# Has physics ever given a physical significance to a mathematically abstract idea?

Consider a fundamental concept in maths that was created to 'solve' a problem that simply couldn't be solved by any other approach (or maybe for some other reason). Now let's assume that this concept made no sense to the contemporary mathematicians whatsoever that they simply brushed it aside as a purely mathematical tool with $$0$$ physical significance. like the square root of $$-1 ,$$ $$i$$ or the Euler number maybe?. Anything that makes no sense to a present-day mathematical outsider really.

A couple of decades later after its birth, baam, a scientist realises that the only way to solve a paradox in physics was to use this very mathematical concept that had to seem to be abstract at the time. Is there any good example of this in history?

Inspiration

This is with respect to the idea of maths being seen as a subject with no real-life application. I wanted to disprove by saying that many mathematical concepts do have a real-life application (if we look back at the history), but it's just that we haven't found any use so far.

I am slightly aware that e and complex numbers like $$i$$ makes it into the world of electrical engineering though I don't remember where exactly. I also vaguely recollect topology space and an array of mathematical concepts coming into the world by the theory of general relativity and quantum mechanics. What I am really looking for is

(1) a concept in maths that initially seems to have had no use
(2) 70 years later it is been used in a formula (please do add the formula too; that's really the reason I came here)

• @MarkSapir: More precisely, Schubert calculus, which was a predecessor of GW invariants (by about 80-90 years). Jun 7, 2021 at 19:45
• George Boole invented an algebra for logic that is now the foundation of our modern world (e.g. the computer I am entering this on, etc). en.wikiquote.org/wiki/George_Boole Jun 8, 2021 at 12:05
• Since you asked... Complex numbers are used in Fourier analysis. They greatly simplify analyzing the frequency domain by representing it has having two signals, a real part and an imaginary part. In electrical engineering, the imaginary part is typically represented by j (and not by the i you learned in high school). The first link I found: dspguide.com/ch30.htm Jun 8, 2021 at 21:53
• You brought up imaginary numbers, but that itself is a good example isn't it? It started as a mathematical abstraction, but in QM it's (arguably) a concept of fundamental importance. (Arguably because I think some people think there are ways of making the same computations without imaginary numbers).
– TKoL
Jun 9, 2021 at 13:05
• @AdilMohammed I'm no expert, but the example I know of in QM that uses imaginary numbers is, I believe, how the amplitude changes as a photon gets bounced off a silver / half-silvered mirror (or beam splitter).
– TKoL
Jun 10, 2021 at 12:59

Physics cannot help giving physical significance to things. But, yes, the first item on your list should have been

Developed as a mathematical "sudoku" game generalizing rotations, in the late 19th century, by Sophus Lie, Felix Klein, Friedrich Engel, Henri Poincare, and, as associated applied mathematical structures by James Joseph Sylvester and Arthur Cayley, they were recognized by physicists in the early 20th century to apply to the new-fangled science of Quantum mechanics, the actual cornerstone of modern physics, by Eugene Wigner and Hermann Weyl, firstly through the rotation group SO(3) ~ SU(2).

In the 1960s, Murray Gell-Mann appreciated that SU(3), both classified and further described, in fact, elementary particle interactions, the Eightfold Way; whence the full-fledged Lie Group theory entered as the mainstay of Quantum Field Theory describing fundamental particles and forces in all of High Energy physics. Now that's an example of abstract math underlying all of nature at its most fundamental, a century before that was recognized!

Today, hardly anything describing electroweak interactions, strong interactions, or gravity is thinkable without crucial and irreplaceable reliance on Lie Group representation theory. The fundamental laws of nature run on Lie groups. Theoretical HEP theorists are part-time Lie-Group applied mathematicians. This is the sort of thing they have to master in school.

• Well, not first in historical order! :-) . Jun 9, 2021 at 18:31
• It might be though that the assumed mathematics of symmetry breaking is not there (in the field of high energies). It might turn out that the weak force is a residual force. And the Higg field a normal field (with no associated false vacuum), thereby not providing a mass for all massive particles. It might be that there are more fundamental massless particles, giving rise to massive particles (because of their interaction in a bound state) and a residual weak force. Eventhough the math of the electroweak unification is impressive, this doesn't mean that it corresponds to reality. Jun 11, 2021 at 6:00
• This is not to say that "Lie math" is of great value. I'm not sure if you can say that high-energy stuff has an inherent mathematical structure though. The stuff it real, but the math? In general relativity, spacetime has mathematical properties and math structures can be imposed, but are they real existing in spacetime? The spacetime is real (as elementary particles), but is the math inherent to (for?) spacetime? Jun 11, 2021 at 6:05

fractal

The so-called Cantor set was described by
Georg Cantor, 1884 (or H. J. S. Smith, 1875?)
Sets with "fractional" dimension were described by
Felix Hausdorff, 1918
Investigated thoroughly by
A. S. Besicovitch, 1930s - 1950s
But these were only mathematical abstractions.

