I currently have six blog posts on the ‘to-be-written’ pile, mostly arising out of conversations I had over the break. I had forgotten just how high-bandwidth face-time can be. Hopefully I can write most of these up before I run out of memory. Obviously most of the ideas in these posts are not due solely to me, though I’d like to think I played some role in their gestation. This post grew out of a discussion with Andy Egan, Eliza Block and Ted Sider during the drive from DC to Princeton.

It’s commonly believed that mereology entails that there could not be an even number of things. In fact, if there are finitely many things, then the number of things is 2ⁿ-1 for some positive integer n. See Antony Eagle’s good paper for a discussion of why this seems to be the case. There has (to my knowledge) been less discussion of what constraints mereology puts on the number of things if there are infinitely many things. But it isn’t obvious that it is completely innocent in this case either.

Question 1: Is it compatible with the axioms of standard mereology that there exist exactly countably many things?

It is certainly compatible with the principle that any two things have a fusion that there exist exactly countably many things. Consider a ‘Russian Doll’ world, consisting of objects o₁, o₂, etc, where for all i, j, if i is greater than j, then o_j is a proper part of o_i. Hence o_i is the fusion of itself with all the later o’s. Hence any set of o’s has a fusion – its least element. So any two things (indeed any set of things) has a fusion.

The Russian Doll world can be extended to show that it is compatible with the principle of fusions that there be exactly n things for any n. But obviously it does not satisfy all the principles of mereology. In particular, there is no object that is the mereological difference between o₁ and o₂, despite the fact that o₂ is a proper part of o₁. So I don’t know what the answer to question 1 is. I suspect the answer is no. It’s certainly impossible for an atomic world to have exactly countably many things, because if there are finitely many atoms there are finitely many things, and if there are infinitely many atoms there are uncountably many things. But I don’t know whether any worlds containing gunk could have exactly countably many things. I suspect not.

Note that I’m assuming here that it’s part of mereology that any things have a fusion. If one only assumes the much weaker principle that any two things have a fusion then it is easy to build a world with countably many things – countably many atoms and finite fusions thereof. If one assumes the slightly weaker principle that any set of things has a fusion it is still impossible for an atomic world to have exactly countably many things. But that assumption may be relevant to the next question.

Question 2: Which cardinalities are such that it is impossible (given standard mereology) that there exist exactly that many things?

I have no idea what the answer to that is. My bet is that (given a plausible version of set theory) there can’t be exactly countably many things, but assuming the continuum hypothesis that will be the only infinite cardinality such that there can’t be that many things.

Question 3: Is there any literature on this question?

If yes, please leave refs in the comments! If no, I guess someone should write up the answers to questions 1 and 2. But would anyone publish it?

The APA Eastern gets a fairly bad press, but I normally look forward to it. It always seems like it should be a good idea to have 2/3rds of my friends in the country gathered in the one place. Most years something manages to go wrong, but this year was a rather enjoyable conference.

The downside of all the partying was that between that and all the interviews, I only attended one session. But it was a very good one – the symposium on Ted Sider’s Four-Dimensionalism. So the average quality of papers I attended was higher than for any other APA I’ve been too, which was nice. Hopefully over the weekend I’ll write some comments on one of the (many) interesting things Sally Haslanger said, but the other three contributors (Mark Heller, Mark Hinchliff and Ted) also made excellent contributions. It might seem like damning with faint praise, but the fact that someone who’d approached the conference the way I had could be kept attentive and interested through a 3-hour symposium on the final day is really a remarkable tribute to the quality of the presentations. Hopefully many of the points the speakers made will be worked into future publications.

One quick comment on some rumours I heard flying around in DC and at various post-conference get-togethers. Brown has not already decided who we’ll be hiring, and we certainly don’t have any offers out. If we had effectively made our minds up already then I’d hope we’d have fewer hiring meetings scheduled than we do. We are still looking at many very good candidates at both junior and senior levels. Whoever we get, whether they be a new PhD or a well-established philosopher, or someone in between, will be a great benefit to the department in philosophy of science, and probably in other areas too. Any prospective grad students who were thinking of not applying to Brown because of our weakness in philosophy of science should rest assured we will be much stronger in philosophy of science going forward. So send in those applications!

I have an inexplicable fondness for college ‘football’, but I’m worried about what will happen to the economy Sunday if this NY Times report is correct.

If the [LSU] Tigers win and claim the Bowl Championship Series title, Saban will be paid one dollar more than the highest-paid college coach in the nation, according to an incentive clause in his contract.

Since Saban is a college coach, it seems he must be paid a dollar more than he is paid. Which can only happen if a dollar is worthless, which I imagine would be rather disasterous for well-established economic relations.

There are a few potential ways out of this problem.

First, one might argue that the above reasoning depends on Saban having a finite salary. If his salary is infinite, then one could argue that very loosely speaking he is paid a dollar more than he is paid. But believing this probably depends on confusing cardinals with ordinals, and in any case having infinite amounts of money sloshing through the system really can’t be good for inflation. So let’s not take that option seriously.

Second, Saban could be fired immediately so that the initial premise, that he is a college coach, is broken. This would be a rather ungrateful reaction to the guy who just won you a (share of the) national championship, but it might be in the best interests of the world economy.

Third, LSU might get beaten. I quite like LSU though, largely because they host my favourite ethics conference, so I don’t want this outcome if it can be avoided.

Obviously this is all meant to be something of a joke, because one presumes that the quantifier domain in Saban’s contract is meant to only include other coaches. Even on that interpretation, as soon as one other coach gets the same clause we could be in trouble. And given how hard it’s been for Nebraska to lure a big name coach, I would not be surprised if they do offer such a clause to prospective candidates.

Thanks to Invisible Adjunct for the original link.

I assume most of you have already seen this, but if you’re as behind the curve as I am, you might be amused by Homestar Runner. Turn sound effects on and read through Strong Bad’s email tray for hours of amusement. (Everything I saw was work-safe, but there could be stuff there I haven’t seen.)

Thanks to Eliza Block for the link.

Here’s a picture of Reuben’s gingerbread house that I mentioned a few posts back.

Much thanks again to Maree, Reuben and Simon for what was easily one of the best Christmases I’ve had in America.

By the way, the card on finger puppet Plato said that his real name was Aristocles, which I never knew.