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In this video about the Banach-Tarski paradox the host stipulates that history is full of examples of abstract mathematical theories that were later found to be applicable to the physical world. Is he right?

(I realise this question may not specific enough for this forum but I am dying to learn more about this subject)

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    $\begingroup$ short answer: yes he is. $\endgroup$ – VicAche Feb 15 '16 at 16:34
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Group theory.

A story I heard (perhaps enhanced over the years):

In 1910, Princeton was about to begin a major in physics. The physicists sat down to decide what would be required for those majors to study. As part of the exercise, they went through a list of mathematics courses to determine which would be used in physics. They came to "Group Theory", they all laughed. "There is a subject that will never be used in physics!"

But of course nowadays group theory is important in physics.*

Another story (you can probably find if you look). An interview with physicist Richard Feynman.

(discussion of group theory used in physics)
Interviewer: What would have happened if group theory had not already been developed by mathematicians?
Feynman: Some physicist would have taken a week off to develop group theory, then progress in physics would have continued.

*J. Cornwell, Group Theory in Physics, volumes I and II

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Essentially the same question was asked on Math Overflow: https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653 , and many examples were given. Such examples are really abundant. It is more difficult to name a major mathematical theory which did NOT eventually find an application in the real world.

Philosophers who think that mathematicians do mathematics just for their own entertainment, were always puzzled by this. But "physical world" somewhat restricts the areas of applications. I would say instead "real world". For example, number theory (prime integers etc.) for two and half millenia was the purest part of the pure mathematics, with no apparent relation to the real world. In the end of 20s century important applications to coding were discovered.

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    $\begingroup$ Interesting, but the answers mostly do not mention connection to physical world, only that the theories are useful today. $\endgroup$ – daniel.sedlacek Feb 14 '16 at 14:07
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    $\begingroup$ @daniel.sedlacek: you have not read all answers. Conic sections which describe the orbits of the planets and satellites, do not they belong to physical world? $\endgroup$ – Alexandre Eremenko Feb 14 '16 at 14:28
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    $\begingroup$ @daniel.sedlacek: And how about complex numbers? Does quantum mechanic belong to "physical world" on your opinion? Or non-Euclidean geometry: general relativity is about physical world, is not it? $\endgroup$ – Alexandre Eremenko Feb 14 '16 at 14:56
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    $\begingroup$ my opinion is not important, as far as I know it does (belong to physical world) in the opinion of lead physicists. So how are complex numbers used in quantum mechanics? $\endgroup$ – daniel.sedlacek Mar 2 '16 at 13:14
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    $\begingroup$ Look at the "Schrodinger equation" in Wikipedia, and then take any book on quantum mechanics. $\endgroup$ – Alexandre Eremenko Mar 2 '16 at 14:19
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Perhaps the answer to that is that a physicist will grab the best (mathematical) tool for the job at hand. Sometimes this means the tool is reshaped (and often extended out any recognition) to fit the use better. Other times it is found it doesn't cut it, and is discarded for one custom-built.

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