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On Non-literal Approaches

  • Ulianov Montano
Chapter
Part of the Synthese Library book series (SYLI, volume 370)

Abstract

This chapter discusses in detail the idea that mathematical beauty should be reinterpreted to preserve mathematics’ stand among the sciences. Three reasons to reinterpret mathematical beauty are examined; the two cultures split, the epistemic character of mathematics, and its rational character. It shall be argued that the reasons for endorsing a non literal interpretation of mathematical beauty are rather weak. The discussion also examines the conceptions of mathematical beauty by Shaftesbury, Hutchenson and Gian-Carlo Rota.

Keywords

Scientific Revolution Mathematical Entity Aesthetic Judgement Epistemic Goal Aesthetic Evaluation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. 1.
    Aaboe, A. (1963). Episodes from the early history of mathematics. New York: MAA.Google Scholar
  2. 2.
    Aigner, M., & Ziegler, G. M. (2004). Proofs from the book. Berlin/New York, Springer.CrossRefGoogle Scholar
  3. 3.
    Alperson, P., & Kivy, P. (2004). The philosophy of music: Formalism and beyond. In P. Kivy (Ed.), The Blackwell guide to aesthetics (pp. 254–275). Malden: BlackwellGoogle Scholar
  4. 4.
    Appel, K., & Haken, W. (1976). Every planar map is four colorable. Bulletin of the American Mathematical Society, 82(5), 711–713.CrossRefGoogle Scholar
  5. 5.
    Appel, K., & Haken, W. (1977). Every planar map is four colorable. Part I: Discharging. Illinois Journal of Mathematics, 21(3), 429–490.Google Scholar
  6. 6.
    Appel, K., & Haken, W. (1986). The four color proof suffices. The Mathematical Intelligencer, 8(1), 10–20.CrossRefGoogle Scholar
  7. 7.
    Appel, K., Haken, W., & Koch, J. (1977). Every planar map is four colorable. Part II: Reducibility. Illinois Journal of Mathematics, 21(3), 491–567.Google Scholar
  8. 8.
    Ax, A. F. (1953). The physiological differentiation between fear and anger in humans. Psychosomatic Medicine, 15(5), 433–442.Google Scholar
  9. 9.
    Balaguer, M. (2001). Platonism and anti-platonism in mathematics. New York: Oxford University Press.Google Scholar
  10. 10.
    Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.CrossRefGoogle Scholar
  11. 11.
    Brady, E., & Levinson, J. (2001). Aesthetic concepts: Essays after Sibley. New York: Oxford University Press.Google Scholar
  12. 12.
    Budd, M. (1992). Music and the emotions: The philosophical theories. Psychology Press. London: Boston.Google Scholar
  13. 13.
    Cohen, T. (1973). Aesthetic/non-aesthetic and the concept of taste: A critique of Sibley’s position. Theoria, 39(1–3), 113–152.Google Scholar
  14. 14.
    Colyvan, M. (2003). The indispensability of mathematics. New York: Oxford University Press.Google Scholar
  15. 15.
    Cooper, A. A. (1964). Third earl of shaftesbury. Characteristics of Men, Manners, Opinions, Times, 1711, 397–400.Google Scholar
  16. 16.
    Damasio, A. (2005). Descartes’ error: Emotion, reason, and the human brain. London/New York: Penguin (Non-Classics).Google Scholar
  17. 17.
    De Clercq, R. (2005). Aesthetic terms, metaphor, and the nature of aesthetic properties. The Journal of Aesthetics and Art Criticism, 63(1), 27–32.CrossRefGoogle Scholar
  18. 18.
    de Villiers, M. (2004). The role and function of quasi-empirical methods in mathematics. Canadian Journal of Math, Science & Technology Education, 4(3), 397–418.CrossRefGoogle Scholar
  19. 19.
    Dirac, P. (1980). Why we believe in the Einstein theory? In B. Gruber & R. S. Millman (Eds.), Symmetries in science. New York: Plenum.Google Scholar
  20. 20.
    Dreyfus, H. L., & Wrathall, M. A. (2006). A companion to phenomenology and existentialism (Vol. 35). Malden: Wiley-Blackwell.Google Scholar
  21. 21.
    Ekman, P., & Davidson, R. J. (Eds.). (1994). The nature of emotion: Fundamental questions (1st ed.). New York: Oxford University Press.Google Scholar
  22. 22.
    Feigl, H. (1970). Beyond peaceful coexistence. In R. H. Stuewer (Ed.), Historical and philosophical perspectives of science (Number 5 in Minnesota studies in the philosophy of science, pp. 3–11). Minneapolis: University of Minnesota Press.Google Scholar
  23. 23.
