I can’t remember where I came across the link to this forum post explaining how math and physics work together, but it’s been open in my browser for a month or so and I finally decided to read it this morning. It’s a really nicely written explanation of why and how physicists use math:
But if you want to know why things are the way they are, and in fact if you really actually want to know how they are, then you have to do more than just describe them, which we call “qualification;” you have to describe them in detail, which we call “quantification.” In other words, “there’s some rocks” is a qualitative description; “there are eighteen rocks located as follows, of the following sizes and compositions and masses,” is a quantitative description. The first is good enough for natural language; the second is only barely good enough for the simplest kind of physics. Physicists get really, really precise about how they describe things, and to do that, they use math.
The post goes on to explain, “But you always have to remember that the math is just a description; it’s the old thing about the map, not the territory. You can look at the map all you want, but until you’ve walked it you really don’t quite know what’s there. And when you have a bunch of math that describes stuff you can’t ever directly sense for yourself, then you have to just trust the math, and look for ways to check it.” Just thought it was all quite nicely put.