November 2017


Someone over at Metafilter had a great question: What syntactic category are mathematical operands? (Their username is notsnot, in case this needs to go in a dissertation someday.)

Let’s start with something like

Three plus four is seven.

For now, we’ll say the numerals are NPs.  In a construction NP <word> NP, the <word> could be various things: a verb, a conjunction, a preposition. We can immediately rule out verbs, since plus and its friends (minus, times, over, etc.) are not conjugated.

We should also look at non-mathematical sentences, like

Determination plus luck means victory.

Let’s do some syntax. There are some standard though fallible tests for prepositions.  For instance, they can usually be modified by right:

He fell right in the river.
She lives right down the street.
Go to the cave right in the forest.

*Determination right plus luck means victory.
*Three right plus four is seven.

Prepositional phrases (PP) can often be fronted:

Up the hill she walked.
*Plus four three is seven.

You can front a PP and replace the NP with an interrogative, or front just the questioned element:

Sam is the king of England.  Seven is three plus four.
Of what is Sam the king? *Plus what is seven three?
What is Sam king of?  *What is seven three plus?

PPs allow gapping:

Sam is king of England, and Joe, of France.
*Seven is three plus four, and eight, plus five.

These tests aren’t definitive, but plus is failing every one of them. A better match might be conjunctions. Plus, like and, can link NPs or sentences, and can be used multiple times:

Bill and Anne and Rahesh came.  Two plus two plus one make five.
Bill came, and Anne left.  Bill came, plus Anne left.

On the other hand, this transformation sure doesn’t work:

Sam is a king and Sam is a dancer.  >> Sam is a king and a dancer.
X is 4, plus X is sin θ >> *X is 4 plus sin θ.

It looks like the construction S, plus S isn’t really the same plus as in two plus two.  And other operators don’t allow it at all:

*Bill left, minus Anne stayed.
*Bill left, over Anne stayed.

A bigger problem is that English sentences allow literally infinite amounts of inserted material.

Sam and Alice are nobles.
Sam and Alice are fine, just nobles.
Sam and Alice are still nobles.
Sam, prince of Florin, and Alice, duchess of Guilder, are nobles of Sylvania.
Sam and possibly Alice are, as of Tuesday, nobles.

How much of this can we do with mathematical expressions?

Three plus four is seven.
Three plus lovely four is seven.
Three plus four is still seven.
Three, square root of nine, plus four, half of eight, are seven.
Three and possibly four are, as of Tuesday, seven.

These are not impossible, but at best they sound jocular.  The additions are not math; they’re intrusions of ordinary English.

There’s also the complication that mathematical plus and minus can be unary: you can say Minus three plus four is one. You can have a conjunction beginning a sentence (And the Lord said to Moses…), but that’s not how minus is working here; it’s obviously a modifier for three.

Not to belabor the obvious, but many of the basic things we can do with a sentence don’t really work in mathematics.  You can’t really put a mathematical expression in the past tense, or use the present perfect, or use pronouns, or passivize, or insert a relative clause, or nominalize, or cleft, or topicalize.

And all this is looking at a very basic expression that probably did arise out of normal syntax.  It’s even harder to apply our notions of normal English syntax to something like

x equals minus b plus or minus square root of b squared minus four a c over two a.

or

e to the i n equals cosine of n plus i times sine of n.

I’ve gone into this much detail to convince you (and myself) that ordinary English syntax doesn’t really explain mathematical expressions.  I hope my conclusion doesn’t shock or appall you: mathematical expressions don’t follow English syntactic rules; they follow mathematical rules.

Now, maybe you could shoehorn the quadratic formula or Euler’s formula into the syntactic framework of your choice. I will bet you, however, that you’ll end up with a pile of very idiosyncratic special rules and special syntactic categories, and a bunch of ad hoc exclusions of normal English rules.

And there’s an alternative formulation that would end up far simpler than that: mathematical expressions have their own cross-linguistic syntax, based on their written form, and languages have conventions on how to say them aloud.

