Occasional reflections on Life, the World, and Mathematics

Posts tagged ‘mathematics’

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.

A dose of mathematics

In Kathryn Hughes’s biography of George Eliot a letter is quoted from the 30-year-old Mary Ann Evans (in 1850, not yet George Eliot)

I take walks, play on the piano, read Voltaire, talk to my friends, and just take a dose of mathematics every day to prevent my brain from becoming quite soft.

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.

Hannah’s sweets

The following problem appeared on one of yesterday’s GCSE maths exams, leading to considerable frustration and media attention:

Hannah has 6 orange sweets and some yellow sweets.

Overall, she has n sweets.

The probability of her taking 2 orange sweets is 1/3.

Prove that: n^2-n-90=0

^ is “to the power of”

Now, I am a professional probabilist, and I wasn’t immediately sure how to do it. Why not? Well, there’s something missing: The problem doesn’t tell us what Hannah’s options are. Did she pick sweets at random from the bag? How many? Are we asked the probability that she took 2 orange sweets rather than 3 yellow, given that she actually prefers the orange?  Did she choose between taking sweets out of the bag and putting it away until after dinner?

There should have been a line that said, “She picks two sweets from the bag, at random, without replacement, with each sweet in the bag equally likely to be taken.”

According to the news reports

Hannah’s was just one of the many supposed “real life” problems that the students were required to tackle.

This is just an example of the ridiculous approach to mathematical “applications” induced by our testing culture. It’s not a “real life” maths problem. It’s a very elementary book problem, decked out with a little story that serves only to confuse the matter. You are supposed to know a standard rule for decoding the chatter. If you try to make use of any actual understanding of the situation being described you will only be misled. (Richard Feynman described this problem, when he was on a commission to examine junior high school maths textbooks in California in the 1960s. His entertaining account is the chapter “Judging Books by their Covers” in Surely You’re Joking, Mr Feynman.)

The Nobel prize in mathematics

There was an interesting article in Der Spiegel about Angela Merkel’s visit to a Berlin secondary school as part of the the “EU-Projekttag”, a national day for teaching about the EU and its institutions. (No surprise that nothing like this happens in Britain.) This school has mostly Muslim immigrant children, and she found that instead of asking about the functions of the European Parliament the children wanted to tell her about discrimination in Germany.

Fatma, eine 15-jährige Jugendliche mit Kopftuch, klagt über Schwierigkeiten beim Praktikum im Kindergarten, weil die Eltern keine Erzieherinnen mit Kopftuch wollen. Das habe ihr Chef ihr gesagt. Ja ja, sagt Merkel, die inmitten der Schüler auf der Bühne Platz genommen hat, man kenne das Problem von Bewerbungen junger Menschen mit komplizierten, ausländisch klingenden Namen. “Viele glauben da nicht, dass jetzt gleich ein Nobelpreisträger in Mathematik um die Ecke kommt.”

[Fatma, a 15-year-old with head-scarf, complains about her difficulties in an internship in a kindergarten, because the parents don’t want a teacher with head-scarf. Her boss told her that. Yes, yes, says Merkel, who is sitting on the podium with the students, we know these problems, as with job applications from young people with complicated, foreign-sounding names. “Many people don’t think, this is a future Nobel-prize winner in mathematics coming around the corner.”]

Never mind this bizarre and nearly incomprehensible stream-of-consciousness from a major world leader asked an uncomfortable question by a 15-year-old. What is it about the chimeric Nobel prize in mathematics? Alfred Nobel established prizes in subjects that were related to the kind of practical science that he made his fortune with (chemistry and physics) and to the kind of selfless causes (medicine, literature, peace) that he hoped would blur the association of his name with weapons manufacture. There are lots of subjects that he did not create prizes in. Mathematics. Geology. Engineering. Astronomy. History. Cooking. No one thinks it odd that any of these subjects don’t have a Nobel prize, except mathematics. They think it so odd, that they either imagine that there actually is one, as above, or they invent outlandish stories to explain this lacuna, generally involving some mathematician — Gosta Magnus Mittag-Leffler, when he is given a name — running off with Nobel’s wife. (This story has the advantage of Mittag-Leffler actually having been Swedish, but the fact that Nobel never married is usually counted against its credibility.)

