– what does it mean for a theory to be “proven” mathematically?
In: Mathematics
If you are talking about pure mathematics, a proof is a sequence of logical steps that get you from a starting point which is already known to be true to an ending point that is often suspected or hypothesized about. The issue usually lies in the logic part, it has to be completely iron clad.
So for a famous example was Fermat’s last theorem. If you remember the Pythagorean theorem which says a²+b² = c², there’s a more general question on whether or not you can do this for other exponents like ³ or ⁴ and the rest. The answer is no and a famous mathematician claimed he could prove it hundreds of years ago but never did. It took several major breakthroughs in other fields of math that needed combining to get to the heart of the problem but it was eventually proven that he was right and those equations don’t work. But not without controversy, right before publishing, one of the most important logical steps actually had an error in it that took ages to fix and find new logic for.
Now if you’re asking about proving say a physics theory with math, this is more a question about theoretical vs experimental physics. A similarly famous example would be the higgs boson. It was theorized through a whole lot of math and quantum mechanics that this particle and phenomenon should exist. It serves as an excellent explanation for the things we observe and predicts several other things we hadn’t quite observed yet, one of which was a particle. It took another 30 years for experimentalists to build a large enough apparatus (and by large I mean a several mile diameter particle accelerator under the mountains in Switzerland) to confirm the theory and the math done. Higgs and his collaborators work was not a mathematical proof, it was a set of equations that made predictions that were found to be true thus we take the equations as accurate.
By contrast, string theory is a similar set of math predictions that has never once been verified in any degree (in fact a few parts have actually been proven impossible). As a result, it’s not given the same level of credence and remains firmly in the realm of theoretical physics.
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