Can pure maths create new theories in physics or does the "idea" ALWAYS come before the math? I am in a debate with a friend about the value of string theory in physics. He is concerned that we are wasting valuable intellectual and financial resources on a path that is fanciful and can't ever hope to be verified by experiment and evidence (11 - 20 dimensions etc). 
My point is not to agree with string theory but to argue that maths is powerful and capable of produce new ideas that can be verified. 
His question to me is
"Can maths provide new and viable ideas without an original idea based on the natural world or through observation"
I can't think of any good examples where the study of maths has resulted in new theory about physics. Can anyone?
 A: Well, it often leads both ways. With Ed Witten, physics has led to new math and a Fields medal for him. And the other way, Galois and group theory has led to all sorts of goodies in physics with gauge theory etc. I had voted to close, but am now apologizing :) Recently, Andrew Hodges, an Oxford mathematician and writer of an excellent biography of Alan Turing, called Enigma wrote a paper with Nima Arkani-Hamed--Twistors, and Alain Connes, another Fields medal mathematician, has been using non-commutative geometry for interesting speculations on physics (even if he has admitted problems with the theory---great man.)
Then, there is Mad Max Tegmark and his theory that all mathematical stuctures have a physical reality which is the ultimate in Platonism :)
A: Physics without math is blind and math without physics is lame. 
Mostly mathematics and physics goes hand in hand. 


