Mark Levi's The Mathematical Mechanic is a book of examples of how physical reasoning can be used to solve mathematical problems; another couple of examples is in this blog post at Concrete Nonsense. Many of these proofs rely on conservation laws and other physical facts which, at least as I understand it, can be made rigorous by appealing to Noether's theorem. So these proofs themselves ought to be presentable in a mathematically rigorous way. But how?
There are many examples, so I will focus my question on two, one of which is in Levi's book and one of which I recently encountered on a friend's homework.
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Problem: prove the Pythagorean theorem. Solution: imagine a fish tank in the shape of a triangular prism. The triangle is a right triangle with side lengths $a, b, c$; call the vertices of the two copies of the triangle $A, B, C$ and $A', B', C'$, where the right angle is at $C$ (resp. $C'$). Drive a rod through $BB'$ and fill the tank with water. This system is at equilibrium (by conservation of… angular momentum?), so the net torque through the rod is zero. On the other hand the net torque through the rod is proportional to $c^2 – a^2 – b^2$.
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Problem: on each face of a tetrahedron, not necessarily regular, place a vector pointing normal to and out from the face with magnitude equal to the area of the face. Prove that the sum of these vectors is zero. Solution: fill the tetrahedron with an ideal gas. This system is at equilibrium (by conservation of… energy?), so the net force exerted by the gas on the tetrahedron is zero. On the other hand the net force exerted on each face is proportional to its area and points normal to and out from each face.
As you can see, I'm not even totally clear what conservation law(s) I'm invoking at the crucial step of the physical arguments. I would really appreciate mathematical insight into what's going on here.
Best Answer
Both proofs in the question are of the following type. Suppose you want to prove a certain identity among numbers $X$, say $f(X)=0$. If you can find another function $g(X,Y)$ such that $f(X)=g(X,Y)$ for some $Y$. Then, proving that $g(X,Y)=0$ for this choice of $Y$, independent of $X$, automatically proves $f(X)=0$.
In the case of the Pythagorean theorem, $X=(a,b,c)$ are the dimensions of the triangular base of the water tank, $Y$ are all the other details of the physical setup and $g(X,Y)$ is the total torque about $BB'$ exerted on the tank. As long as $Y$ contains the fact that there are no torques acting on the tank other than the different parts of the tank acting on each other, physical law dictates that the total torque must be zero, hence $g(X,Y)=0$. The Kut-the-knot page linked from one of jc's comments shows why $g(X,Y)$ is proportional to $c^2-a^2-b^2$.
The case of the tetrahedron is similar, $X$ are the dimensions of the tetrahedron and $g(X,Y)$ is the total force acting on the gas container. The physical situation $Y$ is set up such that no forces act on the container external to the container itself, hence physical law dictates that the total force must be zero.
Now, what is it that allows us to conclude something about the value of $g(X,Y)$ in each case, given only $Y$ and independent of $X$? One way to think about it is to consider the physical world as a giant computer or oracle, $W$. If you set up a physical situation $Z$ and measure some observable quantity $G$, the world will give you a numerical answer $W(G,Z)$. The goal of physics, of course, is to build mathematical models of $W$, so that we don't have to consult the oracle every time we need an to know the value of $W(G,Z)$. But a mathematical model is not necessary if we are happy with just consulting the oracle when needed. Still, observation of the answers that we get from $W$ allows us to identify certain regularities in its output: these are the laws of physics.
Thus, supposing that we can find $G$ and $Z=(X,Y)$ such that $g(X,Y)=W(G,Z)$, the kind of of proof I outlined in the first paragraph can be carried out, appealing to the laws of physics as properties of the oracle $W$. To convert this kind of proof into a sequence of logical deductions, as a usual proof should be, requires a mathematical model for $W$, at least a restricted one that suffices for the purposes of making deductions about $g(X,Y)$. In both cases the model that does the job is plain old Newtonian mechanics of particles, solid bodies and fluids (if desired, solid bodies and fluids may be seen as a limiting cases of large assemblies of particles). The equality $g(X,Y)=f(X)$ follows from direct application of Newton's second law (an axiom of this mathematical model) and the identity $g(X,Y)=0$, independent of $X$, follows from Newton's third law (another axiom).
Personally, I think the appeal of such "physical" proofs is the fact that they rely only on some properties of $W$, and not on the details of a mathematical model for it. However, the validity of the proof only follows if one can show that a detailed mathematical model exists.
Hope this helps.