    Feynman, R. P., & Weinberg, S. (1999). Elementary particles and the laws of physics: The 1986 Dirac memorial lectures. New York/Cambridge: Cambridge University Press.Google Scholar
  24. 24.
    Feynman, R. P., Leighton, R. B., & Sands, M. (2011). The Feynman lectures on physics, Vol. I: The new millennium edition: mainly mechanics, radiation, and heat. New York: Basic Books.Google Scholar
  25. 25.
    Field, H. H. (1980). Science without numbers: A defence of nominalism. Princeton: Princeton University Press.Google Scholar
  26. 26.
    Franks, J. (2010). Cantor’s other proofs that R is uncountable. Mathematics Magazine, 83(4), 283–289.CrossRefGoogle Scholar
  27. 27.
    Frege, G. (1980). Frege against the formalists. In M. Black & P. Geach (Eds.), Philosophical writings: Translations (3rd ed., pp. 162–213; 2nd ed., 1970, 1969, 1966, 1960; 1st ed. 1952). Oxford: Blackwell.Google Scholar
  28. 28.
    Frijda, N. H. (1986). The emotions. Cambridge/New York: Cambridge University Press.Google Scholar
  29. 29.
    Goldman, A. (1995). Aesthetic value (Focus series). Boulder: Westview Press.Google Scholar
  30. 30.
    Goodman, N. (1968). Languages of art: An approach to a theory of symbols. Indianapolis: The Bobbs-Merrill Company.Google Scholar
  31. 31.
    Gow, J. (2010). A short history of Greek mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  32. 32.
    Guyer, P. (2004). The origins of modern aesthetics: 1711–35. In P. Kivy (Ed.), The Blackwell guide to aesthetics (pp. 15–44). Malden: Blackwell.Google Scholar
  33. 33.
    Hardy, G. H. (1992). A mathematician’s apology. Cambridge/New York: Cambridge University Press.CrossRefGoogle Scholar
  34. 34.
    Hillman, D. J. (1962). The measurement of simplicity. Philosophy of Science, 29, 225–252.CrossRefGoogle Scholar
  35. 35.
    Hungerland, I. C. (1962). The logic of aesthetic concepts. Proceedings and Addresses of the American Philosophical Association, 36, 43–66.CrossRefGoogle Scholar
  36. 36.
    Hutcheson, F. (1973). Francis Hutchenson: An inquiry concerning beauty, order, harmony, design (Vol. 9). The Hague: Martinus Nijhoff Publishing.Google Scholar
  37. 37.
    Kaku, M., & Thompson, J. T. (1997). Beyond Einstein: The cosmic quest for the theory of the universe. Oxford: Oxford University Press.Google Scholar
  38. 38.
    Kivy, P. (1975). What makes “Aesthetic” terms aesthetic? Philosophy and Phenomenological Research, 36(2), 197–211.CrossRefGoogle Scholar
  39. 39.
    Kivy, P. (1980). The corded shell: Reflections on musical expression. Princeton: Princeton University Press.Google Scholar
  40. 40.
    Kivy, P. (1989). Sound sentiment: An essay on the musical emotions, including the complete text of the corded shell. Philadelphia: Temple University Press.Google Scholar
  41. 41.
    Kivy, P. (1991). Music alone: Philosophical reflections on the purely musical experience. Ithaca: Cornell University Press.Google Scholar
  42. 42.
    Kivy, P. (1993). The fine art of repetition: Essays in the philosophy of music. Cambridge/New York: Cambridge University Press.Google Scholar
  43. 43.
    Kivy, P. (1997). Philosophies of arts: An essay in differences. Cambridge/New York: Cambridge University Press.Google Scholar
  44. 44.
    Kivy, P. (2002). Introduction to a philosophy of music. Oxford: Oxford University Press.Google Scholar
  45. 45.
    Kline, M. (1990). Mathematical thought from ancient to modern times, Vol. 1 (1st ptg. ed.). New York: Oxford University Press.Google Scholar
  46. 46.
    Kline, M. (1990). Mathematical thought from ancient to modern times, Vol. 2. New York: Oxford University Press.Google Scholar
  47. 47.
    Kline, M. (1990). Mathematical thought from ancient to modern times, Vol. 3. New York: Oxford University Press.Google Scholar
  48. 48.
    Kuipers, T. A. F. (2000). From instrumentalism to constructive realism: On some relations between confirmation, empirical progress, and truth approximation. Dordrecht: Springer.CrossRefGoogle Scholar
  49. 49.
