A book about why do we need abstract math in physics When I study math for physics applications I always have some trouble in understanding why it's necessary to abstract the objects we study (vector spaces, scalar product, manifolds, groups...). In the beginning I thought it was to demonstrate theorems in a way that is valid for many cases but then I noticed it's also useful in order to derive equivalent descriptions of the same things (for example using isomorphisms). Besides, physical laws can change in form when I use strange maps to describe the problems. And in general I realize that the description we start with is just one of many and it isn't special.
However I still have trouble with this topic and I'd like to read a book that explain these things (I don't need a book that explain math for physics)
Do you have any suggestions?
Maybe it is useful to give you an example of the problem I would like to solve, often I don't understand if I'm doing math that already has a physical meaning or I'm just doing math without any physical meaning. Easy speaking, I lose the connection between math and reality. 
 A: This might be not to directly answer your question, but to take on some themes in it. One reason why we need math and models in physics is because we need a formal/rigorous way to talk about physical reality, whereas our usual way of speaking and thinking is rather ambiguous (even though we rarely realize it before we get into a debate with somebody who is obviously wrong). I will therefore suggest below a few directions for expanding general knowledge:


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*Formal languages and the theory of computation or any other good book on theoretical computer science - it is a fascinating domain, and may turn handy in the future. 

*A good book on the theoretical syntax, which would present the same language ambiguities from a more human viewpoint. Real theoretical syntax books, such as Syntax: a generative introduction may be too hard for a beginner, so I suggest more popular level literature, like I-Language: An Introduction to Linguistics as Cognitive Science or Teach yourself linguistics.


These may seem to take one far away from physics... but the question is really not about physics, but science in general. 
A: 
why do we need abstract math in physics

Without abstract math tools, physics will be just a philosophy, which was how things were going on to describe nature in ancient Greece. Some ancient thinkers like Empedocles proposed that humans sees objects by emitting some kind of fire from own eyes which gropes distant objects and transfers "groping" information back to human. Later others have raised a question why "object groping" isn't possible at night ? "Sensing Fire" should be emanating from eyes at night too. But we can't see at total dark, so everybody started to think that there is something wrong with "a sensing fire" theory.
It is just an example, but such kind of unproven philosophical theories were plenty in ancient times. Involving abstract math into Physics we make sure that any theory can be dis-proven faster, because to deny philosophy is very hard, almost impossible. Also math application in Physics encourages critical thinking, which is a-must in good high-quality todays science and scientific method, which is standard de-facto.
