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 don't know if you've heard of this before, but there is an extremely elegant physical proof of the existence and properties of the Fermat point. It even illustrates the degeneracy that starts to occur for largest angle at least $2\pi/3$.

Consider your desired triangle as a flat, sturdy, sheet. Make very small notches at the vertices. Now take three ideal ropes and at one end of each fix identical balls. Join all three free ends. Place the rope-ball configuration on the top of the triangle, and slide each of the ropes through the notches. Let the balls hang off the triangle.

Now, allow the system to assume its minimal potential energy. The intersection of the ropes will move to minimize the sum of the distances to the vertices, as potential energy is linear in height. So the intersection will move to the Fermat point.

But here is the tricky part. As the intersection stabilizes, we know that the forces on it must be zero. The forces all have the same magnitude, so if they are all zero then they must have equal angles between them. So the lines from the Fermat point *X* to the vertices of triangle *ABC* satisfy $\angle AXB = \angle BXC = \angle CXA = 2\pi/3$.

Of course, this suggests that in a suitably obtuse triangle the intersection will fall right through one of the notches, and indeed for angles greater than $2\pi/3$ the Fermat point is at the largest angle.

           ![FermatPoint](https://i.stack.imgur.com/UY6Ax.jpg)
The previous one was so classical!. This one is much less. James Glimm succeeded to prove in 1967 the existence of a global-in-time solution of the Cauchy problem for hyperbolic systems of conservation laws in one space dimension. His proof is probabilistic. The key of the proof is the construction and estimate of the *potential of interaction*. It involves pairs of waves that are approaching each other. Clearly a physical intuition.