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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.

Qmechanic
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SimoBartz
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  • I think what you're describing is a maths textbook for physicists. Riley Hobson and Bence is perfectly good. Once you get to much higher levels of physics you tend to just find the methods in the physics textbooks you look at, because it's not really worth learning whole fields of maths (if you don't care for it) just to learn one thing for physics reasons. On the flip side, abstract mathematics often finds it's way into physics, and so it's actually a rather good idea. – Jake May 14 '20 at 13:01
  • Maybe my question is not clear, I'm sorry. I'm ok with math, I don't need a book about math.. is it useful if I tell that one problem that I often have is that I confuse what is math and what is reality? – SimoBartz May 14 '20 at 13:07
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    Re, "what is math and what is reality?" Math is the language that physicists use to describe reality. Re, "...isomorphisms..." Just as there can be more than one way to describe a thing with words, there can also be more than one mathematical way to describe the same physical phenomenon. – Solomon Slow May 14 '20 at 13:12
  • Yes but there are technique to find different representation, and there are also different ways of describing the physical laws... do you have any book about this topic? – SimoBartz May 14 '20 at 13:13
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    It is an exciting question, and I posted an answer... but it is not specifically about physics. You may also want to look here: https://physics.stackexchange.com/q/538350/ – Roger V. May 14 '20 at 13:15
  • I lose the connection between math and reality This strikes me as a problem that a book cannot solve for you. With math you deliberately abstract things away from reality and then, when you're ready, you undo the abstraction to get a tangible result. But the fact is that some of the results of modern physics don't have everyday human intuitions - they're counter what we expect or outside our experience. You kind of have to trust that math is leading you to something that's at least testable. E.g. spin in Dirac's theory of the electron is not intuitive IMO. – StephenG - Help Ukraine May 14 '20 at 14:04
  • my answer here is relevant to the second part of your title question https://physics.stackexchange.com/questions/262917/why-must-a-physical-theory-be-mathematically-self-consistent/262920#262920 – anna v May 14 '20 at 16:08

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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:

These may seem to take one far away from physics... but the question is really not about physics, but science in general.

Roger V.
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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.

Agnius Vasiliauskas
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