Abstract
This chapter addresses aesthetic experience. Aesthetic experience is interpreted as a subprocess that involves changes in the focus and content of our attention and the eliciting of affective responses. A conception of the characteristic intentional object involved in aesthetic experiences of mathematics is introduced.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In this way we exclude concrete indispensable items, like brains or mathematicians themselves.
- 2.
Rota and McAllister name several types of mathematical entities that are often qualified as beautiful, numbers, theorems, proofs, theories, and so forth. The above definition is adequate to cover those entities and some others not mentioned by them, such as derivations, or axiomatizations.
- 3.
Although the notions of space and dimension I utilize here resemble the ordinary concepts of physical space and dimension, they are rather closer to the formal notions of space and dimension. Unfortunately, a more formal treatment of these notions is beyond the scope of this book.
- 4.
The angle notation, \(\angle \), is very popular in fields like engineering. It is related to the polar form of complex numbers, the expression before the angle symbol represents its modulus and the expression after is the argument. This notation simplifies the visualization of operations: multiplication consists in multiplication of modulus and addition of arguments, exponentiation consists in exponentiation of modulus and multiplication of arguments.
- 5.
We shall see below that calculations, derivations or proofs belong to a different class of experience than formulas or theorems, since they are more “performative”. Furthermore, this example involves not only active attention but also the fact that the person’s history of experiences enables him to see some properties; it thus fits better in a third class comprising evaluations formed by a person’s history of encounters with different mathematical items.
- 6.
It is not unlikely that further types of experience can be identified, but the three discussed here are very relevant for discussing the dynamics of aesthetic value, and the nature of aesthetic terms later on. Discussing further types of experience is a task better suited for future follow-up works.
- 7.
Eulers’ formula was proved in 1714 by Roger Cotes, and published in its current form by Euler in 1748. Wessel introduced his interpretation in 1799 in the Royal Danish Academy of Sciences and Letters but it remained obscure for some time.
- 8.
Although the notation f is usually employed to refer to proper functions, I shall retain it instead of r, for example, in order to avoid confusion with other occurrences of ‘r’ in the discussion.
Bibliography
Aaboe, A. (1963). Episodes from the early history of mathematics. New York: MAA.
Aigner, M., & Ziegler, G. M. (2004). Proofs from the book. Berlin/New York, Springer.
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: Blackwell
Appel, K., & Haken, W. (1976). Every planar map is four colorable. Bulletin of the American Mathematical Society, 82(5), 711–713.
Appel, K., & Haken, W. (1977). Every planar map is four colorable. Part I: Discharging. Illinois Journal of Mathematics, 21(3), 429–490.
Appel, K., & Haken, W. (1986). The four color proof suffices. The Mathematical Intelligencer, 8(1), 10–20.
Appel, K., Haken, W., & Koch, J. (1977). Every planar map is four colorable. Part II: Reducibility. Illinois Journal of Mathematics, 21(3), 491–567.
Ax, A. F. (1953). The physiological differentiation between fear and anger in humans. Psychosomatic Medicine, 15(5), 433–442.
Balaguer, M. (2001). Platonism and anti-platonism in mathematics. New York: Oxford University Press.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.
Brady, E., & Levinson, J. (2001). Aesthetic concepts: Essays after Sibley. New York: Oxford University Press.
Budd, M. (1992). Music and the emotions: The philosophical theories. Psychology Press. London: Boston.
Cohen, T. (1973). Aesthetic/non-aesthetic and the concept of taste: A critique of Sibley’s position. Theoria, 39(1–3), 113–152.
Colyvan, M. (2003). The indispensability of mathematics. New York: Oxford University Press.
Cooper, A. A. (1964). Third earl of shaftesbury. Characteristics of Men, Manners, Opinions, Times, 1711, 397–400.
Damasio, A. (2005). Descartes’ error: Emotion, reason, and the human brain. London/New York: Penguin (Non-Classics).
De Clercq, R. (2005). Aesthetic terms, metaphor, and the nature of aesthetic properties. The Journal of Aesthetics and Art Criticism, 63(1), 27–32.
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.
Dirac, P. (1980). Why we believe in the Einstein theory? In B. Gruber & R. S. Millman (Eds.), Symmetries in science. New York: Plenum.
