TITLE: The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?
QUESTION [51 upvotes]: Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about  "The Unreasonable Effectiveness of Physics in Mathematics".   
What can be the reasons for it ? 
Do physicists have some tools/ideas/techniques which allow them to make insights, which
are not seen for mathematicians? Or it is just because Witten&K are very ... very smart ?
If, yes, what are these tools/ideas ? How to learn/absorb/(put into math. framework) them? 
What can be the further applications of these ideas ? 

Being a mathematician, but working in physicists surrounding for many years,
I have thought on this questions for quite a while. The recent MO question 
Mathematician trying to learn string theory prompts me to ask it here.
I would think, that yes, there are such "ideas". But  from some outstanding mathematicians I've heard an opposite opinion.
My vague feeling is that quantum field theory and string theory it is something like
an analysis/differential geometry on infinite-dimensional manifolds. But these manifolds are not abstract, say Banach modeled manifolds, which theory is not so rich, but kind of maps from one finite-dim. manifold to another, which has certain specific structures which are not fully revealed by mathematicians. E.g. vertex operator algebras, arise from maps of circle to manifold, if we map not circle but something higher dimensional there should be something more complicated. Another issue is about Feynman integral, which
 allows physicist to use integration techniques in geometric problems, it is not well-defined mathematically, but it might be it cannot be defined in very general form of infinite-dimensional integrals, but again physicist have an intuition where it can be defined, where cannot, and proper mathematical theory should first clarify the setup
where it exists, rather than trying to build general theory which might not exist.
These words are probably very vague, so might be answers help to me clarify.

I think everybody knows the influence of physics happened from 80-ies, but
for completeness  let me mention just a few.
Donaldson used instanton moduli spaces in his study of 4-folds.
Faddeev, Drinfeld et. al. created quantum groups
Representation theory of infinite-dimensional algebras have been large influenced by conformal field theory developments.
Witten's contributions are numerous his Fields Medal says more than I can say.
Mirror Symmetry, quantum cohomology etc...
The works of Fields Medalist Kontsevich and Okounkov are largely influenced by physics.
So on an so forth...

REPLY [4 votes]: If you accept Max Tegmark's view that "our physical world is an abstract mathematical structure" (per https://arxiv.org/abs/0704.0646), then it would follow that the "effectiveness" of physics in mathematics is not unreasonable at all. It's exactly what you would expect.

REPLY [2 votes]: It is my experience that physics motivates and allows the consideration of objects, with some degree of rigor, whose nature would be very difficult to understand mathematically with no motivation and prior training.  
Things like manifolds and line integrals are early examples of objects whose involvement in physical theories allowed me to understand their workings through context long before I had developed sufficient mathematical language and machinery to really articulate clearly what it was I understood. 
I spent the last semester and the beginning of this summer constructing $\mathbb{R}$ (and larger real-closed fields up to the Surreals) directly out of cuts in a subclass of $O_n\times O_n\times O_n\times O_n~\;$  using set theoretical axioms (https://arxiv.org/abs/1706.08908*). Two of my math professors were reviewing the paper with me once a week, an analyst and an algebraist by training, and both of them repeatedly wondered what made me think of trying something like this.  I was motivated by the fact that infinities seem to be inextricably working their way into the leading edge of problems in physics -- for example, it is my understanding that one of the disagreements between GR and QM is the existence some nasty divergent series describing a particles position and momentum states in a quantized setting, something that is typically handled through renormalization but which can't be here.
I figured that maybe these series would converge to some value in a larger, denser real-closed field, and I wanted to build those fields in the same way that $\mathbb{R}$ can be built out of $\mathbb{N}$ since so much of quantum mechanics relies on analysis together with indexing over some well defined discrete ordered subset. To do so I constructed larger discrete ordered rings that contained the integers out of $\delta$-numbers greater than $\omega$, then created fields of fractions for them and 'cut up' the fields to real-close them. I also algebraically closed all of these fields so we have an analogous 'non-Archimedean' version of $\mathbb{C}$ to work with. All of this was motivated by happenings in physics, or as another answer elegantly put it 'considerations of Nature'.
*currently under review for publication