Review: The Weil Conjectures by Karen Olsson

The Weil Conjectures by Karen Olsson

I. 

I liked that this was a book full of questions, and I also liked the questions. Here are some of the questions she asks in the book:

  • Why do people like math?
  • Why did I (the author Karen Olsson) like math when I studied it in college? Even though I was an aspiring novelist?
  • Why was Simone Weil — philosopher, writer, mystic — attracted to mathematics?
  • What’s the deal with Simone’s relationship with her brother (famed mathematician) Andre?
  • Where do mathematical ideas come from?
  • What do (the author Karen Olsson) get out of abstract math now that I’m no longer swimming in it?
  • Do analogies for abstract mathematical ideas do a person any good if the math itself isn’t accessible to them?

It wouldn’t be fair to Olsson or the book to reduce it to a neat set of answers to those questions. The book is structured so as to provide an experience that is a lot like the experience of learning abstract math. The Weil Conjectures suppose a connection between two mathematical domains — topology and number theory — and Olsson wants (as far as I understand) to create a literary experience that is analogous to the search for such domain-bridging mathematical connections. So she lays out the Weil biography, her own memoir, mathematics and writing about mathematics for the reader. And I think she really succeeds — as the book goes on it feels a bit like learning some deep bit of theory.

So it’s not fair to reduce the book to a neat set of answers because the book is primarily about the experience of reading it. Some books are like that, and that’s fine. But she does answer some of those questions in interesting ways, and memory is necessarily structured, so it’s worth trying to say a bit about what Olsson says about mathematics itself.

II.

Q: What do people love about abstract mathematics?

A: Attraction to the unknown itself.

That, I think, is as close as we can get to Olsson’s answer in brief. The mathematician is someone who desires to create unknowns and to put obstacles in the way of their knowing so that they can search for answers. So for instance we hear about a Kafka story (“The Top”) about a philosopher who seeks enlightenment by hoping to catch spinning tops in mid-spin (whatever). On this story Anne Carson says “he has become a philosopher (that is one whose profession is to delight in understanding) in order to furnish himself with pretexts for running after tops.” That’s what a mathematician is — they love the chase.

Q: Is this so different from what a writer does?

A: No.

And so there is a connection between Olsson’s 2.5 years studying mathematics in college and her life as a writer.

I love this take on mathematics — that it’s about this love of being in the dark and searching for light. So it’s more about finding light than the light itself, if that makes sense.

This should be seen (I think) in contrast with writers who make much of the beauty of mathematics, or the search for beauty. Olsson is good on this. Twenty years after finishing her degree she decides to go back and watch some online lectures for an Abstract Algebra class. She finds it remote and foreign, but also:

And still, it was beautiful. I’m ambivalent about expressing it that way — “beauty” in math and science is something people tend to honor rather vaguely and pompously–instead maybe I should say that still, it was very cool. (This is something the course’s professor, Benedict Gross, might say himself, upon completing a proof: “Cool? Very cool.”) A quality of both good literature and good mathematics is that they may lead you to a result that is wholly surprising yet seems inevitable once you’ve been shown the way, so that–aha!–you become newly aware of connections you didn’t see before.

Still, the mathematician’s next move is to plunge themselves into darkness. This comes from a desire towards something that cannot be grasped.

This part was hard for me to understand. A key for Olsson seems to be Anne Carson’s Eros the Bittersweet but the theory isn’t entirely clicking for me. “A mood of knowledge is emitted by the spark that leaps in the lover’s soul,” she writes but I don’t quite get. Olsson’s take: “It’s not the knowledge itself, not consummation but the mood, the excitement when you are on the verge of grasping.”

What I understand Olsson to be saying is that the main fun of math isn’t the understanding but the feeling that understanding might be near. And that explains the pleasure we non-experts get out of mathematical analogies. There’s nothing unusual about the idea that analogies give us the thrill of desire — what’s more novel is saying this isn’t so different than the usual state of a mathematician.

For this point she goes to the act of mathematical creation itself, and how unpredictable it is:

What does mathematical creation consist of? asks Poincare, who blazed his way through a large territory of mathematics and physics by relying on his remarkable geometric intuition. It requires not only the combining of existing facts but the avoiding of useless combinations: making the right choices. The facts worthy of study are those that reveal unsuspected relationships between other facts. Moreover, much of this combining and discarding and retrieving goes on without the mathematician’s full awareness, occurring instead behind the scrim of consciousness.

