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Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics

Ulianov Montano
Publication Date: 
Number of Pages: 
Synthese Library 370
[Reviewed by
Felipe Zaldivar
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This book is a developed version of a point of view expressed by James McAllister on the alleged role played by aesthetic considerations on the development of science in general and of mathematics: that, these aesthetic considerations might be rationalized. The author surveys previous attempts to conceptualize the notion of mathematical beauty, from Shaftesbury and Hutchenson in the 17th and 18th centuries, to Gian-Carlo Rota in the 20th century.

The contribution of the author could be summarized as follows: He proposes a mathematical dynamical model of a literal interpretation of aesthetic values in mathematics, borrowing sometimes from some accounts of neuroscience, to address the changing character of our views on beauty. There are some unexplained assumptions on the nature of the variables in the model, but this seems to be typical when authors try to use mathematics to model social phenomena (which sometimes has led some authors astray, as documented, for example, by Serge Lang in a well-known case).

The author discusses at some length Rota’s essay “Phenomenology of Mathematical Beauty” included in Indiscrete Thoughts (Birkhäuser, 1997). To address the author’s attempt to model a theory of beauty, perhaps it suffice to point to Rota’s article (in the same volume) on the pernicious influence of mathematics upon other disciplines.

Contrary to one assertion in the back cover of the book, the mathematics used in this book is of a rather elementary nature: Cantor’s diagonal argument, a few remarks on the derivative and integral of the exponential function or the polar expression for a complex number, etc. There need to be some caveats on the manipulation of some equalities in pages 92 to 94.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is


Part 1. Antecedents
Chapter 1. On Non-literal Approaches
Chapter 2. Beautiful, Literally
Chapter 3. Ugly, Literally
Chapter 4. Problems of the Aesthetic Induction
Chapter 5. Naturalizing the Aesthetic Induction

Part 2. An Aesthetics of Mathematics
Chapter 6. Introduction to a Naturalistic Aesthetic Theory
Chapter 7. Aesthetic Experience
Chapter 8. Aesthetic Value
Chapter 9. Aesthetic Judgement I: Concept
Chapter 10. Aesthetic Judgement II: Functions
Chapter 11. Mathematical Aesthetic Judgements

Part 3. Applications
Chapter 12. Case Analysis I: Beauty
Chapter 13. Case Analysis II: Elegance
Chapter 14. Case Analysis III: Ugliness, Revisited
Chapter 15. Issues of Mathematical Beauty, Revisited. ​