.

17. Parker (2004), p. 288.

18. Personal communication.

19. Ibid.

20. Parker (2004), p. 288.

21. Personal communication (2007).

22. Ibid.

23. Claytor gave a necessary and sufficient condition for a Peano continuum to be homeomorphic to a subset of the surface of a sphere, improving on earlier results by the famous Polish topologist Casimir Kuratowski.

24. Ibid.

25. Ibid.

26. Albers & Alexanderson (1985), p. 23.

27. Megginson (2003), p. 179.

28. Poland, J. (1987). A modern fairy tale? American Mathematical Monthly 94(3), 293.

29. Ibid.

30. Datta, D. K. (1993). Math education at its best: The Potsdam model. Framingham, Mass.: Center for Teaching/Learning of Mathematics, p. 5.

31. Ibid., pp. 65–66.

32. Ibid., p. 9.

33. Ibid.

34. Ibid., p. 23.

35. Personal communication (2006).

CHAPTER 9. LOVING AND HATING SCHOOL MATHEMATICS

1. Cornell, C. (1999). I hate math! I couldn’t learn it, and I can’t teach it! Childhood Education 75(4), p. 1.

2. Tobias, Sheila. (1993). Overcoming math anxiety. New York: W. W. Norton.

3. Lester, W. (2005). Hate mathematics? You are not alone. Associated Press, August 16, 2005.

4. Hardy, G. H. (1948). A mathematician’s apology. New York: Cambridge University Press, pp. 86–87.

5. Slocum, J., & Sonneveld, D. (2006). The 15 puzzle. Beverly Hills, Calif., Slocum Puzzle Foundation, p. 9.

6. Personal communication (2006).

7. Cohen, R. (2006). What is the value of algebra? Washington Post, February 16, 2006.

8. McCarthy, C. (1991). Who needs algebra? Washington Post, April 20, 1991.

9. Ibid.

10. Russell, B. (1957). “The study of mathematics,” in Mysticism and logic. New York: Doubleday, p. 60.

11. Ibid., p. 208.

12. Ibid., p. 68.

13. Zaslavsky, C. (1996), The multicultural classroom. Portsmouth, N.H.: Heinemann, p. 60.

14. Charbonneau M., & John-Steiner, V. (1988). Patterns of experience and the language of mathematics. In R. Cocking & J. P. Mestre (Eds.), Linguistic and cultural influences on learning mathematics. Hillsdale, N.J.: Erlbaum, p. 94.

15. Gerdes, P. (2001). On culture, geometrical thinking and mathematics education. In A. B. Powell & M. Frankenstein (Eds.). Ethnomathematics, challenging Eurocentrism in mathematics education. Albany, New York: State University of New York Press, pp. 231–232.

16. Carraher, T. N., Carraher, D., & Schliemann, A. D. (1985). Mathematics in the streets and in the schools. British Journal of Developmental Psychology 3, 21–29.

17. Schliemann (1995), p. 50.

18. Ibid.

19. PME 19, vol. 1, p. 20.

20. (1988), p. 44.

21. Ibid., p. 165.

22. Semisimple Lie groups are, . . .

23. Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work. Portsmouth, N.H.: Heinemann, p. 124.

24. Schmittau, Jean. (2003). Cultural historical theory in mathematics education. In Kozulin, A. Gindis, B. Ageyev V. S., Miller. S. N. (Eds.) Vygotsky’s Educational Theory in Cultural Context. New York: Cambridge University Press.

25. Devlin, Keith. (2009). MAA On-line, January 2009. Should children learn math by starting with counting? Devlin’s angle: http://www.maa.org/devlin/devlin_01_09.html

26. Umland, K. (2006). Personal communication.

27. Moses, R. P., & Cobb, C. E. Jr. (2001). Radical equations: Math literacy and civil rights. Boston: Beacon Press, pp. 10–11.

28. Ibid., p. 16.

29. Ibid., p. 146.

30. Ibid., p. 177.

31. Ibid., p. 179.

32. Ibid., p. 183.

33. Ibid., p. 18.

34. Treisman, V. (1992). Studying students studying calculus: A look at the lives of student mathematicians. College Mathematics Journal 23, p. 363.

35. Rota, G. C. (1997). Indiscrete thoughts. Boston: Birkhäuser, p. 39.

36. Levin, T. (2006). As math scores lag, a new push for basics. New York Times, November 14, 2006.

37. Ibid., p. 19.

38. Ibid.

39. Pearson, R. S. (1991). Why don’t engineers use undergraduate mathematics in their professional work? UME Trends 3, 8.

40. Dudley, U. (1997). Is mathematics necessary? College Mathematics Journal 28(5), 361–365.

41. Krantz (2002), p. 61.

42. Halmos (1985), p. 261.

43. Noddings, N. (1993). Excellence as a guide to educational conversation. Teachers College Record 94(4), 8, 9.

44. Ibid., p. 13.

45. Noddings, N. (1994). Does everybody count? Journal of Mathematical Behavior 13(1), 10.

46. Ibid.

47. (1999), p. 32.

48. Umland, K. (2006). Personal communication.

CONCLUSIONS

1. Halmos (1985), p. 3.

2. Henrion (1997), p. 228.

3. Albers, D. J., Alexanderson, G. L., & Reid, C. (1990). More mathematical people. Boston: Harcourt Brace Jovanovich, pp. 3–26.

4. Ibid., p. 14.

5. Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction and paradox to create mathematics. Princeton, N.J.: Princeton University Press, p. 78.