On the Importance of Useless Mathematics

  • António MachiaveloEmail author


It is very difficult to convey the purpose of large and important pieces of mathematics to the general public. Even college students of mathematics frequently ask, in an exasperating way: “What is this useful for?” In this short essay it is argued that in order to appropriately answer this question, one must first dispel some common-sense notions that just happen to be wrong. Fundamentally, one has to confront, head on, the question of what mathematics really is and exactly what things it deals with, which can only be satisfactorily understood within an evolutionary perspective.


Natural Selection Mathematical Object Nobel Laureate Wrong Idea Mathematical Community 
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.



This work was partially supported by Fundação para a Ciência e Tecnologia (FCT) through Centro de Matemática da Universidade do Porto. Substantial parts of this paper were adapted from an article by the author, [30], published in Portuguese.


  1. 1.
    Sagan, C.: Cosmos. Random House, New York (1980) Google Scholar
  2. 2.
    Greenberg, M.: Euclidean and Non-Euclidean Geometry: Development and History, 2nd edn. Freeman, New York (1980) zbMATHGoogle Scholar
  3. 3.
    Brian Davies, E.: Let platonism die. Newsletter of the European Mathematical Society, 24–25 (2007) Google Scholar
  4. 4.
    Hersh, R.: On platonism. Newsletter of the European Mathematical Society, 17–18 (2008) Google Scholar
  5. 5.
    Mazur, B.: Mathematical platonism and its opposites. Newsletter of the European Mathematical Society, 19–21 (2008) Google Scholar
  6. 6.
    Mumford, D.: Why I am a platonist. Newsletter of the European Mathematical Society, 27–30 (2008) Google Scholar
  7. 7.
    Davis, P.J.: Why I am a (moderate) social constructivist. Newsletter of the European Mathematical Society, 30–31 (2008) Google Scholar
  8. 8.
    Gardner, M.: Is Reuben Hersh ‘Out there’? Newsletter of the European Mathematical Society, 23–24 (2009) Google Scholar
  9. 9.
    Brian Davies, E.: Some recent articles about platonism. Newsletter of the European Mathematical Society, 24–27 (2009) Google Scholar
  10. 10.
    Heisenberg, W.: Physics and Philosophy: the Revolution in Modern Science. Penguin, Baltimore (1989) (original from 1958) Google Scholar
  11. 11.
    Feynman, R.: What Do You Care What Other People Think? Bantam Books, New York (1989) Google Scholar
  12. 12.
    Rucker, R.: Mind Tools: the Five Levels of Mathematical Reality. Houghton Mifflin, Boston (1987) zbMATHGoogle Scholar
  13. 13.
    Darwin, C.: On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. Murray, London (1859). Available in The Complete Work of Charles Darwin Online at Google Scholar
  14. 14.
    Sagan, C., Druyan, A.: Shadows of Forgotten Ancestors. Ballantine Books, New York (1993) Google Scholar
  15. 15.
    Moore, J.A.: Science as a Way of Knowing: the Foundations of Modern Biology. Harvard University Press, Cambridge (1993) Google Scholar
  16. 16.
    Gregory, T.R.: Understanding natural selection: essential concepts and common misconceptions. Evol. Education Outreach 2, 156–175 (2009) CrossRefGoogle Scholar
  17. 17.
    Penrose, R.: The Road to Reality: a Complete Guide to the Laws of the Universe. Vintage, New York (2005) zbMATHGoogle Scholar
  18. 18.
    Allen, C., Bekoff, M.: Animal play and the evolution of morality: an ethological approach. Topoi 24, 125–135 (2005) CrossRefGoogle Scholar
  19. 19.
    Frank, R.H.: Passions Within Reason: the Strategic Role of Emotions. Norton, New York (1988) Google Scholar
  20. 20.
    Darwin, C.: The Descent of Man, and Selection in Relation to Sex, 2nd edn. Murray, London (1882). Available in The Complete Work of Charles Darwin Online at Google Scholar
  21. 21.
    Sagan, C.: The Dragons of Eden: Speculations on the Evolution of Human Intelligence. Hodder & Stoughton, London (1977) Google Scholar
  22. 22.
    Kline, M.: Mathematics in Western Culture. Oxford University Press, London (1964) (original from 1953) zbMATHGoogle Scholar
  23. 23.
    Osserman, R.: Poetry of the Universe: A Mathematical Exploration of the Cosmos. Anchor (1996) Google Scholar
  24. 24.
    Hon, G., Goldstein, B.: Hertz’s methodology and its influence on Einstein. In: Wolfschmidt, G. (ed.) Heinrich Hertz (1857–1894) and the Development of Communication: Proceedings of the Symposium for History of Science, Hamburg, October 8–12, 2007, pp. 95–105. Books on Demand (2008) Google Scholar
  25. 25.
    Kline, M.: Mathematical Thought from Ancient to Modern Times. Oxford University Press, London (1990) Google Scholar
  26. 26.
    Heisenberg, W.: Development of concepts in the history of quantum theory. Am. J. Phys. 43, 389–394 (1975) CrossRefGoogle Scholar
  27. 27.
    Edwards, H.M.: Fermat’s Last Theorem: a Genetic Introduction to Algebraic Number Theory. Springer, Berlin (1977) zbMATHGoogle Scholar
  28. 28.
    Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 120–126 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Euler, L.: Theoremata circa divisores numerorum (E134). Novi Commentarii academiae scientiarum Petropolitanae 1, 1750, pp. 20–48. Reprinted in Opera Omnia: Series 1, Volume 2, pp. 62–85. Original article available online, along with an English translation by David Zhao, at
  30. 30.
    Machiavelo, A.: A natureza dos objectos matemáticos. Gaz. Mat. 161, 7–16 (2010) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Centro de Matemática da Universidade do PortoPortoPortugal

Personalised recommendations