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Chris Budd answered on 17 Jun 2020:
If the mathematical theory is rigorously derived then I am automatically convinced by it and believe in its value. What I am less convinced about are dodgy non rigorous arguements that are used to ‘prove’ things. A good example is the ‘proof’ that
1 + 2 + 3 + 4 + … = 1/12
which you can find on the Internet.

Richard Pinch answered on 18 Jun 2020:
Yes, there are, but for different reasons.
One is within mathematics itself, in the field of number theory (which happens to be the area that I myself have done most work in). There’s a hard problem called “The ABC problem” which many people have worked hard on, and one of them claims to have solved it. Unfortunately his solution is incomprehensibly difficult and most mathematicians, including me, cannot understand it at all. Of the few who say they can understand it, some say it is correct, and others say that it is wrong, but their explanation of what is wrong are so complicated that the rest of us can’t understand that either. What makes this unusual is that when people publish what they claim to be a solution to a problem, it is usually easy for the rest of us to say whether it’s right or wrong. In this case, it seems very difficult.
Another is about the mathematical foundations of physics. In physics, we have two wonderful theories: the General Theory of Relativity, which is extremely good at describing and predicting the behaviour of large bodies (ranging from space satellites up through planets to galaxies and black holes); and Quantum Theory, which is equally good at describing the behaviour of small particles (such as atoms and electrons). In each case the theories are so accurate that we can and do use them to make gadgets (a satnav uses calculations based on Relativity Theory to locate your car to the nearest metre, for example). But mathematically they are incompatible: you have to decide which of the two to use in any given situation. As a result, many mathematicians and scientists believe that there must be some even more complicated theory that combines the two: Einstein called it the Unified Field Theory, and spent a lot of his life looking for it, without success, and even today nobody has a good answer. There are many possible theories: one called String Theory is very popular among scientists at the moment. But again, I’m not convinced by any of these new theories. i think I should say, though, that unlike the ABC problem, this is not an area that I’m an expert on, so I have to go by what my more expert colleagues say about physics!

Alexandre Borovik answered on 18 Jun 2020:
As an editor of, or a reviewer of papers submitted to mathematical journals, I frequently feel uneasy about certain branches of certain mathematical theories which I find more or less useless (I am not talking “correctness” of results, only about interest to them.. ). But one has to be very careful with value judgements. I also know examples of mathematical theories which were dead in water 20 years ago — but recently are being revived and regenerated because of new problems which came there from other areas of mathematics — one example of that is the socalled matroid theory..Selfisolation of a mathematical theory, lack of connections with the rest of mathematics is usually a sign of a trouble.

Hannah West answered on 22 Jun 2020:
As Richard said below string theory is a popular theory to describe particle behaviour. I studied physics and touched a bit on this. Some scientists think string theory could be used to explain dark matter (mass that we know exists but can’t ‘see’) among other observations. The main problem with this theory is that it is very difficult to prove to be true. Until we can prove it is true using experiments I will remain unconvinced.

Arick Shao answered on 26 Jun 2020:
That is an interesting question, but I think we have to be a bit more careful about what we mean here.
As Chris mentioned earlier, a theory in mathematics is established through rigorous proofs (i.e. formal logical arguments that deduce the conclusions of the theory from the hypotheses). If something has been proved, then it is established as mathematically true, whether we like it or not!
That being said, there are many reasons that you can be “not convinced” about a mathematical theory:
1) *There is a mistake in a rigorous proof.* Mathematicians are humans, and moreover humans doing very complicated things, so we are susceptible to human errors. One famous example is in the celebrated proof of Fermat’s Last Theorem by Andrew Wiles – initial versions had some serious issues that had to be corrected before it was finally considered complete. Ideally, as a community, we would find these mistakes and work to fix them.
2) *A theory may not always reflect the real world.* Mathematical theories are often made to model some part of the real world, but they can be imperfect in that regard. One recent example is the BlackScholes equations in finance, which predicted option prices. There is nothing wrong with the mathematical theory itself, but it was often misapplied without regard for the assumptions behind the model. This played a major role in causing the financial crisis in 2008. Another example is the uncertainty behind how accurate various mathematical predictions of the coronavirus spread are.
3) *A mathematical theory can be misunderstood.* Chris gave an excellent example with “1 + 2 + 3 + … = 1/12”, which became massively viral several years ago. The dodgy “proof” in the famous Internet video is nowhere close to correct.
On the other hand, there is a way in which “1 + 2 + 3 + … = 1/12” is correct, and this has interesting implications in theoretical physics. But, to really make sense of this, the left hand side “1 + 2 + 3 + …” is no longer interpreted as adding numbers together. (In other words, it’s complicated, but if one does not understand the subtleties, then the result one gets is total rubbish.)

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