I always read in modern physics textbooks and articles about the need for physical theories to be mathematically self-consistent, which implies that the theories must not produce contradictions or anomalies. For example, string theorists are proud of the fact that string theory itself is self-consistent.
But what does this really mean? Physical theories are not a collection of mathematical axioms, they are attempts at describing Nature. I understand the need for rigorous foundations in mathematics, but in physics, we have experiments to decide what is true and what isn't.
It's also weird (for me) to say that a theory is mathematically self-consistent. For example, Newton's Laws of Dynamics encode empirically known facts in a mathematical form. What does it mean to say that Newton's Laws are mathematically self-consistent? The same can be said for the Laws of Thermodynamics. There is no logical need for Nature to abhor perpetual motion machines, but from experiments, we believe this is true. Does it make sense to talk about thermodynamics as being self-consistent?
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