This is mistaken. Mathematics and Physics are not so similar. Yes, they both use numbers and fancy symbols, but here are three big differences.
Proof v experiment. Mathematics is all about what can be proved from the axioms, like ZFC. The mathematician seeks 100% certain knowledge, and settles for nothing less. The physicists gains validation by doing experiments. Truth is just a tentative shorthand for explaining some observations.
Infinity. All the interesting mathematics uses infinities. The concept is essential to everything. There are no infinities in the natural world. While they occasionally crop up in some physics theories, they are not essential to anything, and there is no reason for a physicist to believe in them.
Spin. Mathematicians are like fermions, and physicists like bosons. Each mathematician is like a unique piece to a giant jigsaw puzzle of knowledge. Physicists do not see things that way at all, as they replicate the work of others and are susceptible to groupthink.
These are huge differences. They are so large that I don't think that it makes sense to say that the fields overlap.
Sure, math gets applied to physics, and there are some mathematical physicists who are really mathematicians in their outlook. But mostly, mathematicians and physicists are different animals.
Dr. Bee writes on whether infinity is real:
Infinity and zero are everywhere in physics. Even in seemingly innocent things like space, or space-time. The moment you write down the mathematics for space, you assume there are no gaps in it. You assume it’s a perfectly smooth continuum, made of infinitely many infinitely small points.Infinity and zero are everywhere in mathematics, so if you are applying math to physics, they will be there. But they are only mathematically real, and do not exist in the natural world.
Mathematically, that’s a convenient assumption because it’s easy to work with. And it seems to be working just fine. That’s why most physicists do not worry all that much about it. They just use infinity as a useful mathematical tool.
But maybe using infinity and zero in physics brings in mistakes because these assumptions are not only not scientifically justified, they are not scientifically justifiable. And this may play a role in our understanding of the cosmos or quantum mechanics. This is why some physicists, like George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should be formulating physics without using infinities or infinitely precise numbers.