Since the mathematician is dependent on their unconscious associations, mathematical discovery is not entirely in their control. In fact, many suspected relationships don’t work out. And so the mathematician spends most of their time afflicted with that same desire for the unreachable that afflicts we, the non-technical lay audience, who only get analogies.

Analogy becomes a version of eros, a glimpse that sparks desire. “Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often.” This of course, describes more than mathematics; it expresses an aspect of thinking itself–how creative thought rests on the making of unlikely connections. The flash of insight, how often it leads us off course, and still we chase after it.

It’s a neat picture, I think!

I’ve left out all the connections to mysticism, the biographical details of Simone and Andre, and (nearly) all the connections to writing, but that’s in there too. Again, very neat stuff.

III.

I also learned from this book that Brouwer retired early and practiced nudism, that Flannery O’Connor didn’t particularly care for Simone Weil’s writing, and of Hadamard’s fascinating book “The Psychology of Invention in the Mathematical Field.”

One last good quote: “Honestly I think I understand anyone else’s dislike of math better than I understand whatever hold math has had on me.”

Some interesting things I learned from Robert Pondiscio’s book about Success Academy

Success Academy was originally supposed to be a copycat of Robert Slavin’s Success for All program.

Rich person Joel Greenblatt got interested in Slavin, and gave him a call.

“I gave him a call and basically said, ‘I’ve looked at your results, and you’re getting forty or fifty percent of kids reading, but it’s not ‘success for all,'” Greenblatt recalled telling Slavin. “‘I’m just a guy with money. Is it money?'”

Yes, Slavin replied; its just money–if the money is spent in the right way. But it’s difficult to maintain tight control in most public schools. The two agreed to try to find a school they could convince to put “Success for All on steroids,” as Greenblatt put it.

They took over a public school, got a charter, recruited Moskowitz, one thing led to another, etc., here we are today.

Eva Moskowitz hates too much direct instruction.

More than most charter school leaders of her stature, Moskowitz consistently sees and talks about schooling through a child’s lens. Too much direct instruction–explaining the steps to solve a math problem, for example, rather than guiding kids as they struggle and strategize–sets her off. “If you see a teacher, ‘Blah, blah, blah, blah,'” she says, pantomiming extreme boredom. Her tone has been restrained, her deliver pedantic, even when discussing charged topics like race, cheating the Charlotte Dial video. But now she grows animated, even agitated. “Who has seen a kid when their teacher keeps them on the rug for twenty-five minutes? They’re five, for God’s sakes! You try sitting on that rug for twenty-five minutes listening. That is an educational crime!”

I should note that the book is basically Pondiscio’s account of a year he spent observing Bronx 1, a Success Academy elementary school. It is focused entirely on the elementary years, which is too bad. They operate a lot of middle schools (and a few struggling high schools) and you don’t leave the book with any sense of how they work. The direct instruction is one thing, but another is the ra-ra spirit (Ford-ham! Stand! Up!) which every middle schooler I have ever met would hate.

Math at Success Academy is…progressive? Progressive-ish?

Here is the Kindergarten teacher at Bronx 1 speaking to parents:

Writing “27 + 14” on the classroom smartboard, she warns parents to expect “a totally different world of learning math. We all learned math this way,” she says, doing the sum while narrating her work: “Seven plus four is eleven; put the one here and carry the one.” She stops and turns back to the room. “What does ‘carry the one’ mean? I love that you’re giggling, mom,” she says to one parent. “Before I came here, I had no idea why I carried that one. I knew I couldn’t put it here“– she writes “11” below the stacked “7” and “4.” “I did a procedure. No one ever taught me the real reason.”

For now, she begs parents not to teacher their children “procedures.”

This is not a lone teacher. (The book makes clear that there are no lone teachers at SA.)

It’s also reflected in their curriculum which (believe it or not) is built on TERC!

