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Throughout the history of mathematics and physics, there has been many examples where mathematics was discovered first prior to its application in physics. Consider $i=\sqrt {-1}$ as an example, among many others.

I am interested to know about discoveries in physics, for which a mathematical counterpart did not exist at the time and for which new mathematics had to be created. Are there such examples? If so, a few examples would be appreciated (preferably with sources).

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  • $\begingroup$ William Gilbert's discovery of magnetism. $\endgroup$
    – Mauro ALLEGRANZA
    Mar 30, 2021 at 5:47
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    $\begingroup$ Quantum mechanics, for which von Neumann (anticipated by Weyl) developed the abstract formalism of Hilbert spaces and self-adjoint operators, see The "new quantum theory". $\endgroup$
    – Conifold
    Mar 30, 2021 at 8:26
  • $\begingroup$ Discovering phenomena is orthogonal to discovering, or deriving, new concepts in mathematics. You could say that neanderthals "discovered" that things fall down and get faster and faster as they do so, but the equations of motion & derivatives didn't follow for eons. $\endgroup$
    – Carl Witthoft
    Mar 30, 2021 at 11:25

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Potential theory (Green's formulas, Green's function etc.) was discovered by George Green who was doing physics. His work was called "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". Laplace equation was first written in a paper on Saturn rings.

Eigenvalues, eigenvectors and adjoint operators were discovered by mathematicians who were doing physics (namely celestial mechanics). Fourier transform was discovered by Fourier who was doing physics (in his studies of heat flow). Same applies to Bessel functions and theta functions which appear for the first time in Fourier's book on heat. Fourier series were discovered by Daniel Bernoulli who was doing physics (oscillations of a string).

Vector analysis was invented by physicists Herz and Gibbs. Maxwell's book on electromagneitsm is literally packed with mathematical discoveries.

Calculus of operators (a.k.a. operational calculus) was discovered by physicist/electrical engineer Oliver Heaviside, who was studying the "telegrapher's equation".

The theories of unbounded operators, operator algebras and quantum logic were developed by von Neumann to give mathematical foundations of quantum mechanics.

Lee and Yang Theorem was discovered by two physicists who studied phase transitions.

Yang-Mills equations were discovered by physicists Yang and Mills working on quantum field theory.

The list can be made almost infinite.

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    $\begingroup$ A little rectification : Fourier discovered Fourier series, not Fourier Transform (mainly discovered one century later by Norbert Wiener, who was educated as an electrical engineer, which is not a neutral fact) $\endgroup$
    – Jean Marie Becker
    Apr 1, 2021 at 18:20
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    $\begingroup$ @Jean Marie Becker: this is not correct. Fourier series were known to Euler and Bernoulli. Fourier discovered Fourier transform. Open his book on heat to convince yourself. Chap. IX, sections 342-347 of the English translation. $\endgroup$
    – Alexandre Eremenko
    Apr 2, 2021 at 2:43
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Paul Dirac developed the Dirac Delta for physics. Mathematicians said it made no sense. It took the development of the theory of distributions to show they were wrong.

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    $\begingroup$ An electrical engineer Oliver Heaviside Heaviside has to be also mentioned in this connection. $\endgroup$
    – Alexandre Eremenko
    Mar 30, 2021 at 19:15
  • $\begingroup$ Also, I believe Dirac used higher-dimensional distributions, such as "integration over light-cone"... $\endgroup$
    – paul garrett
    Apr 1, 2021 at 17:18
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Joseph Fourier created Fourier series to study the propagation of heat in solid bodies initially but it became an important field in pure mathematics later.

Oliver Heaviside developed Heaviside calculus to investigate electric circuits, which was popular among electrical engineers but was initially rejected by mathematicians for lacking mathematical rigor. Later, Heaviside calculus was reworked by Bromwich into what is known as Laplace transform today.

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