I am studying undergraduate physics and I am always interested in why we consider certain physical models instead of the others. That ultimately leads to the question why we consider one mathematical description over the other in a specific physics problem.
Unfortunately these kinds of questions now make me struggle to learn anything new. In this question I would be happy to receive some advice or some personal experience.
I do not know why but I have the philosophy that we do not understand physics because we do not understand mathematics good enough. Therefore, I took some mathematics courses to make the best of my physics education. I took, for example, set theory, which is assumed to be foundations of mathematics, but I had trouble understanding it as when we wrote something intuitively (such as index sets) then I had great trouble as I wanted for it to be defined well and not just "intuitive". I understood that even in mathematics a lot of proofs are based on our intuition probably assuming that good mathematician could make intuitive argument rigorous. I tried then to learn basics of logic to make my arguments more rigorous and then started spending too much time on understanding why we choose one definition over the other (for example, why we assume vacuous truth). In the end, no matter what subject I take, I get destroyed by the need to try to explain myself why we make definitions the way we do.
For example, the last thing I was thinking about was why do we use real numbers in the physics? For now, the most satisfactory answer would be that we measure some things using some apparatus, say, length with a ruler, and we divide it in equal parts to make it easier (say, in 10 parts). Then we measure length. But probably the object for which the length is being measured does not coincide with the marks on the ruler. So we divide each interval again in 10 parts. Then we make iteration (possibly infinite, whatever that means) until we obtain a number which would be infinite decimal by construction. And real numbers might be constructed as the set of all infinite decimals.
Of course, I could assume that there exists set of real numbers with some properties that we want and then prove all the theorems, for example, in real analysis. But then why do we need exactly these properties?
Anyways, I have several possible answers on why this is happening :
1) I am not mathematically mature enough and this is something that stops being a problem with age,
2) I am not doing mathematics right and this kind of behaviour just has to be stopped by myself because it is destructive (I spend too much time on it instead of studying things that I could apply and to do research on).
I would appreciate your comments!