(cont'd)

OK, so who needs this? Well, this paper is beautiful on its own, but it's not particularly important to many other mathematicians. The reason is that in set theory there's something famously called an "Axiom of Choice" that basically allows you to handle infinite sets and their pairings much more easily than without it. The Axiom of Choice used to be somewhat controversial in the beginning of the 20th century. What happened during that time was that people realized they could look at set theory (and in fact at all of mathematics) much more rigorously than before by coming up with some basic axioms and showing how everything else follows from them. This is something that had been practiced in geometry ever since Euclid, but not really in other parts of mathematics, and around the beginning of the 20th century people started doing the same thing not just with points and lines, but also with numbers and sets and functions. When they tried to _formalize_ set theory - that is, take the usual stuff people have been doing with sets and figure out how to bring it all down to a handful of very obvious axioms - they discovered that all the axioms were completely obvious (as wanted!) except one, the Axiom of Choice, that sort of seems necessary to do all kinds of advanced math, but is not as obviously true as others. See, the Axiom of Choice is not _explicit_: it says something like - under certain conditions - "a matching in pairs between these two sets exists, but I'm not showing you what it is explicitly, I'm just telling you it exists". When it tells you that, it kinds looks reasonable that these two sets should be of the same "size", intuitively, so you're glad that the Axiom of Choice settles the matter by just claiming that a matching exists; but at the same time you may feel a little uneasy about the fact that the matching is not described with some formula or algorithm or anything, you're just told it exists and that's it.

Anyway, with time mathematicians pretty much accepted the Axiom of Choice as a vital part of their mathematical toolkit. Even though some very non-intuitive things can be proved using it (look up the Banach-Tarski paradox if you're interested), it's much better to have it than not. And with the Axiom of Choice, the whole problem of proving that bags of marbles B1 and R1 are of the same size becomes trivial, even when they're infinite. You don't need to carefully engineer a B1-R1 matching out of the larger one between all the six bags; the Axiom of Choice sort of gives you the firepower to just claim that this B1-R1 matching exists, and call it a day. Nearly all mathematicians are satisfied with this, because for them set theory is no more than a convenient tool to carry out their investigations into _other_ things, like numbers, geometric spaces and what not. It's only people who study _set theory itself_ (and call themselves, appropriately enough, set theorists) that are still interested in understanding just how much the Axiom of Choice is needed. These people appreciate much more than other mathematicians a proof of some fact in an _explicit_ way, not requiring the Axiom of Choice, even if it's trivial when using the Axiom. So the paper about "dividing by 3" is an example of this sort of investigation: do we really need the Axiom of Choice to prove that B1 and R1 are of the same size? Turns out we don't, and we can do this explicitly with a rather complicated, yet beautiful, construction. To mathematicians as a whole, this result is largely meaningless; to set theorists, it's a minor theorem of some interest, though even they, I think, probably would look at it more as an interesting puzzle with a beautiful solution. And that, by itself, is nothing to sneer at.