Category Archives: Mathematics

Şirince January 2017

Having enjoyed spending a January week at the Nesin Mathematics Village in 2016, this year I came back for two weeks. My spouse will join me for the second week. Meanwhile, here are some photographs from this amazing place. Continue reading →

Confessions

This is about G. H. Hardy and Sylvia Plath: Hardy quâ author of A Mathematician’s Apology (1940); Plath, The Bell Jar (1963).

Photo: the Hardy and Plath books

I first read Plath only recently, after encountering The Bell Jar by chance in the Istanbul bookshop called Pandora. After I finished reading it next day in Espresso Lab on İstiklâl, a woman who had earlier been speaking Turkish asked in English to look at the book. She pondered the front and the back before handing the book back to me. When I asked whether she knew of it, she simply said yes. She may not have understood my meaning; but I did not put her English (or my Turkish) to the test. Had she been made curious by the cover, showing a woman applying powder with the aid of a compact mirror? Did that cover accurately reflect the novel?

On an airplane once I was reading a paperback whose cover displayed a painting of ruins beneath the Acropolis of Athens. “I love historical fiction!” gushed a flight attendant. The term might be stretched to cover what I was reading; but it was the Oxford World’s Classics edition of Plato’s Republic.

Plato’s Republic

I had first read Hardy’s Apology in high school, thanks to the suggested reading at the end of Spivak’s Calculus. A couple of weeks ago, I somehow found a blog that took its title from the end of Hardy’s opening paragraph. That paragraph reads:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

The blog was called just that: Second Rate Minds. “We quote Hardy with irony,” says one of the two creators,

because we do not agree with him.

I believe there is great importance in communicating mathematics as widely as possible. I think it is important that children are encouraged to enjoy mathematics so that they might take further interest in the subject. Equally important is the view of mathematics held by the general public. Despite Hardy’s disdain for applications, mathematics nevertheless pervades the modern world and benefits from society valuing its role.

This is all fine; except I wonder if the writer has been corrupted by the same culture that made Hardy into somebody he found himself in disagreement with. This is the culture of judging people against one another, in order to rank them. Hardy gives a hint of this culture in the closing section of his essay:

I cannot remember ever having wanted to be anything but a mathematician. I suppose that it was always clear that my specific abilities lay that way, and it never occurred to me to question the verdict of my elders. I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.

I do not remember just what I thought of Hardy’s Apology in high school. I was at a school for boys, where I won prizes for mathematics and other subjects. I did not wish to emulate Hardy, either in pursuing just one thing, or in trying to beat others at it. Nonetheless, at the end of my freshman year at St John’s College in Annapolis, I bought my own copy of Hardy’s Apology in the College bookshop. The manager remarked that the book had decided her against pursuing mathematics. She had had dreams of doing good for the world; by Hardy’s account, mathematics was about personal glory.

I did want to do mathematics, as I ultimately understood. But this final understanding came after four more years: three in college, and one at large. I was working at a farm when I understood in a dream that I must learn modern mathematics. I cannot say that Hardy had any role in this, one way or other. Still, I would suggest now that, if Hardy does discourage you from pursuing mathematics, this may be just as well. You will have to focus like a laser if you want to do mathematics; you will be judged mercilessly, as mathematical truth is merciless; and you will suffer self-doubt, when it seems that the hardest you can work is still not good enough.

I am sorry that Hardy continued to be preoccupied with comparing himself to others:

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one the thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.’

At least Hardy can accept that he was not quite at the level of his two collaborators. The mathematician must guard against all illusions.

In the end, I say, think what you like about Hardy; but give him credit for giving us a window into his life. Reading his essay yet again, I am impressed by the clarity and rhythm of the language, and by the frankness of the writer.

Sylvia Plath reminds me of Hardy. This is not because she ultimately gives up her virginity to a mathematician, at least in her novel. Like Hardy, she appears early on as an unpleasant person.

Plath’s character Esther proposes to Doreen that they ditch a party and have drinks with a man who wears cowboy boots and a lumber shirt. Doreen agrees to go up to Lenny’s apartment, as long as Esther will go. In the apartment, Doreen asks Esther to stick around. Still, Esther slips out; and back at the hotel, when a drunken Doreen pounds on her door, Esther won’t let her in. She allows Doreen to pass out in the corridor, since she won’t remember the incident anyway.

