Category Archives: G. H. Hardy

Boolean Arithmetic

Mathematics can be highly abstract, even when it remains applicable to daily life. I want to show this with the mathematics behind logic puzzles, such as how to derive a conclusion using all of the following premisses:

  1. Babies are illogical.
  2. Nobody is despised who can manage a crocodile.
  3. Illogical persons are despised.

The example, from Terence Tao’s blog, is attributed to Lewis Carroll. By the first and third premisses, babies are despised; by the second premiss then, babies cannot manage crocodiles.

George Boole, The Laws of Thought (1854), Open Court, 1940

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Şirince January 2018

In the Nesin Mathematics Village recently, I was joined at breakfast one morning by a journalist called Jérémie Berlioux. He knew Clément Girardot, the journalist whom I had met in the Village in the summer of 2016. This was before the coup attempt of July 15, but after the terror attack at Atatürk Airport on June 28. I wrote about this attack the next day in “Life in Wartime” on this blog. Then I headed off to Şirince to join a “research group.” My wife and colleague came along, though not to be part of the group; afterwards we headed up the coast for a beach holiday. We were at the beach when the coup attempt happened, as I wrote in my next blog article, “War Continues.” I contrasted politics with mathematics, which was an inherently nonviolent struggle. This was the kind of struggle engaged in by the research group in the Math Village.

Large clay pot against dark vines

Outside the Nişanyan Library

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Women and Men

This began as an update to “Confessions,” which concerns the man called G. H. Hardy and the woman called Sylvia Plath. I had originally included a photograph of the subjects’ respective books. On Hardy’s, the author poses reluctantly; on Plath’s, a woman applies powder in a compact mirror.

Plath’s book was the 2013 Faber and Faber 50th Anniversary Edition of The Bell Jar, and the cover is controversial. See Alexandra Topping, “The Bell Jar’s new cover derided for branding Sylvia Plath novel as chick lit” (The Guardian, Friday 1 February 2013). I learned of the controversy from Emily Van Duyne, “Sylvia Plath Looked Good in a Bikini—Deal With It,” in Electric Literature, hosted by Medium (October 9, 2017). Medium had promoted the essay to me when I read Brian E. Denton, “The World Will Not Quarrel: Day 282 of A Year of War and Peace.

Yesterday I happened upon a tweet juxtaposing the real cover of the British edition of Volume I of The Letters of Sylvia Plath with a fake cover of The Letters of Ted Hughes. Each cover shows the letter-writer posing in revealing swimwear on a beach, though the head of Plath’s husband seems to have been imposed on another man’s topless body. The text reads:

If male writers were marketed in the same way as female writers. Via Christopher Hamilton-Emery.

— Jane Harris (@blablafishcakes) October 4, 2017

(I saw the tweet as retweeted by Jennifer Williams, but Twitter seems not to preserve this valuable information.)

The three real and one fake book cover shown above are all somehow undignified. More precisely, I would feel undignified to be seen reading such books. Hardy did not like having his picture taken, and it shows on the cover of A Mathematician’s Apology. On the cover of The Bell Jar, the compact mirror and powder puff suggest superficiality. But then Hardy’s image does reflect the uptightness that I see in his book. And Sylvia Plath did wear lipstick.

Her favored color was Cherries in the Snow, by Revlon. A devotee of Plath called Patricia Grifasi learned this only after buying the lipstick for herself. She wrote (in “The Rise And Fall Of Sylvia Plath’s Favorite Lipstick,” The Gloss, June 16, 2015),

Forget that Plath’s poetry is terrifyingly intimate, crowded with speakers who say things like, “I eat men like air” (“Lady Lazarus”). Forget that her journal entries describe how satisfying it is to scoop out a pesky glob of snot. In a weird way, wearing Cherries in the Snow allowed me to be even closer to a writer I admired than reading all those very personal things.

