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)
I think, Elliott Sober gives a nice example (cf. Elliott Sober
`Quine's Two Dogmas', Proceedings of the Aristotelian Society,
Supplementary Volumes, Vol. 74 (2000), pp. 268-269):

For logic and mathematics to be tested empirically, one logical or
mathematical statement would have to be pitted against an alternative
and a framework of shared background assumptions would have to be
supplied that permits the two statements to make different predictions
about observations. It is instructive to examine a case in the history
of mathematics in which this actually happened. Plateau was a 19th
century French mathematician who wanted to figure out what the surface
of least area is that fills various closed curves. A simple example of
the type of problem that Plateau had in mind is a curve that has the
shape of an ellipse. An elliptical curve can be filled by the ellipse
it contains; any other surface that fills the curve will bulge into a
third dimension and so must have more surface area. Although the
answer to Plateau's question is obvious for this example, the answers
aren't at all obvious for more complicated three-dimensional
curves. What Plateau did was to dip wires bent in different shapes
into soapy water and observe the resulting soap bubbles that adhered
to the frames when they were removed from the water (Courant, R. and
Robbins, H., 1941, What is Mathematics?, New York: Oxford University
Press). Given the physical assumption that soap bubbles take on the
surface of least area in this experiment, different mathematical
hypotheses ('the surface of least area for curve C is s1' versus 'the
surface of least area for curve C is s2', for example) make different
predictions. This example is interesting in the history of mathematics
precisely because it is so atypical. No such test of '2 + 3 = 5'
against alternative arithmetic hypotheses has ever been carried out,
nor is it remotely clear what this would be like.