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I know that it used to be said, in praise by some and as criticism by others, that Number Theory had no applications. Now it is used in cryptography and Quantum Theory.

Since the mathematics that explain physical systems must be invented before those systems can be so described, the invention/discovery of maths must always precede its utility and therefore the newest math will always be without an application.

However, are there still any major genres of mathematics that lack any applications? If so, what are they?

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    $\begingroup$ Every branch of mathematics is applied to solve problems within the branch. Are you using "applications" as code for "physical applications"? You mentioned cryptography, so I'm guess you're allowing technological/industrial applications as well? $\endgroup$ – rschwieb Dec 30 '13 at 14:01
  • $\begingroup$ Applications as in useful for something besides the development and advancement of pure logic and/or philosophy. I was mostly thinking of science and technology, though something like economics could also suffice. $\endgroup$ – William Muenzinger Dec 30 '13 at 14:31
  • $\begingroup$ Since the mathematics that explain physical systems must be invented before those systems can be so described ... So, you don't believe what Feinman said ... if group theory had not been invented already when physics needed it, then a couple of physicists would have taken a week off and developed group theory, then progress in physics would have continued... $\endgroup$ – GEdgar Dec 30 '13 at 14:47
  • $\begingroup$ That the motivation for developing calculus was physics does not change that you couldn't use calculus for physics until you had calculus. It would be like using a hammer before you created the hammer. I admit that I am not firm on the point of whether creation and use could be concurrent, but use certainly cannot precede creation (though it may precede rigor). $\endgroup$ – William Muenzinger Dec 30 '13 at 15:25
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    $\begingroup$ @GEdgar If non euclidean geometry did not exist in the years 1900, could the relativity been invented anyway ? I doubt it, but I would be happy too if you can explain the opposite. $\endgroup$ – Xoff Dec 30 '13 at 15:59
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Whereas most mathematics did have direct applications in the past (geometry and calculus), now there are branches that have applications on other branches of mathematics. I think about logic, model theory, set theory or computability. Most of this branches have applications and consequences on other mathematics, but not really or directly on any "physical problems".

For example, Continuum hypothesis or the fact that there are non comparable Turing degrees have no direct consequences on physical problem but they speak (loud) about what are the mathematics themselves.

I think this is the revolution of the previous century : Mathematics became a science.

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