Thus the 50th anniversary cover of The Bell Jar may not be inappropriate. As for the bikini on the Letters, says Emily Van Duyne in the aforementioned Electric Lit article,

The reality, and it’s astounding to me that I have to write this sentence down, is that we can take a writer who wears a bikini seriously. I have three in my closet, the most recent of which is a vintage-inspired red-halter. I bought it because I love red; I love red partly because I love Sylvia Plath. I wear “Cherries In The Snow” lipstick to the classes I teach, to parties, to intimidating meetings with condescending men, and when I do, I invoke her, just a little bit—for inspiration. For luck. For permission, which she gave me, which she gives me—to be brave. To try and astound. To say the things no one wants to say, or hear. To be beautiful, and to be smart, and sexual, and to never, ever fall into the foolish trap that these cannot coexist.

I have read The Bell Jar just the once so far. Looking back at my copy with the garish cover, I see that I marked several passages, noting their page numbers at the back. The later passages are uses of the term bell jar, and for now I have nothing more to say about this term than I did at the end of my earlier article. Many essays have no doubt been written about the meaning of the title, and some of these will be available on line, perhaps for a price, so that you can turn them in to your teacher as if they were your own (perhaps after making enough changes to fool automatic plagiarism detectors).

The earlier passages that I marked may perhaps illustrate what Plath meant by the bell jar. In any case, since I found an electronic text on a New York high school English course web page, I am able to quote the passages at length with no trouble.

I hadn’t the heart to tell Jay Cee there wasn’t one scrap of space on my senior year schedule to learn languages in. I was taking one of those honors programs that teach you to think independently, and except for a course in Tolstoy and Dostoevsky and a seminar in advanced poetry composition, I would spend my whole time writing on some obscure theme in the works of James Joyce. I hadn’t picked out my theme yet, because I hadn’t got round to reading Finnegans Wake, but my professor was very excited about my thesis and had promised to give me some leads on images about twins.

When I read this, I had not yet heard about and joined the project of Brian Denton (mentioned above) to read War and Peace over the calendar year, one chapter a day. Plath seems to make an ironical acknowledgment of what Robert Pirsig would write, in a passage of Zen and the Art of Motorcycle Maintenance that I considered also in “One and Many”:

Schools teach you to imitate. If you don’t imitate what the teacher wants you get a bad grade. Here, in college, it was more sophisticated, of course; you were supposed to imitate the teacher in such a way as to convince the teacher you were not imitating, but taking the essence of the instruction and going ahead with it on your own.

Plath either understands this or betrays it. Her first person Esther later looks at herself: the emboldened clause is specifically what I had marked.

I started adding up all the things I couldn’t do.

I began with cooking.

My grandmother and my mother were such good cooks that I left everything to them. They were always trying to teach me one dish or another, but I would just look on and say, “Yes, yes, I see,” while the instructions slid through my head like water, and then I’d always spoil what I did so nobody would ask me to do it again.

I remember Jody, my best and only girlfriend at college in my freshman year, making me scrambled eggs at her house one morning. They tasted unusual, and when I asked her if she had put in anything extra, she said cheese and garlic salt. I asked who told her to do that, and she said nobody, she just thought it up. But then, she was practical and a sociology major.

I didn’t know shorthand either.

This meant I couldn’t get a good job after college. My mother kept telling me nobody wanted a plain English major. But an English major who knew shorthand was something else again. Everybody would want her. She would be in demand among all the up-and-coming young men and she would transcribe letter after thrilling letter.

The trouble was, I hated the idea of serving men in any way. I wanted to dictate my own thrilling letters. Besides, those little shorthand symbols in the book my mother showed me seemed just as bad as let t equal time and let s equal the total distance.

Would somebody call Plath’s Esther a snowflake? “Look, Honey, none of us like the idea of serving men, but we all gotta do it.” This could be said by another woman or by a man. That doesn’t make it fair. Plath is writing the truth, as a confession.

Finally Esther meets a psychiatrist and reveals her fantasy of what such a man might be like.

I hated him the minute I walked in through the door.

I had imagined a kind, ugly, intuitive man looking up and saying “Ah!” in an encouraging way, as if he could see something I couldn’t, and then I would find words to tell him how I was so scared, as if I were being stuffed farther and farther into a black, airless sack with no way out.

