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: 
Kirillov's [orbit method](http://en.wikipedia.org/wiki/Orbit_method) in representation theory establishes a correspondence (which is not exact in general) between irreducible unitary representations of a Lie group $G$ and orbits of the action of $G$ on the dual of its Lie algebra $\mathfrak{g}^{\ast}$. The physical intuition behind the orbit method comes from the notion of quantizing a classical system: the rough idea, as I understand it, is that orbits of the action of $G$ on $\mathfrak{g}^{\ast}$ should be thought of as classical systems with $G$-symmetry, and the corresponding irreducible representations of $G$ should be thought of as the corresponding quantum systems.
There are several examples at the [number theory and physics archive](http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/physics.htm). To get you started let me mention the [statistical-mechanical interpretation of the Riemann zeta function](http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/physics2.htm#partition) as the partition function of a system called the [primon gas](http://en.wikipedia.org/wiki/Primon_gas) or Riemann gas.