The Pope’s Shluchim

I’ve just been reading Amir Alexander’s book Infinitesimal, about the intellectual struggle over the concepts of infinitesimals and the continuum in mathematics and science (and theology) in the 17th century. The early part of the book is a history of the Society of Jesus, presented as a ruthless and intellectually daring force for religious conservatism, strictly hierarchical, devoted to its holy founder, a thoroughly mystical movement that built its reputation and influence on educational outreach. And then it struck me: The Jesuits were just like Chabad-Lubavitch!

Correct me, Lord, but in moderation…

Jeremiah 10:24.

Accounts of error-correcting codes always start with the (3,1)-repetition code — transmit three copies of each bit, and let them vote, choosing the best two out of three when there is disagreement. Apparently this code has been in use for longer than anyone had realised, to judge by this passage from the Jerusalem Talmud:

Three scrolls [of the Torah] did they find in the Temple courtyard. In one of these scrolls they found it written “The eternal God is your dwelling place (maon)“. And in two of the scrolls it was written, “The eternal God is your dwelling place (meonah)”. They confirmed the reading found in the two and abrogated the other.

In one of them they found written “They sent the little ones of the people of Israel”. And in the two it was written, “They sent young men…”. They confirmed the two and abrogated the other.

In one of them they found written “he” [written in the feminine spelling] nine times, and in two they found it written that way eleven times. They confirmed the reading found in the two and abrogated the other. (tractate Ta’anit 4:2, trans. Jacob Neusner)

(h/t Masorti Rabbi Jeremy Gordon, who alluded to this passage in an inter-demominational panel discussion yesterday at the OCHJS. He was making a different point, which for some reason had very little to do with information theory.)

The next war

The BBC reports that education secretary Nicky Morgan “wants England to be in the top five in the world for English and maths by 2020. It is currently 23rd.” They quote her:

Returning us to our rightful place will be a symbol of our success. To achieve this, we will launch a war on illiteracy and innumeracy.

So, I’m thinking about wars that Britain has prosecuted over the past half century or so, often with the goal of “returning us to our rightful place”. Suez. Falkland Islands. Bosnia. Iraq. Yemen. Cyprus. Kenya. Afghanistan. Northern Ireland. Not all disasters, but not an unbroken record of glory either. Not really a set of memories you want to activate if you want your audience to think “overwhelming success” rather than, say “useless drain on national resources”, “antiquated racist ideology”, or “undermining democracy and human rights”.

Putting aside the absurd-sounding ambition for England to be among the top 5 for English, (I’ll just guess this wording reflects the slightly vague British awareness that foreigners tend to speak Foreignish, and so might have literacy skills to be tested that aren’t literally “English”) the battle plan for maths all comes down to tables:
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Math and science

Corey Robin updates us on l’affaire Salaita. I was struck by his comment

Thirty-four heads of departments and academic units at the University of Illinois at Urbana-Champaign wrote a scorching letter to the University of Illinois’s new president[…] Clearly, far from diminishing, the controversy on campus has only expanded.

What’s even more amazing is where it has expanded: three of the signatories are chairs of the departments of chemistry, math, and statistics. The opposition has spilled beyond the walls of the humanities and social sciences. During the summer, lots of folks dismissed this story because the natural sciences weren’t involved. Well, some of them are now.

Math and statistics aren’t really natural sciences, in the crucial economic sense. The people who dismiss the boycott because it’s just the humanities and social sciences are somewhat expressing a sense that those academics are woolly-headed cultural relativists; but even more, I think it’s about the idea that “serious” academics have big grants and big labs and generally deal with big money. Chemistry is the outlier here. Math and statistics are still much more constructed on the same economic model as the humanities, hence barely one step removed from socialism.

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