Then he would lean back in his chair and match the tips of his fingers together in a little steeple and tell me why I couldn’t sleep and why I couldn’t read and why I couldn’t eat and why everything people did seemed so silly, because they only died in the end.

And then, I thought, he would help me, step by step, to be myself again.

But Doctor Gordon wasn’t like that at all. He was young and good-looking, and I could see right away he was conceited.

I would have to read the novel again, to think about how much irony is here.

Recently I was alerted (again by a tweet, as retweeted by Nicholas Christakis) to a theory described in a 2015 article in The Scholar’s Stage blog called “Honor, Dignity, and Victimhood: A Tour Through Three Centuries of American Political Culture.” The Iliad portrays an honor culture, but America passed from such a culture to a culture of dignity. Honor is given us by others; dignity, we can have on our own. Now that many Americans are sensitive to so-called micro-aggressions, the country may have passed to a culture of victimhood.

I confess to being insufficiently interested to read all of the sociology. I know what is happening in America, only from a distance. Plath’s Esther seems to have been hoping for help from a man. In recent tweets, I see women asking men to help fight abuse at the hands of Harvey Weinstein and the like. This is not victimhood, if victimhood is something to be perversely proud of, the way one might be proud of one’s honor or one’s dignity. The requests for help are recognition that we are all connected, and life might be better if we did not have to defend ourselves from attack, all of the time.

This sounds like the sort of thing that I have tried to work out, as in the article “All You Need Is Love,” where the topic is education—as it often is for me, and as it is in a good part of The Bell Jar. For Pirsig, rather than being imposed on you, education is something you ought to demand for yourself. But there is a middle way. Quoting my own email, I wrote,

Perhaps it is best to learn because one understands it as one’s role in the community that one is happy to be a part of. Here again is something that Pirsig may miss. Pirsig is the individualist, the lone wolf, and this is a reason why I like him. But education is not just something one achieves for oneself. It means joining an educated community; it means helping others join that community.

Life can be cooperative and not just adversarial. There are things you can learn on your own, or demand to learn, like a prosecutor examining a witness for the defense. If you can learn them, you may well be smart. Let me then end with one bit of tweeted wisdom:

History of full of stories of catastrophe brought about by people who are smart only in only a narrow domain, and lack humility and remorse.

— Zeynep Tufekci (@zeynep) October 6, 2017

NL XIX: Two Senses of the Word “Society”

Executive summary (below) | Index to this series

After a break of half a year, I return to reading Collingwood’s New Leviathan. Being on holiday at an Aegean beach gives me the opportunity. While here, I may also return to Chapman’s Homer’s Iliad. Last winter I finished Part I of the New Leviathan, the part called “Man.” Here I continue with the first chapter of “Society.” I have reason to look at what Mary Midgley and Albert Einstein say about science. Collingwood’s investigation suggests a way of thinking about prejudice and discrimination.

Part II of the New Leviathan is “Society,” and the first two chapters of this, XIX and XX, concern the distinction between society proper and two more general notions. In Chapter XX, the more general notion will be community. In Chapter XIX, the more general notion has not got its own proper name, and so Collingwood denotes it by writing “society,” in quotation marks.

A “society” of chairs at the beach (Altınova 2017.08.31)

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When is a help a hindrance? The Muses have provoked this question. They did this through their agents, the cicadas, who sang around the European Cultural Center of Delphi, during the 11th Panhellenic Logic Symposium, July 12–5, 2017.

     Cicada, European Cultural Center of Delphi, 2017.07.15     

Cicada, European Cultural Center of Delphi, 2017.07.15

My question has two particular instances.

  1. At a mathematical conference, can theorems “speak for themselves,” or should their presenters be at pains to help the listener appreciate the results?

  2. When the conference is in Greece, even at one of the country’s greatest archeological sites, does this enhance the reading of ancient Greek texts, or is it only a distraction?

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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.”

Learning mathematics

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. The post originally formed one piece with my last article, “Limits.”

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in my last two years at St Albans School, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper.

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