Tag: mathematics

Socialist maths

The recent decision by the state of Florida to ban a slew of mathematics textbooks from its schools because of their links to banned concepts has attracted much attention. The website Popular Information has pored through the banned texts to try and suss out what the verboten ideological content might be. Some books seem to have impermissably encouraged students to work together and treat each other with respect. Another may have set off alarms because it included, among its capsule biographies of mathematicians, some non-white individuals.

I’ve always wondered, though, why it’s not considered problematic that books persistently teach the concept of division with problems that require that a fixed amount of wealth — 10 cookies, say — be allocated equally among a group of children. No consideration of whether some of the children might be smarter, or work harder, or just be closer to the cookie jar, and thus be entitled to a larger share. Pretty much the definition of socialism!

(More generally, it always fascinated me, in my years spent as an observer on the playground, that it was taken for granted that toddlers were always being pressured to share their toys, and learning to share was seen as a developmental milestone, where we do not expect adults to be willing or able to share anything at any time.)

Bergson and Brexit

Once it became clear that I would be staying indefinitely in the UK, I had long planned to apply for UK citizenship. I am a strong believer in democracy, and I thought it would be the good and responsible thing to vote and otherwise take part in politics.

Then came Brexit, and this, naturally, led me to think about Henri Bergson. Born to a Jewish family, Bergson moved gradually toward Christianity in his personal life, he considered himself a Christian from the early 1920s. By the 1930s he was making plans to convert formally to Catholicism, but held off because of solidarity with the increasingly threatened Jewish community. A few weeks before his death, Bergson left his sickbed — having rejected an offered exemption from the anti-Semitic laws of Vichy — to stand in line to register as a Jew.

He wrote in his will:

My reflections have led me closer and closer to Catholicism, in which I see the complete fulfillment of Judaism. I would have become a convert, had I not foreseen for years a formidable wave of anti-Semitism about to break upon the world. I wanted to remain among those who tomorrow were to be persecuted.

For a while, then, I deferred applying for citizenship, out of solidarity with my fellow migrants. And then I went and did it anyway. The stakes are obviously much lower than they were for Bergson. And while I regret having to renounce the migrant identity, which suits me well, I also see that this isn’t an entirely noble inclination, as it also excuses me from taking a citizen’s responsibility for the nation’s xenophobic turn. It’s easy to blame those dastardly “British”. The permission to acquire citizenship reflects the growing responsibility for the society that one acquires merely by living here.

I also can’t resist noting that Bergson’s first publication was the solution of Pascal’s problem in Annales des Mathématiques, for which he won the first prize in mathematics in the Concours Général. On learning that he was preparing for the École normale supérieure entrance examination in the letters and humanities section, his mathematics teacher reportedly exclaimed

You could have been a mathematician; you will be a mere philosopher.

The Brexit formula

A collaborative project with Dr Julia Brettschneider (University of Warwick) has yelded a mathematical formulation of the current range of Brexit proposals coming from the UK, that we hope will help to facilitate a solution:

Numeric calculations seem to confirm the conjecture that the value of the solution tends to zero as t→29/3.

Boris Johnson has an attack of the sads

Haunted Boris Johnson

Now that Theresa May has forced her cabinet to acknowledge a tiny portion of the reality of Brexit, Boris Johnson has apparently taken to moping around Whitehall to make certain that no one will think that he’s happy to now be conspiring with reality to betray his cause.

The best line is this:

Those close to the foreign secretary say that he feels he has been “bounced” into agreeing to a deal that is a world away from the hard Brexit he campaigned for. “He thinks that what’s on the table is so flawed we might even be better off staying in,” one said.

Continue reading “Boris Johnson has an attack of the sads”

The president’s dilemma

In the classic prisoners’ dilemma, two members of a criminal gang have been caught by police. There is enough evidence to convict them of minor crimes, but without testimony from one of them they will receive only a light sentence, say one year in prison. If one of them agrees to cooperate with the investigation, prosecutors will let him out for time served, and be able to send the other to prison for ten years. But if they both cooperate with the investigation, both will go to prison for five years (perhaps because the prosecutors will have their information, but not their testimony). Key to the game is that the players are unable to coordinate their strategy. Clearly the best for both of them would be to keep quiet, but the strategy of cooperating with the investigation is superior, from their private perspective, regardless of what the other player does. So they both talk, and both get heavy sentences.

One weird thing about the story here is that the symmetry really doesn’t make sense. It’s not impossible, but it’s peculiar to imagine prosecutors being so interested in pinning the major crime on someone that they’re willing to let a confederate walk free, but indifferent to who flips on whom. That suggests we consider a less-known hierarchical version of this game, where one player is the powerful boss of a crime syndicate — let’s call him “The President” — and the other one is “The Attorney”, who knows all the details of his crimes, and is sufficiently involved to be criminally liable himself. Let’s call this game “The President’s Dilemma”. Continue reading “The president’s dilemma”

Suspicious precision

Kevin Drum notes some tweets from lawyer Susan Simpson. She was perusing (as one does) the public records of Trump campaign expenditures, and noticed something funny:

To me it looks almost too exact. Is Michael Cohen, in trying to cover up a $130,000 transfer, really incapable of seeing that $129,999.72 looks suspiciously close? Couldn’t he at least swallow, I don’t know, a $20 loss and make it $129,981.34?

Worst mathematics metaphor ever?

