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)
> 
> Mr. McGuire: *Benjamin.*
> 
> 
> Benjamin: *Mr. McGuire.*
> 
> 
> Mr. McGuire: *Benjamin.*
> 
> 
> Benjamin: *Mr. McGuire.*
> 
> 
> Mr. McGuire: *I want to say two words to you.*
> 
> 
> Benjamin: *Yes, sir.*
> 
> 
> Mr. McGuire: *Are you listening?*
> 
> 
> Benjamin: *Yes, I am.*
> 
> 
> Mr. McGuire: [*Path integrals.*](http://www.scholarpedia.org/article/Path_integral)
> 
> 
>