At last, a good audio player I can use!

I've finally found it: Running foobar2000 under Wine.

Wait, you say, didn't you say earlier that this doesn't work very well? That the sound is choppy and that it has other problems, too?

Well it turns out that to fix the choppiness, you just need to change one setting in the Wine configuration. (Under "audio", set "hardware acceleration" to "emulation" instead of "full".) And as for whatever the other problems were... they don't seem to be occurring anymore, so I'm going to assume those were fixed in recent versions of foobar2000 and/or wine.

On top of that, there's now a plugin for wine for playing Organya (the Cave Story music format) files! Except it doesn't loop properly. But that's OK, because by changing that one audio setting now OrgView works properly again! And, leaving the official site and going over to Zophar's domain, I see there's now a USF (N64 music) plugin for it! Which more than makes up for that.

So... yay!

-Harry

(Next entry, if I remember to write it: The finite entry. No, this one is not the "infinite-but-Dedekind-finite" entry.)

(Also, I played a game of Kongai the other day against someone of skill rank 36, with my Gunbjorn/Marquis/Balthazar deck, and won after a long game of attrition; but what makes this worth mentioning is that I did this without ever intercepting and without ever putting in Balthazar. This wasn't deliberate, that was just how my strategy worked out that game. So considering I beat him without even showing one of my characters, I guess I was very ahead; but considering how long it took, it was something of a subtle domination. This is unrelated to anything and I don't even recall it well enough to tell it as a story, I just thought it was neat.)

The infinite entry

...by which I mean "dealing with infinite things". I just wanted to point out some neat infinite things I'd seen recently (mostly due to MathOverflow):

EDIT next day: changed some stuff in paragraph 3 to make obvious inferences that I hadn't previously...

1. In an infinite symmetric group, every element is a commutator. I don't just mean the group is perfect, I mean every element is a commutator. Proof due to Ore in 1951: http://www.jstor.org/pss/2032506

2. Over a noetherian ring which does not decompose as a product, every projective module which is *not* finitely generated is free! Proof due to Hyman Bass, 1962. Still reading through this one. Seems a bit of an odd hypothesis, no? It works by making use of Kaplansky's theorem that any projective module can be written as a direct sum of countably generated modules - I looked that up, for such a scary-sounding statement, the proof is surprisingly simple - and the Eilenberg-Mazur swindle. That, and a bunch of terrible manipulations with infinite matrices to handle the countable case...

3. Speaking of that theorem of Kaplansky, here's something I was trying to find the other day. (Note: Everything in the following paragraphs is just stuff I've read, not stuff I actually know.) I recalled reading somewhere a while ago - I thought it was on MathOverflow, but I can't find it there - some guy arguing that we use the wrong definition of "projective module". His argument was as follows: He was commenting on the fact that it is impossible in ZFC to determine the projective dimension of C(x,y,z) over C[x,y,z] (it's 2 if the continuum hypothesis is true, 3 otherwise). Most people would say, wow, that's a surprisingly non-set-theoretic statement to be independent of ZFC. But he said, it is a set-theoretic statement, because our definition of projective is wrong!

Apparently there is a theorem that a module is projective if and only if it is 1) flat, 2) Mittag-Leffler, and 3) a direct sum of countably generated modules. (Don't ask me what it means for a module to be "Mittag-Leffler". I mean, I know the definition on account of having looked it up, but that's it.) He argued, for all the purposes we use projective modules in algebra, merely being flat and Mittag-Leffler is sufficient! (I have to wonder what purposes he had in mind - surely there's some times when you just want to lift some homomorphisms?) And these two conditions are nice algebraic condition. But condition number 3? Being a direct sum of countably generated modules? That's a set-theoretic condition! And one we never need in algebra! Hence our notion of "projective module" is actually hiding a set-theoretic condition inside it, and we should redefine "projective" to mean "flat and Mittag-Leffler", and if we did that, then with that notion of "projective" it would indeed be possible to determine the projective dimension of C(x,y,z) over C[x,y,z]. (I don't remember if he mentioned what it was - at most 2, I assume, but perhaps less for all I know.)

I guess I don't really have an opinion on this, since I don't really know enough to. But I'm wondering if anyone else has seen this, because I can't seem to find it anymore.

EDIT: See comments for a counterargument.

EDIT 16 December 2011: Found the thing I was talking about; it's here (not sure how I missed it). Seems the idea is due to Vladimir Drinfel'd.

-Harry