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?
Best Answer
Ever since the time of Newton physics is about observing nature, quantifying observations with measurements and finding a mathematical model that not only describes/maps the measurements but, most important, it is predictive. To attain this, physics uses a rigorous self-consistent mathematical model, imposing extra postulates as axioms to relate the connection of measurements to the mathematics, thus picking a subset of the mathematical solutions to the model.
The mathematics is self-consistent by the construction of a mathematical model. Its usefulness in physics is that it can predict new phenomena to be measured. If the mathematics were patched together and inconsistent, how could the predictions of the model have any validity?
It is the demand for self-consistency that allows for falsification of a proposed mathematical model, by its predicting invalid numbers. The consistent Euclidean model of the flat earth is falsified on the globe of the earth, for example. This lead to spherical geometry as the model of the globe. The whole research effort of validating the standard model at LHC, for example, is in the hope that it will be falsified and open a window for new theories.