Maybe this was all part of the Girls’ Code, though it would seem to be a violation. Esther did not seem very nice to me. But then, trying to kill yourself is not very nice either, and Esther will do this repeatedly. There is a lot to investigate and contemplate here, including an academic system that squeezed both Plath and Hardy. It is odd that a bell jar is a place where the pressure is taken off. Now I want just to appreciate both Plath and Hardy, for laying themselves bare.

Written January, 2017. Revisited August 27, 2022. Later in 2017, I wrote more about Plath (and a little more about Hardy) in “Women and Men.”

The geometry of numbers in Euclid

This is about how the Elements of Euclid shed light, even on the most basic mathematical activity, which is counting. I have tried to assume no more in the reader than elementary-school knowledge of how whole numbers are added and multiplied.

How come 7 ⋅ 13 = 13 ⋅ 7? We can understand the product 7 ⋅ 13 as the number of objects that can be arranged into seven rows of thirteen each.

Bottlecaps arranged on a desk in seven rows, thirteen columns
Seven times thirteen

If we turn the rows into columns, then we end up with thirteen rows of seven each; now the number of objects is 13 ⋅ 7.

The same bottlecaps, now seen from another side as thirteen rows, seven columns
Thirteen times seven

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Thales of Miletus

This is about Thales of Miletus and what it means to study him. I am moved to ask what history is in the first place. It is a study of the freedom in which we face our conditions. Thales had his way of understanding the world, and we may benefit from trying to learn it.

“The Thaleses of the future are meeting in Didim, September 24, 2016”

“The Thaleses of the future are meeting in Didim,
September 24, 2016”

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Happiness

If only tangentially sometimes, this is about living in Turkey, especially under the ongoing official state of emergency.

Aristotle, Marx & Engels, and Collingwood
Aristotle, Marx & Engels, and Collingwood

A blog article on Medium recently struck me for its treatment of science. Dated October 3, the article is called “The Purpose Of Life Is Not Happiness: It’s Usefulness,” and its opening section is as follows.

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Beykoz, Istanbul

After five years in Istanbul, we continue to learn how much there is still to discover here. Now we have been to the Asian borough of Beykoz. Much of what we saw there was rural, and the topography and flora reminded me of Appalachia. I have nothing to say about the poverty and ignorance that might be suggested by this term; for me, Appalachia was always a locus for holidays, mostly at my late uncle’s place in West Virginia, but also in the form of bicycle tours. Travelling now to Beykoz, “Country roads,” I could think, “take me home, to the place I belong!” We got there by public bus from our European borough of Şişli.

Polonezköy, Beykoz, 2016.08.14

Polonezköy, Beykoz, 2016.08.14

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Thinking & Feeling

This essay is written as a distraction from current events, though I make some reference to them. I am prompted by questions of analogy provoked by

  1. the similes of Homer and
  2. a recent theater review in Harper’s that mentions the parables of Jesus.

With sea in the background, on the sand of a beach sits Chapman’s Homer: The Iliad, the cover featuring a 16th-century Flemish tapestry (framed by a circle), Scene from Roman History, showing a number of men, the one in the middle armored; a horse is in the foreground

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Academic Freedom

(See also other articles in the Freedom category.)

Yesterday (March 24, 2016) was the first day of the sixth Models and Groups Istanbul meeting. There were participants from the Middle East, Europe, and America. Kıvanç Ersoy was to speak about his own mathematics. He could not speak, because he was in prison. He, Esra Mungan, and Muzaffer Kaya were in prison, because the three of them had publicly insisted that the government of Turkey make peace in the southeast of the country. Absurd, but true. Continue reading →

Nesin Matematik Köyü, Ocak (January) 2016

DSC05976

My first visit to the Nesin Mathematics Village was in the summer of 2007. My first course there was in the summer of 2008. I have been back every summer since, sometimes more than once. Now I have visited for the first time in winter. I am in awe. The photographs here are supposed to give you an idea why. Continue reading →

The Facebook Algorithm

I thank all of the friends who sent me birthday greetings on Facebook this year. [But see note at end.] One friend noted that I was not likely to see his birthday greeting, since I do not pay attention to Facebook these days. I usually do not pay attention; but since so many friends apparently continue to use that medium, I have not closed my account. I recently posted on Facebook a couple of photographs showing a friend from Washington who was visiting Istanbul. These photographs were “liked” by friends of that person or of me. Thus I suppose I used Facebook for its best purpose. Continue reading →