I don’t think this is terribly surprising… it’s like discovering that the Russian of Tolstoy’s War and Peace contains a number of passages which are written in the Roman alphabet and don’t follow ordinary Russian syntax.  Is this a revolutionary discovery about weird undercurrents of Russian?  No, it’s just that Tolstoy included quite a bit of French in the text.  Similarly, English sentences can have embedded mathematics.

Still, I hadn’t thought about it this way, and I find it interesting that a pretty ordinary part of English turns out to be, well, not really English at all.

Now, for historical and practical reasons, there’s a certain overlap, especially with basic arithmetic. People undoubtedly said “Two and two are four” (or “twá and twá sind féower”) long before international mathematics was formalized. So these behave more like ordinary English than the quadratic formula does.

Plus, the conventions for speaking math out loud were, of course, invented by speakers of the language out of existing (or newly borrowed) words, and follow ordinary language conventions– where possible.  So you can read cos (2θ) as “cosine of two times theta”.  On the other hand you can just read it as “cos two theta”, which probably has no non-math analogue in English.

(I should add that programmers are very familiar with the idea that math expressions have a particular syntax.  They don’t bother with linguistic categories at all; they define their own, such as operators, variables, constants, functions, and statements.)

 

 

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I’ve basically redone the Almeopedia. You can find it here.  The help page is here.

ticai-alchemy-shop

The old version (which is still up, for now) has had a speed problem for a long time. Just loading a page took 40 seconds, which made browsing painful and editing almost impossible.

I’ve been redoing pages (like the numbers page) by using Javascript to process big raw text files, and I used the same method here. It runs lightning-fast on my computer. The first few links may be slow when you try the above link, but it should speed up once the files are in your cache.

Since it’s now my code, I was able to do some side features:

  • shortcuts for the Unicode letters I use
  • changes to facilitate the historical atlases
  • SFW mode (makes the pictures tiny and adds business-friendly nonsense titles)
  • improvements to the choose-your-own-adventure section
  • customizations to the Wiki markup to eliminate busywork

There’s about 3 meg of raw text.  I’m amazed at how fast Javascript can plow through it (e.g. if you do a text search).  I skipped some optimizations I could have done simply because it’s good enough as it is.

I also took the opportunity to put the Historical Atlas of Skouras into Almeopedia. The old version was in Flash, which is deprecated these days.

Unfortunately, the new version isn’t externally editable.  Still, if you find errors or think something should be added, shoot me an e-mail.

There are undoubtedly errors and bugs… I copied all the text by hand, but I haven’t yet checked each page.  But, one of the purposes of the project was to make it easy to revise what’s there and put out lots of new material.

 

 

I came across this interview with Eric Weinstein, and it got me thinking. Weinstein is an advisor to Peter Thiel, and– wait, come back!  I know Thiel is the embodiment of everything wrong with late capitalism, but the article is only half that. It’s deeply weird and doesn’t quite know what it wants, but it does have some interesting things to say about the frivolity economy.

Candy_LargeWide

His thesis, more or less, is that capitalism should be more disruptive, and also combined with socialism. But let’s start out with the strong bit: his critique of American education.

The problem is that we have an educational system that’s based on taking our natural penchant for exploration and fashioning it into a willingness to take on mind-numbing routine. This is because our educational system was designed to produce employable products suitable for jobs, but it is jobs that are precisely going to give way to an economy increasingly based on one-off opportunities.

Now, this isn’t entirely right. It’s not that routine is the goal of education; it’s that mass education in any form involves teaching things the kids don’t care to learn at that moment. Kids actually learn quite readily if they happen to love the subject.  Plus, sometimes you can’t get to the good parts without mastering the boring bits. You can’t learn quantum mechanics without learning calculus; you can’t read Sanskrit literature without mastering Sanskrit morphology.

But Weinstein has a point.  Traditional societies didn’t educate the majority at all.  Industrial society educates everyone because manufacturing and service jobs require a high level of literacy, and jobs will resemble school in many respects… often, the unattractive respects: lots of routine and rules; mindless obedience; doing things you are not personally interested in because it’s Your Job.

