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I am looking for mathematical concepts (*) which have been introduced by physicists in a non rigorous way (e.g. without a formal definition, without rigorous proofs of the results, etc.) and used to derive some physical results, but that were later formalized by mathematicians in a rigorous way.

One example that I have heard of is Dirac δ-function, which has been used by physicists before it was rigorously formalized as part of the theory of distributions (do not hesitate to answer/comment with details about this particular example).

For each example, I am looking for those elements:

  • When it was introduced in physics and by whom?
  • Which problem did it solve at the time?
  • When the rigorous mathematical theory was introduced?
  • Did the rigorous definition help derive new physical results or did it just confirm what was already known? Did it prove some results wrong? etc.

Addendum

(*) Someone pointed out in a comment that (as I understand it) the notion of "mathematical concept" is subjective and in a sense a lot of mathematical concepts can be considered to be physical concepts in the first place. Here is my attempt to make things more precise.

I don't want to enter in a philosophical discussion about where the limit between "mathematics" and "physics" lies and whether these concepts make sense in an absolute way. Let's just consider this simplified view:

  • Some people call themselves mathematicians, are interested in defining things precisely and deriving theorems from axioms and logical rules, without making reference to the physical world.
  • Some people call themselves physicists and study the laws of this physical world. To do so, they generally use tools and language considered as part of mathematics.

Nowadays, these two categories of people seem to be distinct and they are part of different categories of the academical organization (it was not always like that).

Some of the tools of mathematics have emerged from pure mathematical thinking (without reference to the physical world) and others from trying to develop models of the physical world.

One interesting thing which I will not develop here is that sometimes tools from the first category have been applied to model the physical world too (e.g. Riemannian Geometry).

I am interested here in tools that went the other way: they were developed to model the physical world, but somehow they ended up being used by the first category of people in a context where no reference to the physical world is made.

More specifically, I am looking for such tools which have been introduced non-rigorously and used by physicist without a precise mathematical formalism for some time before they were made "rigorous" according to the expectations of the first category of people.

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    $\begingroup$ Pretty much every concept of mathematical physics was first introduced non-rigorously and eventually formalized, from velocity to mirror symmetry. You'll have to be more specific. $\endgroup$
    – Conifold
    Commented Apr 6 at 8:53
  • $\begingroup$ @Conifold I am talking about concepts considered as mathematical concepts (e.g. Dirac distribution). I don't consider velocity to qualify as such. $\endgroup$
    – Weier
    Commented Apr 6 at 9:30
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    $\begingroup$ And how are other people supposed to know what you consider "mathematical" unless you explain it beyond a single example? Derivatives came from velocities just as distributions came from δ-functions. Are Hamiltonians "mathematical", or ideal fluids, conservation laws, potentials, quantum fields, energy functionals, optimal control, solitons, strange attractors, or do they only become "mathematical" when mathematicians give them separate names? Because they do not always bother to. $\endgroup$
    – Conifold
    Commented Apr 6 at 9:57
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    $\begingroup$ Also somewhat objecting to the framing: for much of the 18th and 19th centuries, and in many cases well into the 20th, people did not distinguish "math" from "physics" (except maybe for "experimental physics"...?), and notions of "rigor" were not at all the mid-to-late 20th century version. $\endgroup$ Commented Apr 6 at 16:07
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    $\begingroup$ The problem is that the "two categories of people" constantly use each others' concepts and often under the same names. The academic separation of disciplines does not translate into conceptual separation that your question presupposes even today, let alone historically. One thing you can do to save it is explicitly name some "purely mathematical" field (a single one, or the question will be too broad), maybe functional analysis, algebraic geometry or algebraic topology, and ask if initially informal concepts from physics were formalized in it and came to be used there internally. $\endgroup$
    – Conifold
    Commented Apr 7 at 7:03

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Ok, at least as a place-holder:

Much of Euler's work. :)

George Green's 1828 idea now called "Green's function".

The "Sturm-Liouville equation" thing, c. 1830, was very good, but was not made rigorous until the 1890's, by Steklov and others. (But S-and-L "knew" what should happen.)

Heaviside's work 1890's that included what is now called "Dirac delta function" (but/and gave good guidance to the transatlantic telegraph cable, and other things).

Early work on "quantum mechanics", which needed unbounded and not-commuting operators on a Hilbert space... (c. 1930 made rigorous in various ways by Stone and von Neumann...)

The representation theory of specific small Lie groups, by Wigner and Bargmann, for physics purposes... showed that the by-accident-toooo-general representation theory ideas of "pure mathematicians" were too naive... but, yes, at the same time, their mathematics did need some "bolstering" :)

Mirror symmetry!?!

