As to Bill H.'s question (“Why is this even debatable?”) – I’m not a math major, but here’s my take.
A lot of mathematics is intuitive, and that’s no accident. There’s usually an attempt to match up with either a physical phenomena or at least to make things work on a simple and easily understood level. For example, it’s good to have the concept of addition actually correspond to what we perceive as addition, i.e., two + two = four lets us use arithmetic to know[sup]*[/sup] that two monkeys + two monkeys = four monkeys. If instead, a+b = ‘one more than a’ it would not have much (if any) practical use. So we use the mathematics that fits with what we know.
There’s often some point, though, at which things become less intuitive. Infinity is naturally going to be one of those; by its nature it doesn’t have a knowable correspondence with a physical situation. That’s exactly the sort of thing that confuses us in this case. We know that if anyone actually sat down and starting throwing the balls in the urn (and knowing that a person – or an imp – can only put in or remove balls in a finite amount of time) that we would never see the number of balls in the urn shrink.
In order to find out the answer, we ask the mathematicians. And yes, they do come up with a definite answer – sort of. What they say is “starting with this, using these definitions of sets, and limits, etc. you get this” and that answer could not be different. That’s something everyone can agree on [sup]**[/sup]. But in these cases, the answer goes against what we expect.
While nobody disagrees with one particular conclusion, there is real debate. If you look at ZenBeam’s first post in this thread, you’ll notice that there are academic papers written discussing this particular problem[sup]+[/sup]. There’s often papers that argue about a number of mathematical oddities (some of which arise as thread topics here). Not everyone agrees with the solution to the problem.
There’s probably two main causes for debate. In reference to a particular problem, there may be the question of what I’ll call ‘the model’ of the problem. Often a basic question, of great interest when attempting to apply mathematics, is whether or not the mathematical tools used are actually modelling the ‘physical’ (or in this case, the ‘stated’) problem. This is what’s called into question with the second formulation of the problem (the two urns, and relabelling of the balls) – are they in fact equivalent; and why or why not?
The other cause for debate is whether the math is good enough at all. In some way it’s related to physical correspondence but has to do with simplicity and ‘fitting in’ with the rest of mathematics. By good enough roughly what I mean is if despite its non-intuitive behavior in one case, does it follow intuitive behavior in other (simpler) cases? It’s a bit harder to define, and it can get a little wrapped up in technical details, but I suppose that’s why it becomes a topic for debate.
This is the reason why ultrafilter had to have a second long post about his definitions of convergence. He had to show that these generally fit the way convergence ‘should’ behave – in his words “why I think this is a good general notion of convergence for sequences of sets”. For example, I could, if I so desired, say that “all infinite sets converge to being empty”, and solve the problem - it’d look somewhat like ultrafilter’s solution, and it could not be argued that it does not show the urn empty at the end. However, it’s clear that such a definition of convergence is nonsense. It not only doesn’t fit our conception of what should happen to a sequence of sets, but it doesn’t allow some of the basic operations to work that make it a useful tool for math.
So despite the fact that mathematics proceeds in an orderly fashion to a single ‘right’ answer, it is the rigid definitions that shape the process that are called into question and the heart of many debates in mathematics; in more subtler cases, there are conflicting assumptions that both lead to ‘good’ results as we know now, but might be abandoned later.
[sup]*[/sup]According to common usage, but perhaps not by the strictest of epistemological standards.
[sup]**[/sup]Again, this raises a philosophical question (“why must there be universal agreement?”) that doubtless people are working on, but mostly we just accept that mathematics works the way it does because it does, or gives useful results, or whatever we like.
[sup]+[/sup]This is completely unrelated to the rest of my post, but in the paper by John Byl linked to, I noticed what looks like an error when he is attempting to reconcile the continuity conditions : “… assuming that any ball coming to rest at a position which it does not leave before noon is still at that position at noon, then all balls in the urn just before noon should still be there at noon …”
) Anyway…