We know the expression "Tail wags the dog" for when what's supposed to be in charge and driving the direction is not doing its job properly. How often has theory predicted something ground breaking in physics? Here are a few cases 1) Einstein -> relativity verified by gravitational lensing 2) Dirac -> anti particles 3) Josephson -> tunneling through a resistor (aka josephson junction) 4) Higgs -> those bosons. What else has there been? And even in the cases of 1 and 2, was it math leading their thoughts, or was mathematics just a way to communicate it to others? (I read that 3 was all math derived, but this seems like the exception) In physics derivations, we keep Taylor expanding anything we can't solve. But this seems so ad hoc. So what is mathematics role in physics? I mean, It's nice to know that the laws of physics can be approximated by mathematics. But if it's not driving the science, but rather providing ad hoc explanations, we should be informing young prospective physicists of that so they don't get disillusioned.