Something not very intuitive is there's no ab initio (i.e. "blank slate", a priori, purely theoretical, etc.) notion of "legitimacy" in mathematics, or even phy

Something not very intuitive is there's no ab initio (i.e. "blank slate", a priori, purely theoretical, etc.) notion of "legitimacy" in mathematics, or even physics. There was a lot of debate on whether complex numbers were "legitimate". The truth is, they work well in several physical models; that's where their legitimacy comes from. We assume that any crazy axiomatic system can model reality, well, almost any actually. There are some properties excepted of any axiomatic system, like self-consistency, and of course the ability to generate some kind of complete (or locally complete at some domain) model of reality. What we then do is try to come up with models and infer the correct or most useful ones from experiments. This means if a physical model yields a sum like 1+2+3+..., we of course expect the result to be unphysical and the model invalid. However, if experimental data is consistent with a result like -1/12, that would be a clue to explore a different arithmetical system for our model, one which generalizes summations (to Ramanujam summation or other kinds of sums). If this new model reproduces experimental data, you're on a very good track :) [1] In mathematics the situation is even more interesting: the legitimacy is driven by taste of researchers, which will blend physical and philosophical relevance, and even aesthetics, curiosity and interest in part of the mathematician. (This is why we understand mathematics to be almost art, although a very particular art of course) [1] Note: I believe the theory of learning (Machine Learning) developed recently is the best current formal understanding of this procedure, and how not to get it wrong. The overview is that whatever your model is, it shouldn't have too many parameters (certainly not more than data you're trying to fit), and that a reliable test for a theory is (a) either making a novel prediction on unseen data that turns out correct (validation); (b) Use the theory to model a part of the data, and see how it fares on the unseen part (for when you can't conduct new experiments; this is cross-validation). Some references in this area: Computational Learning Theory https://en.wikipedia.org/wiki/Computational_learning_theory https://en.wikipedia.org/wiki/Computational_learning_theory VC Theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_th... Statistical learning theory https://en.wikipedia.org/wiki/Statistical_learning_theory https://en.wikipedia.org/wiki/Statistical_learning_theory