Sunday, December 15, 2013

Fallacious proof of the continuum hypothesis

One of the things amateur mathematicians like to do is come up with fallacious proofs and disproofs of the continuum hypothesis, in much the same way professional logicians like to come up with fallacious rehashings of the ontological argument for the existence of God. Today’s fallacious argument purports to prove that the continuum hypothesis is true. Here we go.

Introduce continuum many names into the language, which are to be names of sets of natural numbers, i.e. subsets of ω. For each new name, add in countably many sentences saying which numbers are and aren’t members of it, and one saying that it is a subset of ω. If all of these sentences are true, then all of the collections of natural numbers form a set. So if the set of these new sentences is consistent with the continuum hypothesis, then the continuum hypothesis is consistent with every collection of natural numbers forming a set. It would only be false in models with a non-standard interpretation of the membership relation, or with non-standard elements as members of ω.

And as it happens, all the new sentences are consistent with the continuum hypothesis. We get this result from the compactness theorem. For every function f from a finite initial segment of the natural numbers into {1, 0}, there will be a constructible subset s of the natural numbers such that n∈s if f(n)=1 and n∉s if f(n)=0. So any finite collection of the new sentences will be true in the Gödel constructible universe. This means that ordinary (ZFC) set theory, the axiom of constructability, and the new sentences are consistent, and since they include ZFC and the axiom of constructability, they entail the continuum hypothesis. This means the continuum hypothesis can be true in a model where every collection of natural numbers form a set. That means it’s true in the intended model, and that means it’s true.

Now, presumably this argument is fallacious. It’s so short! But as with the rehashings of the ontological argument, the fun is in working out what’s wrong with it. So: what’s wrong with it?

Tuesday, September 3, 2013

Chess in a dress update

Regular readers may recall that I once conjectured that I could improve my chess by wearing a dress. Well, you’ll be pleased to hear that I’m finally getting round to testing this. Day 1 was allocated to the control group, and I scored 1.5 out of 4. Day 2 was allocated to the intervention group, and I scored 2.5 out of 4. Pretty promising! After day 40 I’ll see whether the intervention days are significantly more successful than the control days. I’m also gathering some other data to see if there are any more patterns I can test in future work. I’ll let you know how it goes, but don’t put it your diaries as I’m quite busy at the moment and the trial days may not be consecutive. Are you excited to find out? I know I am!

Monday, August 5, 2013

Scrabble: the gathering

I played Magic: The Gathering a few times a while ago, but I never really got into it. I like games and I quite like most of the idea of this one, but there’s one feature that puts me off. It’s not the wizards and magic element; I don’t care about that one way or the other. It’s the building your own deck element. For those of you who don’t know, Magic is a card-based game, where each player uses their own personal deck of cards. You don’t use a standard deck; you put together a deck with good cards in it, to give yourself an advantage. A deck can be good by containing cards which are good in themselves, and by having cards that work well together. This leads to the Magic experience having two parts: playing the game, and buying and trading cards to produce a kick-ass deck. The second part can get quite expensive if you take it seriously, although I don’t see that this in itself should put you off when you compare it with other leisure activities like playing golf, going to the football or drinking in a pub.

Anyway, when I play a game I want it to be fair, either by being roughly symmetrical (like chess) or by having a well-organized handicapping system (like golf). And I suppose that for me the collecting part of the Magic hobby is not only unappealing in itself, but also spoils the playing part. I’d prefer it if the cards were dealt randomly from a communal deck. People could still collect fancy cards to add new elements to the game, although since fancy cards wouldn’t give their owner an advantage people wouldn’t want them so badly and the Wizards of the Coast wouldn’t make so much money. So it won’t happen. Oh well.

When I was thinking about why I don’t like Magic, I thought of how Scrabble could be modified in a similar way, and that I wouldn’t like that either. The corresponding modification to Scrabble would be for each player to have their own customized bag of tiles, instead of both drawing from a standard communal one. There would have to be some restrictions on what a bag could contain to stop people just having a bag full of blanks and laying down bingo after bingo, although thinking about it this would probably lose to a bag with mostly blanks and the odd high-scoring letter, so perhaps the unrestricted version would have some interest. But this would be feasible, and I understand there are restrictions on Magic decks too. Corresponding to the fancy new cards in Magic, you could have fancy new tiles, like a tile that could be any vowel (maybe scoring ½), or a tile which turned its square into a double word score, or increased the values of adjoining tiles. There are all kinds of possibilities.

