# In what ways has physics spurred the invention of new mathematical tools?

I came across this comment:

Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more interesting: their lack of rigor is usually a reflection of the inadequacy of existing mathematical tools for their purposes. When such a theory is read by a mathematician, he is forced to invent new tools or use existing tools in a way he had never thought to before in order to make sense of it. This in turn has spurred many mathematical advances and even created new fields of study. For example, many of the most important PDEs (heat, wave, Laplace, Navier-Stokes, Schrodinger, etc.) were first encountered in physics.

I'm curious in what other ways physics (or other pursuits of physical/scientific/computational truth) have sparked the creation of new mathematical tools to understand and formalize these things. Is one example Turing's formalization of computation, or Shannon's formalization of information?

• This might be a dumb answer but Newton did invent calculus to study planetary motions... Aug 1, 2014 at 1:13
• Another example in @IllegalImmigrant's vein is the development of perturbation theory to study (what else) perturbations in the simple two-body model of planetary orbits (i.e. what kind of weaker influence do other planets have on another planet's orbit.) There's a reason, after all, why Bessel functions are named after someone who was both a mathematician and an astronomer! Aug 1, 2014 at 1:33
• The use of Dirac delta "functions" by Heaviside preceded the rigorous theory of distributions. Dirac's use of spinors in quantum mechanics surely spurred analogous mathematical activity. Another area is the generalization of connections / covariant derivatives (which appear in general relativity) by Koszul. Aug 1, 2014 at 1:33
• Classical mechanics can be translated entirely into the language of symplectic geometry. I don't think this is how the subject was originated but for example symplectic reduction provides a different way of studying Noether's theorem Aug 1, 2014 at 1:46

## 5 Answers

Fourier analysis, which has applications in fields as diverse as PDEs and algebraic number theory, has its origins in physics (which, at the time, really wasn't as distinct from mathematics as it is now). The study of waves and heat flow lead Fourier and others to the theory of trigonometric series.

For example, by working formally you can show that solutions to the heat equation in the unit disc, with boundary function $f(\theta)$, should be of the form $u(r,\theta) = \sum_{n = -\infty}^{\infty} a_{n}r^{|n|}e^{in\theta}$, where $a_{n} = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx} \ dx$. In particular, if $f$ is nice enough then it should be the case that $f(\theta) = u(1,\theta) = \sum_{n = -\infty}^{\infty} a_{n}e^{in\theta}$, which is a Fourier series. This leads to the fundamental question of Fourier analysis: is such a representation possible?

Addressing the Turing Machine:

The Turing machine was not invented due to any physical or computational science. In fact, they were invented to solve a math problem.

Prior to 1930's, computer science (as in theoretical computer science) did not exist. As hard as it may be to believe today, people did not even entirely understand what algorithms were but the intuitions were beginning to form.

In fact, the Turing machine was invented to solve a problem in mathematics. One could even argue a Turing machine was invented to solve a problem in philosophy. One of Hilbert's problems (the so called decision problem) was vaguely stated as: does there exist a mechanical procedure to decide whether statements were provable in some first order theory. Today this may be understood as asking whether there is an algorithm to decide the provability of a statement. But prior to 1930's, numerous people such as Godel, Church, and Kleene were proposing various models for what "mechanical procedure" could possibly mean. The $\lambda$-calculus, $\mu$-recursive functions, etc were various mathematical looking and equation-looking formalization that appeared during this time. However, Godel (and others) was not convinced that philosophically these mathematical-looking models capture what "mechanical meant". Godel was not convinced his own models captured this notion. Then Turing proposed his Turing machine in 1936 in his solution to Hilbert's Decision problem. The Turing machine did not resemble recursively formed functions or equations. The Turing machine actually resembles a person writing, erasing, and moving along a piece of paper. Godel and others were convinced that a Turing machine was a good model of a human being doing calculations. It was later shown that the Turing machine is equivalent to many of the more equational-looking models proposed by Godel, Church, and other researchers.

The Turing was not invented to solve any physical problem in science. It was invented to solve a mathematics problem which played a great role in the modern understanding of algorithm. There is a strong argument to say that the Turing machine is an example of when mathematics created another field of science: the computational science.

• I believe you are correct about Turing machines. Shannon's information theory, on the other hand, seems to have been motivated by (or at least grounded in) telegraphy, so that example seems to hold up (at least under the "other pursuits" heading). Aug 1, 2014 at 1:59

Ok, a semi not stupid answer...

Wavelets and wavelet transform are invented in the same sense as Newton and Leibniz's calculus, by a physicist and a mathematician at around the same time (though many argues that Newton brusquely pilfered the recognition of Leibniz's contribution to calculus after his early death)

In Mallat's book, a Wavelet Tour of Signal Processing, he indicates that the need for further studies in quantum mechanics through time-frequency localization was recognize by Gabor in 1946 where he defined something called a time-frequency atom - a waveform that have minimum spread in time-frequency plane.

His idea led to the development of the wavelet transform, of which was first created by another physicist named Morlet who intended to use it to high frequency seismic waves, and Grossmann, also a physicist, in his study of coherent quantum states.

Doubtlessly, the wavelet transform is now seen in the lights as an engineering tool (JPEG2000 anyone?) that is grounded in pure mathematics from the contribution of Daubechie (famous for the Daubechie family of mother wavelets and Ten Lectures), who stated herself in a recent youtube video that she is a pure mathematician. But when we go through the history of the wavelet, we find that most motivation originally came from physics.

Gauge theory is one example where one sees physics stimulating the development of new mathematics. 15 years ago Seiberg and Witten based on intuitions coming from physics proposed a new powerful way of doing gauge theory and in particular almost trivializing earlier impressive results of Fields medal caliber, such as Donaldson's diagonalisation result for 4-manifolds with definite intersection forms; see this article for a discussion.

Is this question possibly a superset of "when do have physicists have math stuff named after them" ?

Feynman is a physicist, and he and Kac have the Feynman-Kac theorem which link PDEs with stochastic processes. FK Theorem is used in mathematical finance to derive one of the most important results: Black-Scholes-Merton Formula, used in pricing of financial derivatives.

Because I don't understand it well enough, I cannot explain it simply (or the converse?) :(

But I guess Feynman and/or Kac encountered some particles that followed Brownian motion (which stock prices are sometimes assumed to follow) then came up with that.

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