You can see how this sort of education and a certain sort of society fit together nicely in the career of … my Dad.  He was bright and did well in school, though he never went to college.  He started as an assistant pressman at a major printing company, rose through the ranks to management, and ended up as an executive, an expert on technical printing problems.  He not only stayed in one industry, he stayed in one company. It was a pretty good life for him, and a definite step up from his father, who was a carpenter.

Now, late capitalism has nearly destroyed that sort of career– especially for those who never make it to management.  Weinstein takes the view that predictable, medium-affluent working class jobs are over, presumably because those are ripe to be automated.

At this point many give up and conclude that humans are worthless, or at best should just be given a stipend while the robots do all the work.  Weinstein says instead that humans should do what they’re best at: handling one-time events. Or with less jargon: doing things that aren’t easily automated, because they require adaptability and creativity.  For him this means technologists and finance people, but also inventors, artists, and writers.

I’d actually make this category far broader– see my list of professions unlikely to be taken over by AIs. It’s interesting to speculate what sort of education would focus on these skills, rather than those required to be an accountant or an assistant pressman.

This is also about the point where Weinstein dissolves into a messy set of contradictions. He’s obviously been talking to too many venture capitalists: he tells us “certain fields will need to undergo a process of radical deregulation to give the minority of minds that are capable of our greatest feats of creation the leeway to experiment and to play.”  Unreflective Thiel-worship, in other words.  Haven’t those people done enough damage?  But!  He also thinks we need to recognize the “dignity, well-being, and health” of the 90%.  He mentions a universal basic income, but so far as I can see, his bright idea is that this is tied to having jobs that aren’t necessarily marketable.  So, you can be a playwright or something, and still get paid enough to live on, and this is somehow valuable dignified recompense but it’s not, y’know, welfare.

The interview ends with Weinstein’s assurances that the Very Powerful are thinking very hard about inequality these days. And this of course is complete nonsense.  His boss, Thiel, supports the GOP, whose first order of business right now is not subsidizing playwrights, but handing the super-rich another five trillion dollars.  Whatever the super-rich say to Weinstein at cocktail parties is not the truth of their secret benevolent hearts; it’s the bullshit they tell the 10% to hide what they are really doing to the rest of us.

(As a side point, I’m not saying that the super-rich are all evil plutocrats.  Only half of them are!  Once they have all the money, quite a few rich men develop other interests, including improving the world. The class to be terrified of is the moderately rich: the mere millionaire.  They’re the ones who need more money and can’t understand any value but money and vote consistently Republican.)

What’s interesting is that there’s a bright future we could have— if we chose to pursue it. Thinkers of the early 20th century couldn’t see past the mass of uneducated peasants in their societies.  Aldous Huxley couldn’t see any alternative to keeping around a class of “Epsilon Semi-Morons”; Orwell couldn’t imagine anything nicer than spreading around the wealth such that everyone could eat but no one could go to a restaurant.  But postwar America and Europe turned out very differently from either vision.  It turned out that everyone could have a way better job than peasant, and everyone could go to restaurants.

And similarly, lots of people can’t see a future right now that doesn’t include masses of factory jobs.  They fixate on what ex-factory workers could do, and all that comes to mind is, well, factory work.  But the world is actually quite rich, and AI could make it far richer.  You really could put everyone to work doing things that are today only open to the 10%.

The thing is, you don’t do this by deregulation, disruption, and giving the super-rich more money.  Weinstein’s bosses won’t give us that bright future: no ruling class voluntarily disinherits itself.  They won’t even give us a UBI, because it would cost money.

The tricky bit is, of course, how do we get there from here? Well, it’s not going to be a quick process.  But we can’t even start on it till we 1) shut down the current wave of downright reactionaries; and 2) get rid of plutocracy.

(If you want an idea to rally around, though: UBI isn’t a bad idea, but combine it with Piketty’s 0.5% tax on wealth. You can’t get to the bright future and also keep increasing inequality.)