Now, I’ve already said that I wouldn’t welcome this modification to Scrabble myself, but the kind of people who like Magic might, and I wouldn’t mind them sending some of their money my way. I could call it Words with Enemies. I doubt the makers of Words with Friends would have the front to sue me for making a minor modification to their idea and marketing it as my own. That would be ridiculous.

Wednesday, June 26, 2013

Weird expression

I’m a philosopher of language, and if you don’t know much about what philosophers of language do then you might think that means I know all about grammar and syntax and subordinate clauses and that sort of thing. Sadly it doesn’t: philosophers of language can get by without knowing much grammar at all. (You should see some of the things we say about the word ‘that’.) Nonetheless, today I’m writing about grammar. I do hope I’m not becoming one of those ill-informed windbags that Geoff Pullum so entertainingly excoriates on Language Log, but time will tell.

Anyway, the expression I’ve been wondering about is ‘try and’, as in ‘I’ll try and get home in time for tea’. I understand that some people don’t like this expression, and pretend to misunderstand it as (roughly) ‘both try and’. If it meant that then you couldn’t try and do something without succeeding, and this isn’t what people mean. They use it to mean what ‘try to’ uncontroversially means. Maybe people pretend to misunderstand it because they think it’s new and they don’t like new things. Or maybe they dislike it because it’s weird.

The OED gives six examples of ‘try and’ being used like this, between 1686 and 1883, and I saw another example on the BBC website yesterday, even though I wasn’t consciously looking for one. I’ve forgotten where it was though, so I deliberately found another example. It's in the text under this video. It’s even easier to find examples in reported speech, which I suppose suggests it’s a bit informal. The OED says it’s colloquial, so I guess they agree.

Anyway, the grammar they give for ‘try and’ is that it’s followed by a co-ordinate verb, whereas ‘try to’ is followed by an infinitive. I don’t know whether that’s right. Consider these two:
  • ?I try and be the best.
  • *I try and am the best.

To me, the second sounds totally ungrammatical if it’s meant to mean ‘I try to be the best’, and the first sounds a lot better. It seems to be the one people use, anyway. But if I’ve understood the OED’s rule correctly (I blush to confess that I’m a bit unfamiliar with the terminology and after looking it up I’m still not quite sure), they predict that the second is better. It isn’t.

Leaving ‘be’ out of it for the moment though, I think the only times ‘try and’ really sounds fine are when the second verb is the same in the infinitive form and in the form ‘try’ takes in the sentence. These all sound pretty bad to me, where ‘try and’ is meant to mean ‘try to’:

  • *I am trying and find my keys.
  • *I am trying and finding my keys.
  • *He tries and find his keys.
  • *He tries and finds his keys.
  • *I tried and find my keys.
  • *I tried and found my keys.

I don’t actually know whether I use ‘try and’ in my own speech, and a search of my blog seems to reveal that I don’t use it there, but I'm used to reading and hearing it so my grammaticality reactions shouldn't be too far off. I’m fairly sure that people who do use it only do so when ‘try’ is in the plain-looking form, so (except with ‘be’) it doesn’t matter whether the verb following ‘and’ is supposed to get modified or not. All the OED’s examples are like that. I guess when people are using a form with 'trying', 'tries' or 'tried', they go with ‘try to’.

This seems an odd state of affairs. Maybe what’s going on is that there are two rules, which in a lot of constructions can’t both be obeyed: use an infinitive verb and use a co-ordinate verb. If you can’t do both, it sounds wrong. Is that plausible? It doesn’t seem terribly plausible, especially when we’ve had at least 327 years to get used to it. Our language-processing machinery has had plenty of time to work out how it’s analysing the construction. But I can’t see what else might be happening.

Another problem is that this explanation doesn’t predict that ‘try and be’ sounds OK. On the other hand, while Google Ngrams has no results for either ‘he tries and be’ or ‘she tries and be’, the former gets plenty of genuine results from a standard Google search and the latter gets none. That’s pretty weird, and I’m not sure what kind of theory would predict it. But it’s how we talk. If you know of any proper linguists who know what the deal is with ‘try and’, do let me know, and if the OED really do have it wrong then maybe let them know too.