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There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as general mathematical tools.

One example is the Dirac delta "function" which was used to simplify integrals, but at the time was perhaps not very well-defined to any mathematica standard. However, it now fits well within the theory of distributions. Perhaps another example is Newton's calculus, inspired by fundamental questions in physics.

Are there any other examples of mathematical concepts being inspired by work in physics?

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    $\begingroup$ One of the most famous papers in 20th century mathematics, Yang-Mills Equations on Riemann Surfaces: csee.wvu.edu/~xinl/library/papers/math/geometry/Atiyah1983.pdf $\endgroup$
    – Moya
    Jan 8, 2016 at 16:29
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    $\begingroup$ Gauss invented the FFT to study orbits of asteroids. $\endgroup$
    – WimC
    Jan 8, 2016 at 16:38
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    $\begingroup$ The dot product and cross product of vectors in $\mathbb R^3$ were invented by Heaviside and Gibbs to simplify Maxwell's equations of electrodynamics, which were originally expressed in terms of quaternions. $\endgroup$
    – user856
    Jan 8, 2016 at 22:26
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    $\begingroup$ O. Heaviside also introduced the delta function as an impulse function in signal analysis, and defined it as the derivative of the unit step function. www-theory.lbl.gov/jdj/ZerothT_AJP.pdf . The delta function was definitely motivated by Physics/Engineering, but is incorrectly attributed to Dirac. Heaviside confused people with his many new objects, including his operator calculus to solve ODEs of signal analysis. $\endgroup$
    – Disintegrating By Parts
    Jan 9, 2016 at 14:35
  • $\begingroup$ Not geometry. Geo+metry is derived from earth+measuring, as it had its origins in real estate! Surveying, measuring fields etc, with string and protractors. The ancient egyptians, even before the greeks, knew pythagoras's therom and with the aid of calibrated string used it to build rectangular pyramids with nice square corners. Bricklayers still use string to mark out straigh edges today $\endgroup$
    – Level River St
    Jan 9, 2016 at 16:35

7 Answers 7

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Until about 150-200 years ago, Math and Physics were not even considered separate disciplines. Before then, Mathematics was just the language one used to describe the natural world. Thus one can reasonably claim that all mathematics older than this has its origin in physics.

Physics is the science of measurement. Mathematics was developed as a tool for discussing those measurements. The Babylonians and Egyptians in particular had collected a large cache of algorithms for calculating various measurements, particularly areas and volumes, which were used as physical laws: "If you have this shape and size of container, how much water will it hold?" They arrived at these rules by some basic reasoning, but mostly from experimentation. They had no means to distinguish between fully accurate and approximation. The Greek Thales of Miletus saw that certain parts of their accumulated knowledge could be deduced from a few simple principles, which was the birth of mathematics as we know it today.

All of ancient Greek mathematics was about describing the real world. They disdained the idea of testing their ideas by physical experiment, but this was from a belief that is wasn't necessary to test - that logic was sufficient to divine all principles. It was not from any idea that mathematics was detached from the real world. Rather the real world was considered a corruption of the perfect world of their conception.

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Concepts that are first introduced in Physics:

Product vector (internal and external)
Fourier Transforms
Tensors, rotors and spinors

And succintly almost all directly concerned about physical phenomenons.

As said before, even the whole numbers appear because they were needed for count physical objects.

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    $\begingroup$ This is the only answer that I think is really in the spirit of the question. +1 $\endgroup$
    – user137731
    Jan 8, 2016 at 22:37
  • $\begingroup$ I've heard of tensors and spinors, but what is a "rotor" in this context? $\endgroup$
    – David Zhang
    Jan 9, 2016 at 9:40
  • $\begingroup$ @DavidZhang: I think it's the concept from geometric algebra: en.wikipedia.org/wiki/Rotor_(mathematics) $\endgroup$
    – Dietrich Epp
    Jan 9, 2016 at 23:01
  • $\begingroup$ I believe spinors were discovered by mathematicians (Cartan) before physicists (Pauli/Dirac). $\endgroup$
    – akhmeteli
    Jan 10, 2016 at 1:12
  • $\begingroup$ Surely tensor calculus was created by the mathematition Gregorio Ricci-Curbastro and then used by Einstein and others in physics. $\endgroup$
    – Paul Evans
    Jan 12, 2016 at 9:01
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The notion of $\color{red}{\text{Derivatives}}$ and more generally calculus.