Endless ideas of Ed Witten, though, well, maybe, as a Fields Medalist, there's some "resonance" here. :)

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    $\begingroup$ One can argue whether Euler can be labeled as "physicist". The case of Leibniz is even weaker. $\endgroup$ Commented Apr 7 at 12:23
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    $\begingroup$ @AlexandreEremenko, I was thinking of Euler's "Mechanics"... some of which seems to me to involve implicit-and-reasonable assumptions about smoothness, completeness, and so on. Of course, Lagrange did both mechanics and number theory, and then there's Hamilton... $\endgroup$ Commented Apr 7 at 16:23
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    $\begingroup$ Most of mathematicians before 19 century can be also called physicists. $\endgroup$ Commented Apr 8 at 1:57
  • $\begingroup$ @AlexandreEremenko, indeed! :) $\endgroup$ Commented Apr 8 at 18:28
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    $\begingroup$ Indeed, as in L. Bernstein's "Westside Story", there are "gangs" who live in the same neighborhood, but/and want to have "naming rights" for the back alleys, and so on. If only contemporary mathematicians and physicists could get a choreographer to help them work out some details. :) $\endgroup$ Commented Apr 8 at 22:27
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The prime example is Newton's Calculus. The notions of limit, convergent series, derivative and integral were made rigorous (to our modern standard) only in 19th century.

In Newton's time, even the concept of a real number was not rigorous, and one can argue whether real numbers were introduced by physicists or mathematicians.

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    $\begingroup$ "the concept of a real number was not rigorous": this is a popular (perhaps even dominant) ideology, but what basis is there for claiming such a thing, seeing that Stevin already envisioned representing every number (rational or not) by an unending decimal? Stevin was before Newton. I understand that people are naturally excited about Cantor's and Dedekind's constructions of R, but they are more abstract glosses on the same old unending decimals. One can argue whether unending decimals are practical when actually proving theorems about the reals, but it is indisputably a "rigorous" approach. $\endgroup$ Commented Apr 16 at 11:10
  • $\begingroup$ If you read Newton's Principia, you may notice that he uses systematically the ancient Greek notion of proportion, which was the ancient substitute of real numbers. He never mentions "infinite decimals". But the theory of proportions is not exactly the same as the modern theory of real numbers. $\endgroup$ Commented Apr 16 at 13:07
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    $\begingroup$ Newton's Principia also tends to avoid using the calculus even though Newton had already invented it by then :-) The fact that it is an old-fashioned text does not change the fact that Stevin's ideas were widely accepted. As far as Leibniz is concerned, I don't recall seeing him emphasize the theory of proportions when dealing with quantities. $\endgroup$ Commented Apr 16 at 17:45
  • $\begingroup$ With these remark I agree. $\endgroup$ Commented Apr 16 at 18:46
  • $\begingroup$ Then you agree that there did exist a reasonable theory of what a real number is in Newton's time? Note that Stevin was many decades before Newton. If so, you should correct your answer accordingly. $\endgroup$ Commented Apr 16 at 18:57
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Leibniz applied his infinitesimal calculus to solve numerous problems that today would be considered elementary physics. Several such applications are analyzed by McDonough in his book

Jeffrey K. McDonough, A Miracle Creed: The Principle of Optimality in Leibniz's Physics and Philosophy, Oxford University Press, 2022

Chapter 1 is on Leibniz's derivation of the laws of reflection and refraction by applying a minimization principle to derive the easiest path; Chapter 2 is on his provision of a derivation of the correct formula for the fracture strength of a beam to match Edme Mariotte's experimental results; Chapter 3 is on his promotion of the conservation of vis viva; Chapter 4 is on the catenary or hanging chain problem; and Chapter 5 is on Johann Bernoulli's problem of the brachistochrone, that is, the path of most rapid descent. Each of these studies was published by Leibniz as an article in the Acta eruditorum of Leipzig.

Leibniz's infinitesimals were formalized centuries later in Abraham Robinson's nonstandard analysis:

Robinson, Abraham Non-standard analysis. North-Holland Publishing Co., Amsterdam, 1966.

Unending decimals for representing every number (rational or not) were popularized already by Simon Stevin decades before either Leibniz or Newton.

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    $\begingroup$ Leading to this there is also Galileo's indivisibles $\endgroup$
    – Mauricio
    Commented Apr 9 at 13:06
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    $\begingroup$ @Mauricio, the issue is not merely indivibles and infinitesimals (which are not the same thing), but rather that Leibniz was able to turn infinitesimals into a powerful tool known as the infinitesimal calculus, applicable in particular in physics. Galileo did not have such a tool. $\endgroup$ Commented Apr 9 at 13:13
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    $\begingroup$ Isn't one the precursor of the other? Also this is about physics concepts that were made rigorous later, one could say that Galileo (and others) came up with it and Leibniz made it rigorous. $\endgroup$
    – Mauricio
    Commented Apr 10 at 13:38
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    $\begingroup$ @Mauricio, one could look at it that way, but note that Kepler used infinitesimals even earlier. He used them to calculate volume of wine barrels but I am not sure if this counts as a physics application :-) $\endgroup$ Commented Apr 10 at 13:43
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Wavelets were first studied by physicists in the 1970s. Their study by mathematicians took off in the 1980s.

Some work decades earlier by Haar, a mathematician, can be viewed as belonging to the study of wavelets, but that is more like pre-history since it was linked to wavelets only looking back.

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I am not sure about the following, but maybe Gibbs measures fit the OP's question, as they are a generalization of Gibbs' statistical ensembles, such as the canonical ensemble of statistical physics, to infinite systems.

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