I’ve come to accept “growing exponentially” — though I once had to bite my tongue at a cancer researcher claiming that “exponential growth” of cancer rates began at age 50, because earlier the rates were just generally low — and didn’t say anything when someone recently referred to having “lots of circles to square”. But here’s a really new bad mathematics metaphor: the Guardian editorialises that after Brexit

Europe will be less than the sum of its remaining parts.

“More than the sum of its parts” or “less than” is something you say when you’re adding things together, and pointing out either that you don’t actually get as much extra as you’d think or, on the contrary, that you get more. That you get less when you take something away really doesn’t need much explanation and, in any case, it’s not about the sum of the parts. Whether the remains of Europe are more or less than the sum of the other parts seems kind of irrelevant to whatever argument is being suggested.

Condorcet method for choosing a religion

Making choices is hard! Particularly when there are multiple possibilities, differing in multiple dimensions. Like choosing the best religion.

There are many possible methods, leading to a variety of outcomes. The 18th century French mathematician Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet, advocated privileging methods of deciding elections that will always grant victory to a candidate who would win one-on-one against each other candidate individually. (Of course, there need not be such a candidate.) Such methods are referred to as “Condorcet methods”.

I’ve just been reading The Jews of Khazaria, about the seventh to tenth-century kingdom in central Asia that converted to Judaism around the middle of the ninth century.

According to the Reply of King Joseph to Hasdai ibn Shaprut, one of the few surviving contemporaneous texts to describe the internal workings of the Khazar kingdom,

Each of the three theological leaders tried to explain the benefits of his own system of belief to King Bulan. There were significant disagreements between the debaters, so Bulan went a step farther by asking the Christian and Muslim representatives which of the other two religions they believed to be superior. The Christian priest preferred Judaism over Islam, and likewise the Muslim mullah preferred Judaism over Christianity. Bulan therefore saw that Judaism was the root of the other two major monotheistic religions and adopted it for himself and his people.

You can’t have your pocket money and save it too

My 13-year old child received the following maths problem in school:

Paul saves 4/15 of his pocket money and spends 5/12 on topping up his mobile phone. What fraction of his pocket money does he have left?

(The question was part of a sheet from Cambridge Essentials.) With a PhD in mathematics, I usually feel myself adequately qualified to deal with school maths questions, but this one stymied me. I have decided to stop blaming myself, though. This question is

  1. misleadingly formulated;
  2. ambiguous;
  3. exceptionally dependent on hidden cultural assumptions.

Let’s start with number 1. Who counts fractions of pocket money? This makes about as much sense as asking

Paul and Paulina order a pizza together. Paul eats 0.375 pizza. Paulina eats 0.5 pizza. How much pizza do they take home?

It’s like you were trying to teach children about toothbrushes, and showing them how useful they are by having them use the toothbrush to clean the floor. Sure, you can do it, but it’s really not the tool anyone would choose to use, and it doesn’t give them a fair impression of what it could really be good for.

Okay, maybe Paul lives in a socialist country, where “from each according to his ability”, so that prices are stated as fractions of your income. But it gets worse. Point 2: My first thought was that Paul had spent 11/15 of his money on other things — probably drugs — and now had to top up his phone, which cost 5/12 of his pocket money. But he only has 4/15, which is smaller, so he needs to go into debt by 5/12-4/15=3/20. Okay, that didn’t seem likely. So then I figured that the 5/12 was intended to be a proportion of the 4/15 that he has remaining. Then it would at least make a little bit of sense to express it as a fraction. (Extreme socialism: Prices are all formulated as a fraction of the money you have in your pocket. Customer: How much? Merchant: How much you got?) So the amount remaining is 4/15*7/12=7/45.

But on further discussion with my partner I recognized that neither of these versions was what was intended by the people who set the question. I was thinking in terms of a model of sequential spending: The money you “save” is the money you have available to spend the next time an expense arises. The question, though, presumes that money that is “saved” is being saved from yourself. Whereas I would think that the money you “save” is part of — or possibly identical with — the money you “have left”, you were supposed to think of spending and saving as just two different ways of losing money. You add the two together to get a total loss of 4/15+5/12=17/30, leaving Paul with 13/30 pocket money units to spend on non-mobile-phone and non-banking expenses. (Probably drugs.)

Of course, I’m overthinking this. The point is that you’re not supposed to think. You’re just supposed to see two fractions and add them, because that’s what you’ve been learning to do. It’s a kind of pseudo-applied maths problem that is quite common — even at university level — where any actual thought about the issues involved will only penalise you. It’s a puzzle, where you’re supposed to read through the irrelevant verbiage to get to the maths problem that has been concealed there.

I call this “adding up the temperatures”, after the story by Richard Feynman (in Surely You’re Joking, Mr Feynman) about his time evaluating textbooks for the state of California. He describes a problem from one elementary school textbook:

Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees, Green stars have a temperature of seven thousand degrees, blue stars have a temperature of ten thousand degrees, and violet stars have a temperature of … (some big number).

John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?

Feynman points out that the temperatures aren’t really right, and that there is no such thing as green and violet stars, which he is willing to tolerate, but then blows up at the sheer pointlessness of adding up temperatures. Like the above, it only looks like an application of the mathematical tool being presented (in this case addition).

But I’m even more amazed at the absurdity of the story. How is it possible that John sees only 3 stars, his father sees 4, and they see completely different stars? But the point is, in school mathematics you’re supposed to do, not think.