*

*Group theory 

*Tensor calculus

*Stokes theorem in Vector calculus 


These mathematical theories came first and then utilized later on by physicists. 
A: Perhaps the most drastic mathematical concept that was developed without any reference to physics is the idea of imaginary numbers, which came to their full glory in physics only through quantum mechanics. This is one of the reasons why Feynman called the Euler identity $e^{i\pi}+1=0$ "the most remarkable formula in mathematics". 
See also Feynman's popular book "QED: The Strange Theory of Light and Matter", which places emphasis on complex numbers and their physical importance in quantum mechanics.
A more recent example would be the Atiyah-Singer index theorem, which started out as pure math and now is a valuable tool in physics (see also the question Where is the Atiyah-Singer index theorem used in physics?).
A: The question is where you draw the line separating the "original idea about the natural world" from ensuing mathematics.
But I think a particularly good example is non-euclidean geometry and Gauss: After Gauss understood that the axiom of paralles is indeed an axiom that can be replaced by other axioms, and does not follow from the other axioms, he tried to measure the angles of large triangles during is work on land surveying. He wanted to find out if physical space is Euclidean or if it is not, motivated by the pure mathematical insight that other geometries are possible.
From our viewpoint the question was incomplete, of course, Riemann and Gauss should have thought about the geometry of spacetime instead of that of space only, but I think it is safe to say that these purely mathematically motivated line of reasoning paved the path for Einstein.
Another example is Maxwell's prediction of electromagnetic waves and their velocity, which he got from analyzing his equations, but in this case you could of course argue that his equations were extracted from observational research done by Faraday and others.
A: The successful historical relationship between Mathematics and Physics seems to be when there is some pre-existing Mathematical Theory M (e.g. Group Theory, Topology, Non-Euclidean Geometry - or special cases of these) and an evolving Physical theory P. P might be expressed in a different kind of mathematics, if at all. Then a mapping is found which maps P onto M, except that M has more equations or components than this mapping includes. So the question becomes:
"Do the missing components of M map onto some (so far) unobserved properties of P?"
When this is successful (as in many of the other examples cited) then historians say that mathematics has proven valuable for Physics (yet again).
Another example of this phenomenon might be the SU(3) classification of some particles which had a gap in the representation when mapped onto known particles; the gap mapped onto the $\Omega$ particle. Didnt Gell-Mann get a Nobel prize for that one?
String Theory seems a little bit different from this classic scenario (but it is not the only such) where there is a deliberate attempt to develop Mathematics to model known (and perhaps unknown) Physics. This might be seen as a form of anticipation.
Some mathematicians take a stronger view than the account given here, in that they believe that the Physical Universe is fundamentally mathematical in character. Often, like Penrose, they may have specific types of mathematics in mind with that claim. So from this perspective the development of that mathematics is valuable over and above any current experimental data. A similar belief seems to underlay the String Theory efforts.
A: I think this question is too philosophical and I am not sure whether it is within the scope of this site or not. However, I think the question you have asked is very important and deserve serious discussion.
Traditionally mathematics always played the role of a tool. Since nature is written in the language of mathematics any physical idea found its natural expression in it. It has also realized that no pure idea (pure mathematical ideas in case of physics), however beautiful, by itself can lead to any real physical truth. One needs to have empirical knowledge about the world. Science should start with observation, go with it and ultimately suggest/lead us to new realm of experiences. Since a scientific theory is essentially a guess, experimental verification should always be the ultimate judge of a scientific theory. That's how science has progressed always.
However, one can not help observing a key feature about the relationship between mathematics and physics. The more physics has advanced the possible scope of mathematical manipulation became ever more constrained. Theories became evermore rigid and unique. You just try to tinker with a minor element of a theory and the whole structure immediately collapses. This has brought a new insight. The insight that in an advanced stage of science one can rely with far more confidence on the purely formal nature of inquiry than when it was primitive. Mathematical consistency may itself become a powerful guide in the search of the laws of nature. Experimental verification of ideas are still the ultimate judge but mathematics may play the role of a very strict judge on initial validation checks of physical ideas.
A: Well, it often leads both ways. With Ed Witten, physics has led to new math and a Fields medal for him. And the other way, Galois and group theory has led to all sorts of goodies in physics with gauge theory etc. I had voted to close, but am now apologizing :) Recently, Andrew Hodges, an Oxford mathematician and writer of an excellent biography of Alan Turing, called Enigma wrote a paper with Nima Arkani-Hamed--Twistors, and Alain Connes, another Fields medal mathematician, has been using non-commutative geometry for interesting speculations on physics (even if he has admitted problems with the theory---great man.)
Then, there is Mad Max Tegmark and his theory that all mathematical stuctures have a physical reality which is the ultimate in Platonism :)
A: It is implicit in the wording of the question that string theory is an example of math coming first, but this is false. String theory grew out of Regge theory which was and is a phenomenological theory of the strong interactions which applies to scattering processes at high energies but small momentum transfer. This in turn is connected to the Regge trajectories observed in the spectrum of QCD, that is mesons and baryons tend to lie on straight lines in a plot of angular momentum $J$ vs mass squared $M^2$. This is the same as the spectrum of a relativistic string. This is where string theory came from, a phenomenological model of the strong interactions, not from math.
A: Speaking of Dirac, he also came up with the Dirac equation, creation and annihilation operators, magnetic monopoles, and the theory of constraints, in particular second class constraints and the Dirac bracket.
A: I'll  add the quote to the noise.

The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.

