Is mathematics invented or discovered?

One of the most fascinating aspects of modern mathematics is the extent to which developments in “pure” mathematics are subsequently, and often quite unexpectedly, found to have direct relevance to the physical world. Albert Einstein asked, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” [Jammer1921, pg. 124].

One source that is often cited in this context is Eugene Wigner’s 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” [Wigner1960]. He cites numerous examples:

  1. Newton’s laws and planetary motion. Wigner notes that Newton’s laws of motion involved abstract notions such as the second derivative, which was “simple only to the mathematician, not to common sense or non-mathematically-minded freshmen”. Yet the laws explained planetary motions to very high accuracy, and then were utilized over the following decades and centuries to explain many, many other phenomena. Indeed, only in the late 19th and early 20th century did scientific measurement technology become so precise as to detect deviations from Newton’s laws (as in the perihelion motion of Mercury).
  2. Quantum mechanics. Max Born first noticed that some ad-hoc rules of computation given by Heisenberg were formally identical with certain rules of matrix computation established many years earlier by pure mathematicians. Later this matrix formalism was applied to situations, such as the analysis hydrogen and helium atoms, that were well beyond the scope of Heisenberg’s reckoning, and yet the results were still accurate beyond any reasonable expectation.
  3. Quantum electrodynamics (QED). Wigner notes that this purely mathematical theory agrees with experiment to “better than one part in a thousand”. He was being very modest. For example, a more recent application of the theory to find the magnetic moment of the electron found the value 1.001159652201 (plus or minus 30 in the last two digits), compared with 1.001159652188 (plus or minus 4 in the last two digits) — better than one part in a billion agreement! [Sokal1998, pg. 57]. Is this merely a coincidence?

Wigner closed his essay by saying “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” [Wigner1960].

A new perspective has recently been offered by astrophysicist Mario Livio. In a August 2011 article in Scientific American [Livio2011], he addresses two questions: “Is mathematics invented or discovered?” and “What gives mathematics its explanatory and predictive powers?” Livio believes that we know the answer to the first question: mathematics is an “intricate fusion” of both inventions and discoveries, and even though concepts are first invented, humans still choose which ones to study. The second question, though, is more complex. Although there is no doubt that the selection of topics we address mathematically has been key to mathematics’ effectiveness, these mathematical principles would not work if there were no universal truths to be discovered.

Livio closes by asking:

Why are there universal laws at all? Or equivalently: Why is our universe governed by certain symmetries and by locality? I truly do not know the answers, except to note that perhaps in a universe without these properties, complexity and life would have never emerged, and we would not be here to ask the question.

Readers may also wish to view an interesting slide show of mathematical structures, available on the Scientific American website: Slide show.


  1. [Jammer1921] Max Jammer, Einstein and Religion, Princeton University Press, 1921.
  2. [Livio2011] Mario Livio, “Why Math Works,” Scientific American, Aug 2011, pg. 82-83.
  3. [Sokal1998] Alan Sokal and Jean Bricmont, Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science, Picador, New York, 1998.
  4. [Wigner1960] Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Communications in Pure and Applied Mathematics, vol. 13, No. 1 (Feb 1960), John Wiley and Sons, New York.

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