# Can mathematics lead to a result which is physically untenable?

Here is a mathematical theorem: the internal angles of a triangle add up to 180 degrees (i.e. half a complete rotation). To be a little more thorough, let's define a triangle: it is a closed figure consisting of three straight lines, and a straight line is the line of shortest distance between two points. Ok so we have a nice mathematical theorem.

Now we go out into the world and start measuring triangles. They all have internal angles adding up to 180 degrees, to the precision of our instruments, so we are reassured. But then we get more precise instruments and larger triangles, and something happens: the angles are no longer adding up right! Oh no! What has happened? Is it a contradiction? Or perhaps our lines were not straight? We check that the lines were indeed of minimum distance. Eventually we go back to our mathematical theorem and realise that it had a hidden assumption. It was an assumption lying in a subtle way right at the heart of geometry and it turns out that it is an assumption that need not necessarily hold. One to do with parallel lines, called Euclid's fifth postulate. Then we discover a more general way of doing geometry and we can make sense of our measurements again---using the theory of general relativity and the geometry of curved spaces.

So, to answer your question, what happens when physical observations contradict a mathematical statement has, up to now, always turned out to be like the above. What happens is that we find the mathematical statement was true in its own proper context, with the assumptions underling the concepts it was using, but that context is not the one that applies to the physical world. So, up till now at least, physics has never contradicted mathematics, but it has repeatedly shown that certain mathematical ideas which were thought to apply to the physical world in fact do not, or only do in a restricted sense or in some limiting case.

The Banach-Tarski paradox seems like an obvious candidate. It's possible to cut a sphere up into finitely-many pieces, then glue it back together into two spheres each identical to the original The math is correct, but this is obviously not possible in the real world, so what's going on?

Every mathematical proof is based on some set of "axioms", or assumptions. If the logic of the proof is sound, but we reach some outcome that's impossible in the real world, that must mean that at least one of our axioms does not hold in the real world. In this case, it's probably the axiom of infinity (or possibly the axiom of choice).

So to answer the question explicitly, if we assume some equation like $$\nabla \cdot B = 0$$ holds, but that allows us to prove something that doesn't hold in the real world, then that necessarily means one of the assumptions used in the proof does not hold in the real world.

The most likely candidate would be the original equation itself, although it could be something more subtle, like "in step 12 we assume the geometry of space to be Euclidean". It could even be that the laws of (first-order) logic do not hold in our universe, though if that were the case I think we'd be in trouble!

If you have a physical theory, expressed as mathematics, then if, based on the premises of the theory, you prove a theorem which, when translated back into physics, contradicts experiment, then the physical theory is wrong.

So no, it is not possible that both the premise (the physical theory) and the theorem (a thing with a correct proof in other words) derived from that premise are correct, but the conclusion is wrong, and in this case the premise (the physical theory) is wrong.