Dreyfus, H. L., & Wrathall, M. A. (2006). A companion to phenomenology and existentialism (Vol. 35). Malden: Wiley-Blackwell.
Ekman, P., & Davidson, R. J. (Eds.). (1994). The nature of emotion: Fundamental questions (1st ed.). New York: Oxford University Press.
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.
Feynman, R. P., & Weinberg, S. (1999). Elementary particles and the laws of physics: The 1986 Dirac memorial lectures. New York/Cambridge: Cambridge University Press.
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.
Field, H. H. (1980). Science without numbers: A defence of nominalism. Princeton: Princeton University Press.
Franks, J. (2010). Cantor’s other proofs that R is uncountable. Mathematics Magazine, 83(4), 283–289.
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.
Frijda, N. H. (1986). The emotions. Cambridge/New York: Cambridge University Press.
Goldman, A. (1995). Aesthetic value (Focus series). Boulder: Westview Press.
Goodman, N. (1968). Languages of art: An approach to a theory of symbols. Indianapolis: The Bobbs-Merrill Company.
Gow, J. (2010). A short history of Greek mathematics. Cambridge: Cambridge University Press.
Guyer, P. (2004). The origins of modern aesthetics: 1711–35. In P. Kivy (Ed.), The Blackwell guide to aesthetics (pp. 15–44). Malden: Blackwell.
Hardy, G. H. (1992). A mathematician’s apology. Cambridge/New York: Cambridge University Press.
Hillman, D. J. (1962). The measurement of simplicity. Philosophy of Science, 29, 225–252.
Hungerland, I. C. (1962). The logic of aesthetic concepts. Proceedings and Addresses of the American Philosophical Association, 36, 43–66.
Hutcheson, F. (1973). Francis Hutchenson: An inquiry concerning beauty, order, harmony, design (Vol. 9). The Hague: Martinus Nijhoff Publishing.
Kaku, M., & Thompson, J. T. (1997). Beyond Einstein: The cosmic quest for the theory of the universe. Oxford: Oxford University Press.
Kivy, P. (1975). What makes “Aesthetic” terms aesthetic? Philosophy and Phenomenological Research, 36(2), 197–211.
Kivy, P. (1980). The corded shell: Reflections on musical expression. Princeton: Princeton University Press.
Kivy, P. (1989). Sound sentiment: An essay on the musical emotions, including the complete text of the corded shell. Philadelphia: Temple University Press.
Kivy, P. (1991). Music alone: Philosophical reflections on the purely musical experience. Ithaca: Cornell University Press.
Kivy, P. (1993). The fine art of repetition: Essays in the philosophy of music. Cambridge/New York: Cambridge University Press.
Kivy, P. (1997). Philosophies of arts: An essay in differences. Cambridge/New York: Cambridge University Press.
Kivy, P. (2002). Introduction to a philosophy of music. Oxford: Oxford University Press.
Kline, M. (1990). Mathematical thought from ancient to modern times, Vol. 1 (1st ptg. ed.). New York: Oxford University Press.
Kline, M. (1990). Mathematical thought from ancient to modern times, Vol. 2. New York: Oxford University Press.
Kline, M. (1990). Mathematical thought from ancient to modern times, Vol. 3. New York: Oxford University Press.
Kuipers, T. A. F. (2000). From instrumentalism to constructive realism: On some relations between confirmation, empirical progress, and truth approximation. Dordrecht: Springer.
Kuipers, T. A. F. (2002). Beauty, a road to the truth. Synthese, 131(3), 291–328.
Lakatos, I., Worrall, J., & Zahar, E. (Eds.). (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge/New York: Cambridge University Press.
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.
Laudan, L. (1978). Progress and its problems: Towards a theory of scientific growth. Berkeley: University of California Press.
Lazarus, R. S. (1991). Emotion and adaptation. New York: Springer.
LeDoux, J. (1996). Emotional networks and motor control: A fearful view. Progress in Brain Research, 107, 437–446.
LeDoux, J. E. (1998). The emotional brain: The mysterious underpinnings of emotional life. New York: Simon & Schuster.
LeDoux, J., & Bemporad, J. R. (1997). The emotional brain. Journal of the American Academy of Psychoanalysis, 25(3), 525–528.
