Examples of physics to maths and vice versa

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I sometimes think math can become kind of disconnected from reality and be self sustaining.

Hence the stuff invented by mathematicians long time ago find application in physics. I think it's the case of pseudo riemannian geometry and relativity used the results ?

Are there cases where physicists invented new maths ?
 

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Ed Witten, maybe?
 
  • #3
Ibix
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Newton.
 
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fresh_42
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The Bernoullis, Gauß, ... But these are examples of people who were both, not examples where physical calculi have been new to mathematicians. I think the closest are graded Lie algebras. It's not really an invention by physicists, but earlier to SUSY nobody was interested in these structures.
 
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Klystron
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Joseph Fourier was both physicist and mathematician who provided insights into, and some invention of, new mathematics including analysis. Gauss, as @fresh_42 stated, might be first among a pantheon of polymaths who practiced physics while advancing and inventing new mathematics such as Jean Baptiste D'Alembert.
 
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atyy
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Dirac delta function -> Schwartz distributions
Dirac notation -> Gelfand Rigged Hilbert Spaces
Einstein summation notation (does that count as new mathematics?)
Kardar-Parisi-Zhang solutions for nonlinear stochastic partial differential equations -> Hairer formalism for nonlinear stochastic partial differential equations
 
  • #7
Astronuc
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Leonhard Euler (15 April 1707 – 18 September 1783), a Swiss mathematician, physicist, astronomer, geographer, logician and engineer, made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.
https://en.wikipedia.org/wiki/Leonhard_Euler

Johannes Kepler (27 December 1571 – 15 November 1630) was a astronomer and mathematician; let's forget the bit about being an astrologer. He is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonices Mundi, and Epitome Astronomiae Copernicanae.
https://en.wikipedia.org/wiki/Johannes_Kepler
https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

In astronomy, Kepler's laws of planet motion are three scientific laws describing the motion of planets around the Sun, published by Johannes Kepler between 1609 and 1619. These modified the heliocentric theory of [URL='https://www.physicsforums.com/insights/an-introduction-to-theorema-primum/']Nicolaus Copernicus[/URL], replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The laws state that:
  1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
Joseph-Louis Lagrange (25 January 1736 – 10 April 1813), an Italian mathematician and astronomer, later naturalized French, made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange

The mathematical insights came through observation.

References from Wikipedia.
 
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  • #8
collinsmark
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Let's not forget about engineers.

Nearly the whole of information theory, and specifically the field of Forward Error Correction (your cell phone, WiFi devices, BlueTooth devices, CD & DVD player, etcetera use FEC as part of their communication), begins and ends in the world of mathematics. It can be applied to smoke signals or drum beats just as well as it can be to radio signals.

While it's technically a field of mathematics, it was developed for practical reasons, by many of whom were engineers, although they might have also had mathematics degrees too. (Claude Shannon, Irving Stoy Reed, Gustave Solomon, Andrew Viterbi, Claude Berrou, to name a few.)
 
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fresh_42
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The development of physics and mathematics took place in parallel most of the time in history. One posed the questions which the other one has been challenged to answer. The term pure mathematics is comparably new. Mathematics on the other hand had been considered a part of philosophy, rather than natural science.

And here is the most important difference: physics is a descriptive science, mathematics a deductive. This automatically implies that physics cannot provide any mathematical tools, it can only create a demand! The question has to be answered by a strict 'no'. Any exceptions merely reflect their parallels and shroud cause and effect. They do not hold strong science theoretic rigor.
Einstein summation notation (does that count as new mathematics?)
This is the third important difference between mathematics and physics. Einstein notation couldn't be more unmathematical! Mathematicians hate coordinates, they muddy the principles and structures behind!

Btw., the confusion that physicists created by their questionable usage of co- and contravariance is the second most important difference, still ahead of ##(-1,1,1,1)## versus ##(1,-1,-1,-1)## and ##\langle \lambda\vec{a},\vec{b}\rangle = \bar{\lambda}\langle\vec{a},\vec{b} \rangle## versus ##\langle \vec{a},\lambda\vec{b}\rangle = \bar{\lambda}\langle\vec{a},\vec{b} \rangle## .
 
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Btw., the confusion that physicists created by their questionable usage of co- and contravariance is the second most important difference, still ahead of ##(-1,1,1,1)## versus ##(1,-1,-1,-1)## and ##\langle \lambda\vec{a},\vec{b}\rangle = \bar{\lambda}\langle\vec{a},\vec{b} \rangle## versus ##\langle \vec{a},\lambda\vec{b}\rangle = \bar{\lambda}\langle\vec{a},\vec{b} \rangle## .

lol, don't also forget 'passive' vs 'active' transformations!
 
  • #11
epenguin
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I have only knowledge of lower reaches of this and don't even know what a lot of what's mentioned is. A lot of the guys mentioned, Euler, D'Alembert, Lagrange, Fourier etc. didn't much care whether their stuff was called maths or physics did they? - probably called it Natural Philosophy.

At my elementary level, things are often gone through twice in the courses, in the math and the physics courses. Sometimes they feel uncomfortable in both. For example - coupled vibrations. On the one hand, springs and masses are not mathematical concepts. On the other hand apart from F=ma, Hooke's law, which is almost bound to be widely true for small enough displacement from equilibrium is all the physics that comes into it. And I noticed in courses I attended and in books, once the mathematical results had been got there was not much tendency to linger on examples and applications.

This classification anxiety has been perhaps relieved by the creation of the category "Applied Mathematics" which in most places was little or nothing else but physics? Perhaps the worthy, laudable, noble aim has been the creation of extra University departments and posts?
 
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jk22
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Suppose new maths were generated by computers (that human could understand), could this be considered physics (material, electronics) -> maths ?
 
  • #13
BillTre
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By that logic, current math could be considered biology (or psychology) -> math (math being a product of the human brain).
 
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fresh_42
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Suppose new maths were generated by computers (that human could understand), could this be considered physics (material, electronics) -> maths ?
This works only partially, as it has been done for the four colour theorem already, where positive results can be expected. You cannot prove non-existence by computers. Beside that, logic is logic, it doesn't matter who set up the string of conclusions. The problem I see is, whether it makes any sense: a string of logical correct deductions doesn't make a theorem, only an oddity.
 
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Astronuc
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Not new math, but a realization of a previously seemingly unnoticed, or unpublished, relationship in established math.

Three physicists wanted to calculate how neutrinos change. They ended up discovering an unexpected relationship between some of the most ubiquitous objects in math.
https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/

Their joint paper with Terence Tao.
Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra
https://arxiv.org/abs/1908.03795

Peter B. Denton, Stephen J. Parke, Xining Zhang
Eigenvalues: the Rosetta Stone for Neutrino Oscillations in Matter
https://arxiv.org/abs/1907.02534
 
  • #16
atyy
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Btw., the confusion that physicists created by their questionable usage of co- and contravariance is the second most important difference, still ahead of ##(-1,1,1,1)## versus ##(1,-1,-1,-1)## and ##\langle \lambda\vec{a},\vec{b}\rangle = \bar{\lambda}\langle\vec{a},\vec{b} \rangle## versus ##\langle \vec{a},\lambda\vec{b}\rangle = \bar{\lambda}\langle\vec{a},\vec{b} \rangle## .

:oldlaugh:

Hmmm, what is wrong with any of the above "physics" things?
 

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