George David Birkhoff

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George David Birkhoff (1884—1944) was an American mathematician. He was a professor at Harvard University in Boston (USA) and worked in different mathematical areas – from dynamic systems (formulating here his important Ergodic theorem) to gravitation (developing a mathematical gravitation model) to art and rational aesthetics. In 1937-39 he served as Dean of the Faculty of Arts and Sciences at Harvard.

Mathematical theory of aesthetics

Birkhoff was already speculating on the possibility of a mathematical theory of music while he was in Princeton (until 1912). [1]

In 1928 he published his first attempts to formalize aesthetics. He introduced the concept of the Esthetic Measure (M), defined as the ratio between Order (O) and Complexity (C): M = O/C. The Complexity is roughly the number elements that the image consists of; the Order is a measure for the number of regularities found in the image. For different artistic genres, Birkhoff has indicated specific rules to actually compute precise values for Order and Complexity. For polygons he thus defines Complexity as the number of edges, while the numerical value for Order depends among other things on the presence of vertical symmetry, point symmetry, and mechanical stability with respect to an imaginary horizontal plane. [2]

Following a yearlong trip around the world devoted to studying art, music and poetry, Birkhoff expanded his theory in the 1932 article "A Mathematical Theory of Aesthetics" and the book Aesthetic Measure (1933).


Birkhoff on aesthetics and ethics
  • "Quelques eléments mathématiques de l’art", in Atti Congressi Bologna I, 1928, pp 315-333. An address given at an mathematical congress in the Salon del Cinquecento of the Palazzo Vecchio at Florence.
  • "A Mathematical Theory of Aesthetics", 1932.
  • Aesthetic Measure, Harvard University Press, 1933, 226 pp; selection repr. as "Mathematics of Aesthetics", in The World of Mathematics, Vol. 4, ed. James R. Newman, New York: Simon and Schuster, 1956, pp 2185-2195; repr. Kessinger Publishing, 2003, 292 pp. Reviews: Garabedian (1934).
  • "A Mathematical Approach to Ethics", Rice Institute Pamphlets (vol. 28), 1942; selection repr. in The World of Mathematics, Vol. 4, ed. James R. Newman, New York: Simon and Schuster, 1956, pp 2198-2208.
On Birkhoff's aesthetics and ethics
  • James R. Newman, "Commentary on George David Birkhoff", in The World of Mathematics, Vol. 4, ed. James R. Newman, New York: Simon and Schuster, 1956, pp 2182-2184.
  • Remko Scha, Rens Bod, "Computational Esthetics", c1993.
  • Rul Gunzenhäuser, Ästhetisches Mass und ästhetische Information. Einführung in die Theorie G. D. Birkhoffs und die Redundanztheorie ästhetischer Prozesse, Quickborn bei Hamburg: Schnelle, 1962, 164 pp; expanded as Maß und Information als ästhetische Kategorien. Einführung in die ästhetische Theorie G. D. Birkhoffs und die Informationsästhetik, Baden-Baden: Agis, 1975, 216 pp. Based on the author's dissertation defended at the Technische Hochschule, Stuttgart. (German) [3]
  • Tomas Staudek, On Birkhoff's Aesthetic Measure of Vases, Brno: Masaryk University, September 1999, 8 pp.
  • Jaume Rigau, Miquel Feixas, Mateu Sbert, "Conceptualizing Birkhoff’s Aesthetic Measure Using Shannon Entropy and Kolmogorov Complexity", in Computational Aesthetics in Graphics, Visualization, and Imaging, eds. D. W. Cunningham, G. Meyer and L. Neumann, 2007.
  • Christoph Klütsch, "Information Aesthetics and the Stuttgart School", in Mainframe Experimentalism: Early Computing and the Foundation of the Digital Arts, eds. Hannah Higgins and Douglas Kahn, University of California Press, 2012, pp 65-89.