    Kuipers, T. A. F. (2002). Beauty, a road to the truth. Synthese, 131(3), 291–328.CrossRefGoogle Scholar
  50. 50.
    Lakatos, I., Worrall, J., & Zahar, E. (Eds.). (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge/New York: Cambridge University Press.Google Scholar
  51. 51.
    Langer, S. K. K. (1957). Philosophy in a new key: A study in the symbolism of reason, rite, and art (Vol. 17). Cambridge: Harvard University Press.Google Scholar
  52. 52.
    Laudan, L. (1978). Progress and its problems: Towards a theory of scientific growth. Berkeley: University of California Press.Google Scholar
  53. 53.
    Lazarus, R. S. (1991). Emotion and adaptation. New York: Springer.Google Scholar
  54. 54.
    LeDoux, J. (1996). Emotional networks and motor control: A fearful view. Progress in Brain Research, 107, 437–446.Google Scholar
  55. 55.
    LeDoux, J. E. (1998). The emotional brain: The mysterious underpinnings of emotional life. New York: Simon & Schuster.Google Scholar
  56. 56.
    LeDoux, J., & Bemporad, J. R. (1997). The emotional brain. Journal of the American Academy of Psychoanalysis, 25(3), 525–528.Google Scholar
  57. 57.
    Le Lionnais, F. (1971). Les grands courants de la pensée mathématique. Marseille: Cahiers du Sud.Google Scholar
  58. 58.
    Le Lionnais, F. (2004). Great currents of mathematical thought: Mathematics: Concepts and development (Vol. 1). Mineola: Dover.Google Scholar
  59. 59.
    Levenson, R. W. (1994). The search for autonomic specificity. In P. Ekman & R. J. Davidson (Eds.), The nature of emotion: Fundamental questions (pp. 252–257). New York: Oxford University Press.Google Scholar
  60. 60.
    Livio, M., & Grunwald, E. (2006). The equation that couldn’t be solved: How mathematical genius discovered the language of symmetry. The Mathematical Intelligencer, 28(4), 63–64.CrossRefGoogle Scholar
  61. 61.
    Marenbon, J. (2003). Boethius. Oxford/New York: Oxford University Press.CrossRefGoogle Scholar
  62. 62.
    McAllister, J. (1996). Beauty and revolution in science. Ithaca: Cornell University Press.Google Scholar
  63. 63.
    McAllister, J. (1998). Is beauty a sign of truth in scientific theories? American Scientist, 86(2), 174–183.Google Scholar
  64. 64.
    McAllister, J. W. (2005). Mathematical beauty and the evolution of the standards of mathematical proof. In M. Emmer (Ed.), The visual mind II (Vol. 2). Cambridge: MIT.Google Scholar
  65. 65.
    McWeeny, R. (2002). Symmetry: An introduction to group theory and its applications. Mineola: Dover.Google Scholar
  66. 66.
    Montano, U. (2010). Beauty in mathematics. PhD thesis, [s.n.], S.l. University of Groningen, The Netherlands.Google Scholar
  67. 67.
    Montano, U. (2012). Ugly mathematics: Why do mathematicians dislike computer-assisted proofs? The Mathematical Intelligencer, 34(4), 21–28.CrossRefGoogle Scholar
  68. 68.
    Nahin, P. J. (1998). An imaginary tale: The Story of i [the square root of minus one]. Princeton: Princeton University Press.Google Scholar
  69. 69.
    Nahin, P. J. (2006). Dr. Euler’s fabulous formula: Cures many mathematical ills. Princeton: Princeton University Press.Google Scholar
  70. 70.
    Nahin, P. J. (2011). Dr. Euler’s fabulous formula: Cures many mathematical ills. Princeton: Princeton University Press.Google Scholar
  71. 71.
    Newton, I. (2003). Newton’s philosophy of nature selections from his writings. Whitefish, Mont. Kessinger Publishing.Google Scholar
  72. 72.
    Newton-Smith, W. (1981). The rationality of science. Boston: Routledge.CrossRefGoogle Scholar
  73. 73.
    Ortony, A. (1991). Value and emotion. In G. Mandler, W. Kessen, A. Ortony, & F. I. M. Craik (Eds.), Memories, thoughts, and emotions: Essays in the honor of George Mandler (pp. 337–353). Hillsdale: Erlbaum.Google Scholar
  74. 74.
    Ortony, A., Clore, G. L., & Collins, A. (1990). The cognitive structure of emotions. Cambridge/New York: Cambridge University Press.Google Scholar
  75. 75.