Benoit B. Mandelbrot, 1960s and later
claimed relevance of these sets in natural sciences.
Mandelbrot reported that, in the early days, when he would submit a paper to a physics journal with a Cantor set in it, that it would be rejected out-of-hand as "unphysical".

Largely due to Mandelbrot's efforts, that situation has been completely reversed.
In 1992, I counted that physics papers with the word "fractal" in the title were appearing more frequently than once a week.

The following quote is from C. N. Yang, delivered at a 1979 symposium dedicated to the geometer Chern: "When I met Chern, I told him that I finally understood the beauty of the theory of fibre bundles and the elegant Chern​–Weil theorem. I was struck that gauge fields, in particular connections on fibre bundles, were studied by mathematicians without any appeal to physical realities. I added that it is mysterious and incomprehensible how you mathematicians could think this up out of nothing. To this Chern immediately objected. 'No, no, this concept is not invented — it is natural and real.'"

• That is true. Also the concept of "reality" is one of the most difficult concepts in philosophy. Jun 8, 2021 at 5:17

Abelian and non-abelian group theory -> quantum chromodynamics

Noneuclidean geometry -> general relativity

Sorry, I cannot write you any equations as examples.

I don't know if this can be counted for as physics, but to my knowledge the radon transformation was mostly something mathematicians thought about without any application. Now, it is widely used (and necessary) to transform images in tomography.

1917 - Radon transform introduced by Johann Radon

1970s - First Computer Tomograph Scan

There are many examples of a physical significance given to abstract mathematical ideas.
One of the most prominent is the spacetime as introduced in general relativity. In general relativity a mathematical framework is given that informs you about the structure of space. It even relates space with time, thereby introducing new physical stuff, i.e., spacetime. The theory is inspired by real physical things (space and time) but the extra quality that came up was a connection between those two, given by the math. Matter and energy already existed before the theory but general relativity proposed a connection between these and the new kind of spacetime that it introduced. This connection was absent in the earlier theory of Newton who said that space and time are not connected and are not influenced by the matter and energy that is contained in space.
New things like spacetime curvature, an absolute limit on speed (which actually gave rise to the concept of relativity), the equivalence of mass and energy, etc., were introduced as mathematical expressions, changing the physical aspects of the universe.
Einstein, for formulating his ideas, made use of in his time already existing mathematical ideas introduced by mathematicians like Riemann, Ricci, Levi-Civita, and Killing.
New discoveries were made, inspired by Einstein's ideas. Though physical things like space and time already existed, the mathematics of Riemannian manifolds was used to make a connection between them, thereby postulating the new physical entity, i.e., spacetime.
So, a physical significance was indeed given to the ideas of real manifolds and tensor calculus.

Riemann geometry: Einstein had to look around for the mathematics that could describe General Relativity. Before that nobody would have guessed that curved geometry has anything to do with space-time and gravity.

I think one of the best example is the binary system, the representing of numbers in 0 and 1 by diving them by 2. This is how every processor, ram, HDD, SSD,.. etc keep data and how digital communications work. Also the whole Boole algebra is the basics of how computer processors calculate numbers.

In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text. Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature".

https://en.wikipedia.org/wiki/Binary_number

• See also George Boole's contribution with Boolean Algebra, and its notation. Jun 10, 2021 at 16:35

Spinors were introduced by E.Cartan and found applications in quantum theory 15 years later.

The most striking examples are the special and general relativity theories. Einstein gave physical meaning to abstract ideas of Riemann, Poincare, Lorenz, Lie and others.

Not really physics, but: I think that number theory (edit: specifically research on primes) was an entirely idle mathematical pastime until cryptography became a thing.

• That is not true, en.wikipedia.org/wiki/Number_theory#Applications Jun 8, 2021 at 17:33
• @MarkSapir You mean "Thank God that number theory is unsullied by any application"? OK; that changed in the middle of the 20th century, so maybe before cryptography but with the advent of computers. And computer based cryptography is still one of the main applications. Jun 8, 2021 at 20:50
• Physicists would disagree. Antient astronomers would be insulted too. You are safe insulting them now: they are long dead. Jun 8, 2021 at 21:31
• Yes, crypto is a thing. But/and the modern (post 1945?) significance of representation theory and functional analysis in the theory of automorphic forms, etc., has been intertwined-with, and both motivated by and motivating, various parts of physics. A big story... Jun 8, 2021 at 21:37
• Cryptography doesn't have physical reality in terms of describing physical systems. Information theory in general has connections to thermodynamics, but your answer is going to need to be more specific to work as an answer to the question of "physical significance". That's different from "practical applications". Jun 10, 2021 at 1:03