We don’t have much experimental data in theoretical physics, so perhaps math is acting as the driving force.
Perhaps, it is just math driving math?
Has AdS/CFT made any contact with reality in strongly coupled condensed matter systems?
Some contact is being made in the theory of cuprates. In general, this approach helps one organize the ideas but we are still far from a quantitative understanding.
Story about Friedan (?) getting into physics by being impressed by Witten’s work on knot invariants. A correspondence between the two followed which lead to a fruitful collaboration. The story illustrates the usefulness of communication between mathematicians and physicists.
At the beginning of the twnetieth century, physics and math were in a happy marriage, which lead to a divorce in the forties due to quantum field theory. Recent decades have seen some reconcilliation, also due to quantum field theory.
Math has given more to physics: We learned a lot from applying mathematical techniques to string theory and quantum gravity.
It is a symbiotic relation. Physicists have learned a lot from mathematicians and mathematicians have learned a lot from physicists.
It feels there is a language barrier sometimes, but we can learn a lot from each other.
Inspiration can go both ways. Sometimes physics problems can give the spark to mathematicians and vice versa.
The question seems to be aiming at the following. In physics, we use simplicity and elegance when we don't have experiment to guide us. Sometimes there is a physical phenomenon that we don't understand and then we need new mathematical tools to understand the physics. In this paradigm, the mathematics serves the physics as a tool or language. The mathematical thinking is not a driving force of a new physical picture.
The question could also be the following. How does existing mathematics shape the physics that we do? Physics is shaped by how much is defined by mathematics. For example, in quantum information, traditional math is not so proper. You need new math and new languages. We wouldn't have quantum computational complexity theory unless we had quantum computing. Similarly, category theory has existed for a long time, but the category physicists use is quite recent. And there's feedback. Physicists are interested in symmetry so we get a category theory for symmetric manifolds. Physicists are interested in Fermions so we get a category theory for Fermions.
Quantum theory has mostly been focussed on 2 or 3 systems. So we have not looked at what happens in the limit of many systems. And there the math is not there. Also it's not clear what the questions for large systems even are.
For the most part physicists are using older math. There is also a fundamental distinction between how a physicists and mathematicians view mathematics. The former are principally concerned with practicality, what is it good for. The latter care more about posterity, have a correctly proved a result.
Some areas of physics have decoupled from data and therefore are principally driven by mathematics.
There is a risk of mistaking the mathematical efficacy of a construct for physical reality. The fact that one has a mathematical formalism to describe something doesn’t make it real.
Who uses math from the last decade? Is it interesting mathematics to mathematicians? Are string theorists doing mathematics?
Is the distinction sociological? Physics is what physics graduate students are useful for, and likewise for math. I.e. fields are defined by the different toolboxes that scientists use, not by the specific question they tackle.
Consider Arnold, using physicist-style thinking for math.
Mathematicians later polish some rough estimates that physicists have done.
Delta function example. And what I say about GR/PDEs etc.
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