    Popper, K. R. (2002). The logic of scientific discovery. London/New York: Routledge.Google Scholar
  76. 76.
    Robinson, J. (2005). Deeper than reason: Emotion and its role in literature, music, and art. New York: Oxford University Press.CrossRefGoogle Scholar
  77. 77.
    Robinson, J. (2005). Deeper than reason: Emotion and its role in music literature and art. London: Claredon.CrossRefGoogle Scholar
  78. 78.
    Rota, G. (1997). The phenomenology of mathematical proof. Synthese, 111(2), 183–196.CrossRefGoogle Scholar
  79. 79.
    Rota, G.-C. (1997). The phenomenology of mathematical beauty. Synthese, 111(2), 171–182.CrossRefGoogle Scholar
  80. 80.
    Rota, G. C. (2005). The phenomenology of mathematical beauty. In M. Emmer (Ed.), The visual mind II (pp. 2–13). Cambridge: MIT.Google Scholar
  81. 81.
    Russell, B. (2004). Mysticism and logic. Mineola: Dover.Google Scholar
  82. 82.
    Sibley, F. (1959). Aesthetic concepts. The Philosophical Review, 68(4), 421–450.CrossRefGoogle Scholar
  83. 83.
    Snow, C. P. (2012). The two cultures. Cambridge/New York: Cambridge University Press.Google Scholar
  84. 84.
    Solomon, R. C. (2005). Subjectivity. In T. Honderich (Ed.), The Oxford companion to philosophy new edition (2nd ed.). Oxford/New York: Oxford University Press.Google Scholar
  85. 85.
    Stuewer, R. H. (Ed.). (1970). Historical and philosophical perspectives of science (Volume 5 of Minnesota studies in the philosophy of science). Minneapolis: University of Minnesota Press.Google Scholar
  86. 86.
    Swinnerton-Dyer, P. (2005). The justification of mathematical statements. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 363(1835), 2437–2447.CrossRefGoogle Scholar
  87. 87.
    Thagard, P. (2005). Why is beauty a road to the truth? Poznan Studies in the Philosophy of the Sciences and the Humanities, 84(1), 365–370.Google Scholar
  88. 88.
    Thomas, R. S. D. (2007). The comparison of mathematics with narrative. In B. Kerkhove & J. P. Bendegem (Eds.), Perspectives on mathematical practices (Volume 5 of Logic, epistemology, and the unity of science, pp. 43–59). Dordrecht: Springer.Google Scholar
  89. 89.
    Tymoczko, T. (1979). The four-color problem and its philosophical significance. The Journal of Philosophy, 76(2), 57–83.CrossRefGoogle Scholar
  90. 90.
    Weber, K. (2009). How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. The Journal of Mathematical Behavior, 28(2), 200–208.CrossRefGoogle Scholar
  91. 91.
    Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2), 209–234.CrossRefGoogle Scholar
  92. 92.
    Weinberg, S. (1994). Dreams of a final theory. New York: Vintage Books.Google Scholar
  93. 93.
    Wells, D. (1988). Which is the most beautiful? The Mathematical Intelligencer, 10(4), 30–31.CrossRefGoogle Scholar
  94. 94.
    Wells, D. (1990). Are these the most beautiful? The Mathematical Intelligencer, 12(3), 37–41.CrossRefGoogle Scholar
  95. 95.
    Wright, C. (1994). Truth and objectivity. Cambridge/London: Harvard University Press.Google Scholar
  96. 96.
    Zajonc, R. B. (1984). On the primacy of affect. American Psychologist, 39(2), 117–123.CrossRefGoogle Scholar
  97. 97.
    Zajonc, R. B. (1998). Emotions. In D. T. Gilbert, S. T. Fiske, & G. Lindzey (Eds.), The handbook of social psychology (Vol. 2, pp. 591–632). Boston: McGraw-HillGoogle Scholar
  98. 98.
    Zajonc, R. B. (2001). Mere exposure: A gateway to the subliminal. Current Directions in Psychological Science, 10(6), 224–228.CrossRefGoogle Scholar
  99. 99.
    Zangwill, N. (2003). Beauty. In J. Levinson (Ed.), The Oxford handbook of aesthetics. Oxford/New York: Oxford University Press.Google Scholar
  100. 100.
    Zangwill, N. (2010). Aesthetic judgment. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2010 ed.). http://plato.stanford.edu/archives/fall2010/entries/aesthetic-judgment/.
  101. 101.
    Zangwill, N. (2011). Music, essential metaphor, and private language. American Philosophical Quarterly, 48(1), 1.Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ulianov Montano
    • 1
  1. 1.Mexico CityMexico

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