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: 
I view the [Gauss-Bonnet theorem](http://en.wikipedia.org/wiki/Gauss-Bonnet_theorem) for a surface without boundary
via the physical analogy that if you push a dent into a maleable surface, or squeeze it,
the surface must compensate by curving elsewhere
to maintain $2 \pi \chi$.

![sphere blobs](https://3dideas.files.wordpress.com/2008/09/s.jpg?w=319&h=200)
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.