On translation of games

At Language Log recent discussion has gone on about how you can translate from one language to another, but you can't translate from one game to another. For example, you can't take a game of chess and translate it into poker.

I'm reminded of the Subjunc-TV in Douglas Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid, which has characters tuning into a baseball game that has been made to look like a football game. Of course this doesn't work perfectly, which is intended to illustrate Hofstadter's points about the imperfection of analogies.

At Language Log, I learned that there are certain "logical games" for which a notion of translation is possible. These are apparently of interest to logicians; you can read more at the Stanford Encyclopedia of Philosophy.

But in combinatorial game theory, we can associate each position in certain games with a "number"; is it meaningful to say that positions in different games which have the same number are the "same position"? In this case, translations between games would become possible, except that those numbers are apparently difficult to calculate.

Derivation is not destiny

Arnold Zwicky at Language Log points out that the "derivative" of "financial derivatives", a word we've been hearing lots in the news lately, is not derived from the "derivative" of calculus. (This is commenting on a misunderstanding in a recently published letter to the New York Times.)

It never made sense to me that while the verb for "find the integral" is "integrate", the verb for "find the derivative" is "differentiate" -- in one case the forms are parallel and in the other they aren't. Of course the derivative involves finding a difference, but the language seems a bit inconsistent. And I've had the occasional student refer to "derivating" a function.

It probably doesn't help that they both start with d, and that every d-like symbol (off the top of my head, at least d, D, δ, Δ, and ∂) gets used for some sort of derivative/difference-like thing in some context.)

Open and closed?

At Language Log there's a post about how English-speakers use "open" and "closed", which are not grammatically the same sort of thing, in opposition to each other -- "open"/"close" or "opened"/"closed" would, on the surface, make more sense. (Compare French ouvert and fermé, which are both past participles.)

I won't try to summarize the linguistic content of the post; I'm not a linguist, although I did go through a phase where that seemed interesting.

But in mathematics-land, open and closed aren't even opposites, in the sense that open means not-closed and closed means not-open. Of course the complement of an open set is closed, and vice versa, but that's a more complicated relationship, because now we're talking about two sets, not one. This is one of about a zillion examples of how we take perfectly good natural-language words and give them specific meanings (group, ring, field, set, class, ...), which may or may not be preferable to making up entirely new words as some other fields (biology comes to mind) prefer.

Searching for mathification

Over at Language Log (which a friend of mine claims is very widely read among grad students at Penn -- and it does come from Penn -- but I read it before I came here), one often finds references to "linguification". This is the rhetorical device of expressing true things about the world in terms of false statements about language. "It snows a lot where Eskimos live" is true. "The Eskimos have eight zillion words for snow" is false, but if you say it, or in general "The language spoken by X has lots of words for Y", everyone knows that you mean "Y is important to X-speakers" -- even though it doesn't matter that such a fact might not actually be true. (The snow example is both linguification and a snowclone).

A frequent example is claims that certain words are often followed by certain other words, like "It's difficult to find a piece of writing in the mainstream press which mentions the word 'bisexual' without finding that it is immediately followed by the word 'chic'." The folks at Language Log don't like this much, in part because the write word there is not "difficult" but "trivial", especially in the age of Google.

Anyway, around the same time I came across that Language Log post, I came across "The knights who say "nerd": 20 pop-cultural obsessions even geekier than Monty Python", from The Onion's AV Club. It begins:

It's the elephant in the nerdy-obsessions room, and in the Venn diagram of nerd-dom, it may be the meeting point for everything else on this list, with good reason.

. Venn diagrams surface again later on in the article, when they're talking about Joss Whedon: "We need a Venn diagram for this one, too. (Maybe diagram-making deserves its own entry?)" Why do Venn diagrams always come up as what seems like a bad example of something that the author seems to think is mathematical? And who was Venn, anyway? So I'm starting to keep an eye out for what one might call "mathification" -- mathematical statements which occur in the popular media which clearly intend to get across a true point about the real world by making a false point about mathematics. I could swear I see this a lot, but I don't think I've seen it since seeing the Onion article ten days ago.