Le Lionnais, F. (1971). Les grands courants de la pensée mathématique. Marseille: Cahiers du Sud.
Le Lionnais, F. (2004). Great currents of mathematical thought: Mathematics: Concepts and development (Vol. 1). Mineola: Dover.
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.
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.
Marenbon, J. (2003). Boethius. Oxford/New York: Oxford University Press.
McAllister, J. (1996). Beauty and revolution in science. Ithaca: Cornell University Press.
McAllister, J. (1998). Is beauty a sign of truth in scientific theories? American Scientist, 86(2), 174–183.
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.
McWeeny, R. (2002). Symmetry: An introduction to group theory and its applications. Mineola: Dover.
Montano, U. (2010). Beauty in mathematics. PhD thesis, [s.n.], S.l. University of Groningen, The Netherlands.
Montano, U. (2012). Ugly mathematics: Why do mathematicians dislike computer-assisted proofs? The Mathematical Intelligencer, 34(4), 21–28.
Nahin, P. J. (1998). An imaginary tale: The Story of i [the square root of minus one]. Princeton: Princeton University Press.
Nahin, P. J. (2006). Dr. Euler’s fabulous formula: Cures many mathematical ills. Princeton: Princeton University Press.
Nahin, P. J. (2011). Dr. Euler’s fabulous formula: Cures many mathematical ills. Princeton: Princeton University Press.
Newton, I. (2003). Newton’s philosophy of nature selections from his writings. Whitefish, Mont. Kessinger Publishing.
Newton-Smith, W. (1981). The rationality of science. Boston: Routledge.
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.
Ortony, A., Clore, G. L., & Collins, A. (1990). The cognitive structure of emotions. Cambridge/New York: Cambridge University Press.
Popper, K. R. (2002). The logic of scientific discovery. London/New York: Routledge.
Robinson, J. (2005). Deeper than reason: Emotion and its role in literature, music, and art. New York: Oxford University Press.
Robinson, J. (2005). Deeper than reason: Emotion and its role in music literature and art. London: Claredon.
Rota, G. (1997). The phenomenology of mathematical proof. Synthese, 111(2), 183–196.
Rota, G.-C. (1997). The phenomenology of mathematical beauty. Synthese, 111(2), 171–182.
Rota, G. C. (2005). The phenomenology of mathematical beauty. In M. Emmer (Ed.), The visual mind II (pp. 2–13). Cambridge: MIT.
Russell, B. (2004). Mysticism and logic. Mineola: Dover.
Sibley, F. (1959). Aesthetic concepts. The Philosophical Review, 68(4), 421–450.
Snow, C. P. (2012). The two cultures. Cambridge/New York: Cambridge University Press.
Solomon, R. C. (2005). Subjectivity. In T. Honderich (Ed.), The Oxford companion to philosophy new edition (2nd ed.). Oxford/New York: Oxford University Press.
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.
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.
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.
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.
Tymoczko, T. (1979). The four-color problem and its philosophical significance. The Journal of Philosophy, 76(2), 57–83.
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.
Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2), 209–234.
Weinberg, S. (1994). Dreams of a final theory. New York: Vintage Books.
Wells, D. (1988). Which is the most beautiful? The Mathematical Intelligencer, 10(4), 30–31.
Wells, D. (1990). Are these the most beautiful? The Mathematical Intelligencer, 12(3), 37–41.
Wright, C. (1994). Truth and objectivity. Cambridge/London: Harvard University Press.
Zajonc, R. B. (1984). On the primacy of affect. American Psychologist, 39(2), 117–123.
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-Hill
Zajonc, R. B. (2001). Mere exposure: A gateway to the subliminal. Current Directions in Psychological Science, 10(6), 224–228.
Zangwill, N. (2003). Beauty. In J. Levinson (Ed.), The Oxford handbook of aesthetics. Oxford/New York: Oxford University Press.
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/.
Zangwill, N. (2011). Music, essential metaphor, and private language. American Philosophical Quarterly, 48(1), 1.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Montano, U. (2014). Aesthetic Experience. In: Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics. Synthese Library, vol 370. Springer, Cham. https://doi.org/10.1007/978-3-319-03452-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-03452-2_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03451-5
Online ISBN: 978-3-319-03452-2
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)