Towards? This kind of pedagogy is nothing new. Isn't this just part (maybe the main part, along with being able to evaluate proofs) of learning higher mathematics? Being forced to develop an intuition like this?
In an ironic twist, the pigeonhole principle itself breaks down for large enough sets (if we phrase it as bijection between set and set with one more element). Well the sets have to be infinitely sized for the principle to break down, but still. So, in the vein of the article, let me ask: how do we really know the pigeonhole principle works for large but finite sets?
We know that because it is rigorously proven within some axioms. A large, but finite, set falls into the given axiomatic system where it applies, so we know.
If you're going to start questioning whether math is real (is the whole universe just a dream type stuff) then sure, do that, but don't try to apply that for math which such fuzzy junk.
There is a world of difference between infinite and finite. "large but finite" is not in any way comparable to "infinite". Any attempt to make such a parallel between infinite and large finite is fundamentally flawed to begin with.
The pigeonhole principle has been proven and it's by that proof, and the history of math it relies on, and the fact that that math has a higher predictive power than anything else (especially within its own axioms where it's p-much perfect!)... that's why we really know that it works.
If you want to question whether it works, then the burden is on you to provide evidence of some form more than some misguided "feeling" that large numbers are relateable to infinite. There already exists an argument in the form of the original proof and it is your burden to make a more compelling one.
Also, I'm not sure why you're saying that the principle breaks down with infinitely sized sets. Infinitely sized sets.. Firstly, if both sets are infinite (and your plurality of sets there indicates you mean that), then it doesn't apply at all since m = n. If the sets are of different orders of infinity and you can, in fact, create a bijection between then then it does still hold.
However, you state it as "a set and a set with one more element" and then say "infinitely sized", which when combined reads as complete and utter nonsense.
Basically, I think you're starting with a flawed premise, leading into a flawed question, and there is no good answer.
Base case: There is no injection from a non-empty set to the empty set.
Inductive case: Let A be a set of size (m+1) and B be a set of size (n+1), and let f be an injection from A to B. Pick x in A.
Since f(y) is not equal to f(x) for any y not equal to x in A, the restriction of f to A\{x} is an injection into B\{f(x)}; and presumably you believe that A\{x} has size m and B\{f(x)} has size n. Then we've produced an injection from a set of size m to a set of size n.
Thus if there is no injection from a set of size m to a set of size n, then there isn't one from a set of size (m+1) to one of size (n+1).
It certainly isn't ironic nor is it a breakdown, the pigeonhole principle is stated on finite number of containers and a finite number of pigeons.
However, if you just arbitrarily change it around, you may well find that the new statement you just generated is not true any more, if it even makes sense at all.
It's ironic from my point of view because the article cautions against hasty generalizations, and that was a generalization I implicitly did upon first hearing about Hilbert's paradox of the Grand Hotel.
But you can easily answer of course, which is because induction can't not work.
For it to not work there has to be a step N where the N-1 step satisfies the claim proven and the Nth step violates the claim. By hypothesis this is exactly what you have proven to never happen in the inductive step of the proof.
Probably GP will then answer with something like "how do you know that N-1 works" or something equally obnoxious. (It's not obnoxious because it's an invalid question, it's obnoxious because GP is just being a pain at that point, and coming up with objections to be annoying but without actually caring about the answer. It's like if you answer every statement anyone ever makes with "I would like to respond, but I can't be sure you aren't an evil demon sent to trick me, or a foreigner who speaks a language that sounds exactly like english but with a completely different meaning.")