Are we so different after all?

Mark Liberman writes about The Pirahã and us, from Language Log:

The Pirahã language and culture seem to lack not only the words but also the concepts for numbers, using instead less precise terms like "small size", "large size" and "collection". And the Pirahã people themselves seem to be suprisingly uninterested in learning about numbers, and even actively resistant to doing so, despite the fact that in their frequent dealings with traders they have a practical need to evaluate and compare numerical expressions.

And we're like this too, he claims; to a very good approximation, we don't have words for information about the distribution of representative samples.

We have these words (the examples he submits are "percentile", "histogram", "standard deviation", "frequency distribution", "variance", and "confidence intervals") but perhaps a hundred thousand or so people in the USA actually understand what these words mean. There are three hundred Piraha; if you pick three hundred Americans, chances are you won't get one who understands this stuff. (I am not quite bored enough to go out on the street and do this study, and even if I did, I live near a university so the data would be skewed.) I suppose that this is true of any specialized field, though, not just statistics.

Notably not among that set of people are the journalists whose job it is to inform other people of these things; as readers of this blog know, this causes much comedy for those of us who do know a little bit about these things.

Although I haven't seriously thought this through, it seems plausible to me that instead of teaching college students who will take one math course calculus, we should teach them statistics; statistics might actually be useful.

Other, larger, languages have the same issue as the Pirahã do, though. It's my impression that there are very few things you can't talk about in English due to a lack of vocabulary. However, I admit that I might not know about these things if they exist -- all the languages I know are either English or have a large number of people who speak the language and English. I've heard that in smaller European languages this isn't the case -- there are things that, say, Norwegian or Catalan just doesn't have the words for, that speakers of those languages might want to talk about. (For example, what do they call the Catalan numbers in Catalan?) This is because in a large language community like English-speakers, someone will want to talk about thing X, even if thing X is rare, but this is less likely to be true in a smaller language community. The usual solution seems to be to borrow words from other languages -- but perhaps small numbers are the sort of things you just can't borrow. (Remember that I'm not a linguist.)

first let's kill all the lawyers

Language Log writes about the semantics of pork, inspired by this New York Times article. A current health care bill includes statements like

"any hospital that is co-located in Marinette, Wis., and Menominee, Mich., is deemed to be located in Chicago"

and other similar references that sound innocuous -- they're just defining a metropolitan area -- until you think, wait a minute, is that anywhere near Chicago? (I checked a map; it's 259 miles away.) And how many hospitals like that could there be? (As it turns out, exactly one.) Apparently hospitals in metropolitan areas get bigger reimbursements from Medicare for various procedures, on the theory that the cost of living for their employees is greater than that for rural hospitals.

Language Log goes on to point out that is an example of how there are two ways to define any particular set: by listing its elements, or by specifying a set of constraints. A mathematical example that immediately comes to mind is "the set of even primes". Or, somewhat more innocuously, the statement "let p be an even prime". You don't see this that often, because it's easier to say "let p equal 2". But I have seen it, in proofs of the form: "Theorem: All primes p have property x. Proof: let p be an odd prime. Then (proof). Alternatively, let p be an even prime. Then (simpler proof)."

Of course, when one specifies a set by giving constraints, there's always the problem that the set might be empty. What would happen if the health care bill in question said, say, "$100,000 should be distributed evenly between all hospitals within a quarter-mile of Isabel's apartment?" There are none. Who gets the money? (I suspect there is some conventional interpretation for this sort of thing. I don't think you'd see it in this sort of bill, but I can imagine, for example, a doting grandparent writing in their will "my grandchildren shall equally split my [large sum of money]" and then the grandchildren tragically die before the grandparent.) I've heard the apocryphal story of a student who goes into his PhD defense and says that he will be presenting results on a certain sort of algebraic structure satisfying the following eight conditions. One member of the committee interrupts and says that he can prove there are no such groups. The student doesn't get the PhD. Proving things about something that doesn't exist is considered worthless, no matter how ingenious the proofs might be.

bracketings and triple negatives

From Language Log: some commentary on yesterday's Doonesbury. The text is as follows:

Some guy: I just don't get it, Jorge. Why do you get the big bucks and not me?
Jorge: We have different work styles, man.
Some guy: Like how?
Jorge: Well, for one thing, I'm not stoned half the time.
Some guy: What are you saying -- that I am?
Jorge: I'm not saying that you're not.
Some guy: Don't try your tricky double negatives on me, señor!
Jorge: Enjoy your break.
Some guy: What?
Jorge: Never mind.

