Question: I believe I.M. Gelfand said that when beginning to learn a new subject, one should learn it like a physicist.

In this spirit, what are some helpful and surprising physical intuitions accompanying mathematical objects?

I especially am interested in intuitions that suggest nice proofs...and helpful new ways of thinking.

Answer: 
Well, there are of course many examples but perhaps the Dirac $\delta$-function is the most striking one. With quite some violence to the math of usual calculus, Dirac's idea turned out to be exremely useful: the "function" everywhere zero and very high at $0$ so that the integral is $1$. Applications are ranging from Fourier analysis, (linear) PDE, quantum mechanics, and many more. Now of course we know how to make things rigorous in the framework of Schwartz's test function and distribution theory. But the intuition is clearly from physics.

Similar and also in this framework is the notion of oscillatory integrals...
My favourite example is that of braided fusion categories (a fusion category has finitely many equivalence classes of irreducible objects, but that is not essential here). This is basically the idea behind topological quantum computing. Without going into details, a nice interpretation of the equivalence classes of irreducible objects is as "particle types" or "charges". In a fusion category there is a monoidal operation $\otimes$, which can be interpreted as "composition of charges". Moreover, if $X$,$Y$ are two irreducible objects, the product $X \otimes Y$ can be decomposed into a direct sum $\bigoplus\_i X\_i$ of irreducible objects. This is called *fusion*, and can be interpreted as "fusing" particles. The irreducibles appearing in the decomposition are the particle types that can appear after fusing two particles.

The braiding of the category describes what happens if we interchange two particles. If the braiding is in fact a symmetry, that is, we have a *symmetric* monoidal category, the situation reduces to the well-known Bose/Fermi statistics in physics. The existence of dual objects corresponds to anti-particles. In particular, there is a morphism from the tensor unit (the "vacuum") to $X \otimes \overline{X}$ for every object $X$, interpreted as creating a particle together with it's anti-particle form the vacuum. Note that charge is conserved!

Pushing this analogy a bit further, one can interpret the "tangle diagrams" that are commonly used to describe morphisms in braided monoidal categories as the wordlines of particles. For example, a diagram with one input and two outputs can be interpreted as a particle splitting into two. Although this whole picture of worldlines should not be taken to literally, I find it a helpful when thinking about such diagrams.