This idea of cutting the length of the proof in half and applying induction is probably an allusion to Zeno's Paradox. In fact, the end of the story explicitly brings Zeno into it, though the analogy here is not a perfect one as the number of lines in a proof can only be divided in half a finite number of times.

From a purely literary point of view, there is nothing wrong with this story. But, it really does not try to do anything interesting in that regard either. It is mostly just a brief dialogue between the two characters followed by the slightly surreal effect on the character falling to the floor.

So, I am guessing that this story is trying to be deep or philosophical. (This is further justified by the fact that it was published not in a literary journal but in the "Futures" column of the scientific journal Nature [28 May 2009].) However, anyone who has considered questions of the foundations of mathematics, from either a philosophical or mathematical point of view, has to have considered this idea that "math might be wrong" before, and this story does not really add anything new.

Again, there is nothing wrong with the philosophy either. I think any reasonable person would have to admit that there is a possibility that any given "proof" is invalid and (since Gödel) that math itself may be inconsistent. But, I would strongly recommend readers look at Division by Zero by Ted Chiang which addresses the same idea but is spectacular both from a literary and philosophical point of view.

Further Notes:

1. The author is a geologist at Duke University.
2. The illustration is apparently an illustration of the Four Color Map Theorem applied to a region divided by the boundaries of the symbols "a", "b", "0" and "?".
3. The proof of the Four Color Theorem was controversial in the 1970's when it was proved with the assistance of computers. Strangely, the story does not mention this aspect of the controversy, even though it fits into its notion of a "physical" problem with the proof. (Instead, it focuses on the length.) However, in the decades since, both because mathematicians are more familiar with programming languages and other developments in this particular proof, I do not think it is particularly controversial anymore. So, although I agree in principle with Haff's idea that mathematicians could be mistaken about any given proof, I do not see any reason to pick on this one in particular.
4. In fact, there is no reason to focus on math here in particular either. Math seems to be absolutely certain, and it is arguably the branch of human intellectual development about which we can be the most certain. But, if you want to get all epistemological about it, you end up rediscovering Descartes' old maxim that the only thing I can be certain of is that I exist, since I know I am thinking. Anything else could be the result of either faulty reasoning or illusion. (Unfortunately, that line of reasoning -- as valid as it may be -- does not get you very far.)