Multiple negatives are indeed tricky. As the post at Language Log points out, a lot of the time a double negative is indeed a negative, and triple negatives often turn out to be positives. The triple negatives they give examples of are generally obtained by taking a double negative which has negative semantics and then negating it again.

And our main character in this particular strip, whose name I don't know, could argue that even if he is stoned half the time, he's also not-stoned half the time. The following conversation could take place:

Jorge: Well, for one thing, [I'm not] [stoned half the time].
Some guy: I'm [not stoned] half the time.

where the same string of words occurs, but with two different meanings. In general there is quite a large number of ways to put parentheses around the words in a string like this, which is a semi-standard tactic for linguistic analysis. (It seems to me that linguists, when talking about syntax, like to draw trees when they want to explain how a whole sentence works, but they do ad-hoc bracketings like the one I did above when they just want to make a simple point about which words go with which words, because trees are annoying to draw.) The number of ways to bracket an n-word sentence completely turns out to be the (n-1)st Catalan number; a lot of problems involving trees (and various other recursive structures!) end up involving Catalan numbers. Of course, most of these bracketings are illogical: anything that looked like

I'm not [stoned half] the time

just isn't going to correspond to a grammatical interpretation of that sentence, because "stoned half" doesn't mean anything. But I suspect that longer sentences have more possible interpretations, on average; it's kind of surprising that a six-word sentence can actually be parsed in multiple ways.

In general, I suspect mathematicians tend to parse natural-language sentences differently than "ordinary people", because we are more used to dealing with subjects where the precise meaning of some statement is what matters, and an interpretation which is off by a little bit might as well be no interpretation at all. (Or perhaps worse than no interpretation -- it is better to not understand a complicated definition and know you don't understand it than to not understand it but think you do. The truly wise are aware of the limits of their knowledge.) The Jargon File, in referring to the speech style of the hacker community (which has some overlap and contact with the mathematical community), states that "...English-speaking hackers almost never use double negatives, even if they live in a region where colloquial usage allows them. The thought of uttering something that logically ought to be an affirmative knowing it will be misparsed as a negative tends to disturb them."

Names for large numbers

Language Log on number delimitation. In the United States, we put commas between every three digits: 123,456,789. In Europe they use periods instead of commas, but in the same places.

In China, they group into sets of four digits instead of three: 1,2345,6789. (I've seen this a few times written with the commas, in English-language texts; it's very disorienting.) The ancient Greeks also did something like this: they referred to "myriad", "myriad myriad", and so on, where "myriad" is 10⁴. ("Myrio-" and "myria-" are also obsolete metric prefixes for 10⁴ and 10^-4 respectively.)

In India, they break into a low-order group of 3 and then groups of 2: 12,34,56,789. This reflects the structure of the language -- certain odd powers of 10 have names, with 10⁵ being "lakh" and 10⁷ being "crore".

In a way, though, we do the same thing, and least if you consider the etymology of the number names thousand, million, billion, trillion, and so on. (By "billion" I mean an American billion, 10⁹.) "Thousand" is somehow special; we're treating the first power of 10³ differently than all the others. The British system, where successive powers of 1000 are named thousand, million, milliard, billion, billiard, trillion, ... is more "logical". But shouldn't "thousand" be "thousard" or something like that? And "billiard" is also a name for that game where you hit the balls with the sticks.

There's also the Knuth -yllion notation, which answers a question that Poser asked: are there systems with groups that double in size? The answer is yes, although I don't think anyone seriously uses this system.