  • The ancient Egyptians and Greeks (in particular Archimedes) used the notions of infinitesimals to study the areas and volumes of objects.
  • Indian mathematicians (in particular Aryabhatta) used infinitesimals to study the motion of moon and planets.
  • These notions were later extended and formalised by Newton and Leibniz.
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    $\begingroup$ Area and volume are geometry concepts, not physics concepts. $\endgroup$
    – djechlin
    Jan 8, 2016 at 17:38
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    $\begingroup$ @djechlin but objects are physical things $\endgroup$
    – njzk2
    Jan 8, 2016 at 19:21
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    $\begingroup$ @djechlin Among many other things, Archimedes compared the volumes of a sphere, cone, and cylinder by placing them on a fulcrum. It's a remarkable proof, and shows a concrete physical understanding of the geometrical objects. $\endgroup$
    – Théophile
    Jan 8, 2016 at 19:47
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    $\begingroup$ @njzk2 insofar as you are making any distinction between mathematics and the physical word, area and volume fall squarely under mathematics. If you're not making any distinction then the question may be rephrased as "What are some math concepts?" so no I would not count this. $\endgroup$
    – djechlin
    Jan 8, 2016 at 19:51
  • $\begingroup$ @Théophile this answer would certainly be less vacuous if it included that information. $\endgroup$
    – djechlin
    Jan 8, 2016 at 19:51
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Lets go with the most basic one first: $\color{red}{\text{Numbers}}$

  • First man wanted to count his community members, prey animals, etc. giving birth to natural numbers.
  • He then wanted to borrow and lend, giving birth to negative numbers and zero.
  • He wanted to share and divide his earnings among his family members and community, giving birth to rationals.
  • He then wanted to measure length of sticks, area of land, volume of water, giving birth to irrationals and a whole host of transendentals...
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    $\begingroup$ It's a little bit of a stretch IMO to call those examples of physics (with the exception of the last one). $\endgroup$
    – user137731
    Jan 8, 2016 at 16:37
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    $\begingroup$ Any activity related to physical world is physics. Measuring areas, lengths are very much part of physics... $\endgroup$
    – Adhvaitha
    Jan 8, 2016 at 16:39
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    $\begingroup$ I'd say any activity related to learning about the rules that govern the physical world is physics. Counting people in your tribe or knowing how much money you owe your prehistoric-version-of-a-bookie doesn't really fit that description to me. But like said, it's just MO. $\endgroup$
    – user137731
    Jan 8, 2016 at 16:40
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    $\begingroup$ @Bye_World conservation and consistency of measure = number is the most fundamental learning of the rules that govern the physical world there is. $\endgroup$
    – fleablood
    Jan 8, 2016 at 16:48
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    $\begingroup$ This answer presents imaginary events that are completely disconnected from actual historic developments. Notably if you go and check when negative quantities were first accepted as numbers, you will find that this was thousands of years after positive rational and (certain) irrational quantities were being used (by the ancient Greeks, maybe even before) as what we would now consider numbers. And certainly the notion of transcendental number has no roots in natural science, but is the result of a purely mathematical reflection. $\endgroup$
    – Marc van Leeuwen
    Jan 9, 2016 at 10:15
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The string theory is a big source of new mathematical ideas. Abstract of the link:

String theory, ot its modern incarnation M-theory, gives a huge generalization of classical geometry. I indicate how it can be considered as a two-parameter deformation, where one parameter controls the generalization from points to loops, and the other parameter controls the sum over topologies of Riemann surfaces. The final mathematical formulation of M-theory will have to make contact with the theory of vector bundles, K-theory and noncommutative geometry.
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The 13th Century Persian astronomer, scientist and mathematician Nasir Al-Din Al-Tusi was perhaps the first to treat trigonometry as a separate mathematical discipline, distinct from astronomy. Building on earlier work by Greek mathematicians such as Menelaus of Alexandria and Indian work on the sine function, he gave the first extensive exposition of spherical trigonometry, including listing the six distinct cases of a right triangle in spherical trigonometry. One of his major mathematical contributions was the formulation of the famous law of sines for plane triangles.

Source: Story Of Mathematics.

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One that came to mind is the concept of the soliton, which is a self-reinforcing solitary wave, whose discovery eventually led to the Korteweg–de Vries equation and other applications in differential systems, field theory, etc.

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