How important are proofs in physics? If something is mathematically proven to follow from something we know is true, does it still require experimental verification? Are there examples of things that have been mathematically proven to some reasonable degree of rigor (eg satisfy a mathematician) that turned out to be false based on experiment?
Mathematical proofs relate to exactly how a MODEL will behave. They don't have much to do with how the real world behaves. If the mathematics is carried out correctly, then one has "proven" how the model will behave.
The reason for experimentation, is to find out if the completely fictional MODEL that somebody simply made up, behaves in any way the same, as observations suggest the real world behaves. If it doesn't, that is not an indication of a mathematical error, it simply means the fictional model is not a good description of the real world, and must be changed.
Mathematical proof is to physics roughly what syllogism (or some other fundamental inference rule) is to logic. Namely, it begins from assumptions modelling our conception of some physical reality and shows what must be so if the assumptions hold, but it cannot say anything about the underlying assumptions themselves. A simple example was given by dmckee in his comment:
If a mathematical proof disagrees with experiment it doesn't say anything about the proof or the physics. It says that the physics is not well modeled by that math. If you proved a fact about vector fields and the physical behavior differs from the prediction then your physical object is not a vector field.
One must always test results from physics "mathematical proofs" with experiment. Indeed it could be argued that the building of such mathematical proofs is the main job done by theoretical physicists, and the sole reason for building them is to discover what the theory in question foretells that is falsifiable (see Wiki page on falsifiability). There are two "experiments" that need to be done on such a "proof":
- I think of the peer review process as well as a history of independent testing of a theoretical argument as an important "experiment" to test the mathematical soundness of the inferences made in such "proofs". Many theoretical musings nowadays rely on many mathematical results at once and mathematical proofs themselves (witness, for example, Wiles's Theorem) can be the size (in bits, measured, say by Mizar or MetaMath encodings of such proofs) of small to medium software libraries. As such, they are not too far behind, complexity-wise, things like operating systems and telecommunication network control software that rank as the highest complexity objects ever constructed by humans (see footnote). If the "proof" fails this experimental test, it is by definition, NOT a mathematical proof.
- Once the proof is deemed sound by peer review and independent reproduction, the experimental results following from the proof's conclusion must the reproduced in experiment.
So, in short, the process of experimental verification is what sets mathematics and physics apart.
Notice that even if the proof "fails" in step 2, it has been invaluable to physics because it then becomes a proof by contradiction, i.e. it tells us that one or more assumptions underlying the proof must be wrong and therefore our ideas about what really is going on in experiment need to be revised. The proof itself if shown to be such by step 1 above cannot be "wrong".
To answer your question:
Are there examples of things that have been mathematically proven to some reasonable degree of rigor (eg satisfy a mathematician) that turned out to be false based on experiment?
As discussed in step 1. above, there are no "mathematical proofs" in physics which are wrong. If their foretellings and experimental results gainsay one another, then the assumptions are wrong. However, here are two famous examples of how proof "failure" and physics interact:
- For an example of something that eventually fails step 1. above, think of Euclid's parallel postulate (see Wiki page of this name, particularly under the headin "history"). Many mathematicians came up with "proofs" that this postulate followed from Euclid's other postulates, because it seemed to them that the parallel postulate was not self-evident and should follow from the others. Sometimes the proofs concerned were accepted by the scientific community as sound for a long time until someone found the flaw in them. After Lobachevsky showed the hyperbolic geometry was a sound axiom system that fulfilled all Euclid's postulates but NOT the parallel one, nineteenth century mathematicians, such as Gauss, Riemann and Clifford, took the independence of the parallel postulate so seriously that they thought of it as being theoretical physics as well as mathematics, i.e. that it was like a proof passing step 1. but failing step 2. i.e. it challenged the notion that Euclid modelled our physical World. Gauss even made sensitive surveys to check that the angles in big triangles experimentally summed to half a turn. Nowadays not only is General Relativity with its non-Euclidean, generally curved geometry mainstream physics, but non-Euclidean geometrical interpretations of gauge theories together with the panoply of techniques the language of differential geometry brings with it are part of basic theoretical physics;
- A good example of the subtle interplay between mathematical reasoning and experimental testing is the history of the idea of local realism and quantum mechanics, as tested by von Neumann's argument against the idea of hidden variables and the EPR paradox leading onto the work of John Bell and the experimentally tested violations of his inequality, therefore making the idea of local realism harder and harder to uphold. All of these ideas involved logical and mathematical reasoning to foretell experimental results that people had formerly thought to be absurd, and therefore greatly advanced our understanding of quantum physics. See the following Wikipedia pages for a summary: Bell's Theorem, John Stewart Bell, the Einstein-Podolsky-Rosen Paradox as well as the paper David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", Physics Today, April 1985.
Mathematics through all these processes helps us clarify the fine minutiae of meanings present in our sometimes taken for granted physical assumptions.
Lastly of course, Mathematical Proof can be seen as a kind of "investment adviser": it tells us where to put our hardest work and other resources in experiment. Unless you have reason to finely question a physical assumption A that seems already backed up by experiment, an experiment that tests the logical outcomes of combining assumption A and assumption B by mathematical reasoning is a much better use of time and work than a foretold result which can be shown to be logically equivalent to the already experimentally supported assumption A.
Footnote: I foresee automated proof development and checking by things like software proof systems as important to physics in the future. As I understand it, many parts of String Theory suffer from this kind of problem, that they are hard to review by one or few reviewers alone. Fortunately although proof development systems are themselves astoundingly complex, the proof checking software itself is a simple parser that can be reduced to one or two pages of code and thus can be thoroughly experimentally debugged, so it doesn't matter how proofs are constructed as long as they are deemed valid by the parser.