David Deutsch. "The Fabric of Reality"
I've brought that up because Deutsch argues in his book that brain is a computer (and physical object). And, by doing any kind of math, we are studying a kind of "virtual reality" that we created by our brain. Therefore, he argues, idea that math is somehow "disconnected" from a physical reality is naive and there is no such separation at all.
A: This question is the analogous of : Is Romeo and Juliet the result of grammar and syntax or is there an external input necessary? In my opinion mathematics as far as physics goes, is a tool. A beautiful tool, tools are very important for creating stuff, but still physics is a meta level to mathematics. Mathematics is necessary for rigorous physics but not sufficient.
There is of course the school of Pythagoras, "everything is the music of the spheres" can be replaced by " everything can be described with mathematics",  and in this point of view mathematics comes first. In this view everything exists in potentia as a mathematical concept waiting to be born.
If the Theory of Everything is found, maybe the latter will be true. Until then, I vote that physics uses mathematics as a necessary tool.  I suspect that Godel's theorem ( can there be a TOE?) in some form or another will somehow still set mathematics in a tool position, necessary but not sufficient. 
A: Mathematical develoments can lead to new physics, this is how anti-particles were predicted by Dirac. See also Why beauty is a good guide in physics?
A: Riemann developed Riemannian geometry for purely mathematical, geometrical, logical reasons.  But after he did so, he went on in print to make theoretical physics speculations. (To be specific, that the curvature of space was caused by the matter occupying it.) They were not much understood at the time, but Einstein's work verified those speculations, and also filled them in considerably, giving them greater specificity, extending them to four dimensions instead of three, an indefinite metric instead of a definite one,  and other developments and alterations. Weyl gives an account of this in his book, Space-Time-Matter. 
Although Maxwell used the previous physics of Faraday and others, there was one small part missing which he supplied himself purely by mathematical analogy with their work, after having mathematized their work.  In a small way, then, this is also an example.  But more importantly, although the earlier scientists did indeed have the idea of the field and current, it was Maxwell's examination of the maths that led him to the physical idea of an electromagnetic wave.  This is huge: a wave without anything material which it is a wave of.  (It took a long time for physicists to accept this.)  Our modern physical notion of a wave came from this maths.
Less clear is Hamilton's purely mathematical examination of the relation between geometric optics and the wave theory of light, which he then extended to Newtonian Mechanics.  But I think this counts as well: the mathematical structure of Hamiltonian Mechanics and its duality between wave theory and Newtonian particle theory is quite consciously what led Schroedinger to discover Wave Mechanics (in the Quantum Theory) and for a couple of years no one knew what the physics of the wave function was: they worked with $\psi(x,y,z,t)$ based purely on the Hamiltonian maths, and only later did Born discover the accepted physical meaning of this wave.  So I think this is again an idea of the maths, the wave function, leading to the discovery of new physics later, by Schroedinger and Born.
Even less clear is Hilbert's discovery of linear operators and their spectra.  He named the 'spectrum' of a linear operator the spectrum deliberately because it looked like the atomic spectra then being studied, but put in a footnote that of course this was only a figure of speech, an analogy.  Later, Born (a physicist who was not exactly his student but someone who had worked with him) pointed out to Heisenberg that this was the maths that described Heisenberg's quantum mechanics.  But this is not exactly the physics (in the sense of physics ideas) growing out of the pure maths.  It is rather the mathematicians having invented, in advance, and purely for their own reasons, exactly the maths needed for the physicists to formulate the physical law with.  
This has happened over and over, as pointed out in some of the other posts, but is may not be quite what the OP was asking about, which seems to be whether someone, for purely mathematical reasons, discovers a physical idea, a physics concept.  Hilbert did not have any such, nor did Levi-Civita.  But Riemann did.
A: Maybe a good one for your debate is in this question.
What is interesting about Universal Sequence is that, after some math/nature round trips in Chaos Theory this pattern was found and after that, math inspired what should be looked after in nature. This sequence is a product of repetitive applying output of a simple math function to itself and according to the professor who I learned from, for the first time math inspires experiment.
Personally agree with anna v. What about extensionality when thinking of many world interpretation or string theory.
Math and reality relationship is like when in movies someone goes to police station and an album of criminal is shown to him. If he had seen the criminal in charge he can identify the picture, but if not and yet later somewhere else seeing one of those bad guys, recalls him something suspicious.
A: Math is infinite in size so given any finite number of data points that each is taken to a finite accuracy, there has to be an infinite number of theories that are consistent with the data. Another way of saying the same thing is that there are a lot more beautiful, but incorrect, theories of physics than correct physics.
A: Mathematical ideas serve to do mathematics. Often some mathematical ideas are related to those one uses in physics; then one deals with the "mathematical physics". The true theory that describes some experimental facts should be based on them. In other words, any physical theory should be phenomenological first of all. Otherwise it is a mathematical physics branch.
This question is rather practical. Those who think they are capable of inventing a Theory of Our Everything (TOE) are involved in a mad rush for fame and Nobel prize and they make too many groundless promises. It became difficult to raise a voice of reason. Even clear failures are represented as "achievements" or "insights" now.