The math curriculum is cobbled together from different sources: an off-the-shelf curriculum called TERC; Contexts for Learning, a “conceptual math” approach pioneered by Catherine Fosnot, an education professor at City College of New York; and “a variety of things we found on the Internet,” according to Stacey Gershkovich, who oversees math at Success.

That’s pretty progressive stuff! Though Pondiscio emphasizes that they do drill facts a lot.

I will quote extensively, because I was surprised by this whole passage.

In the fourth-grade wing, Kerri Lynch is teaching fractions. Because of its test scores, it is commonly assumed that math lessons at Success Academy resemble a Chinese cram school’s, with instruction focused exclusively on “drill and kill” to prepare kids for high-stakes tests. However, Moskowitz, the daughter of a mathematician, is a proponent of a conceptual approach; she derides “direct instruction” and other standard explanatory pedagogies, where kids learn and practice algorithms and formulas, as “math by card tricks.”

Sure, sure you say. Show me the classrooms.

Lynch’s students sit on assigned spots on the rug. “One thing we’ve worked on is to be able to compare fractions to landmark fractions, such as one half or one whole,” she begins. “Today, when I put up the two fractions you’re comparing, I don’t want you to show any work. I only want you to write if it’s less than, greater than, or equal to. Just the symbol.” With those minimal instructions, Lynch writes 1/8 and 1/10 on the board and watches as her students bend silently over their whiteboards. “I’m seeing that some of us very quickly know it,” she observes. “Go ahead and turn and talk with your partner. What knowledge of fractions did you apply to solve this question?” The chatter rises as Lynch circulates, asking questions, drawing students out, and listening to their explanations, making mental notes about which students she will ask to “share out” with the class.

As the discussion goes on, the explanations gradually grow more economical and precise…Lynch raises the rigor. She writes 3/4 and 7/8 on the board. With no common denominator or numinator the answer is less obvious, but she offers a hint: “Matthew, what you said about ‘closer to one whole’ might be helpful. Turn and talk with your partner.” The room breaks into passionate arguments: “They’re equal!” “Seven-eighths is more!” “Only one piece is left to get it to one whole!” “No they are equal! They’re both only one pieces to a whole!” “But this one’s a smaller piece to a whole.”

Lynch’s ears perk up. “What did you just say? They’re both one away from a whole? How much further away from a whole? I want you to share that,” she says to Matthew.

Jo Boaler would be proud! (Steve Leinwand is.)

Eva Moskowitz is fond of saying that Success Academy is “Catholic School on the outside, Bank Street on the inside.” That is not true, because all the behavioral reinforcements and super-obsessive rules are very not Bank Street. But you can sort of see what she means.

Success Academy uses Fountas and Pinnell’s Leveled Reading System, and they generally spend a lot of time on reading skills (something researchy-types don’t care for)

Success Academy follows the “leveled reading’ system developed by reading researchers Irene Fountas and Gay Su Pinnell, which starts at Level A early in kindergarten and reaches Level Z, usually by the end of eighth grade.

They also use it in a way not recommended by Fountas and Pinnell, which is to give each kid a specific and highly publicized reading level. There’s a scene where a kid excitedly goes around telling teachers that she is “Level L” and again I wonder what their middle school looks like because this culture? Doesn’t work for older kids, right?

Systematically building background knowledge is not an explicit aim of Success Academy’s ELA curriculum. I had long assumed that its ELA curriculum played a significant role, even a dominant one, in the network’s standardized test results. But when the network made its curriculum available for free online in 2017, the response among experts who emphasize the importance of knowledge-rich curriculum was muted. “Kindergartners spend all but one of their seven units supposedly developing skills–like the ‘skill’ of reading nonfiction–and those at higher grade levels get only one or two more units per year that are content-based,” observed education journalist Natalie Wexler, who noted that the lessons posted on the website “sound very similar to the fruitless exercises that are found in classrooms across the country. Teachers jump from one topic to another, using content merely as a delivery mechanism for skills, and then students are sent off to ‘practice’ the skills on books they choose themselves.”

These are the most important quotes in the book.

There’s a meeting for parents who either won or were declared “likely” to gain an admissions spot for their kid. The meeting asks parents to consider whether Success is right for them.

Then, almost as an afterthought, Reeder mentions transportation. Success Academy does not offer buses. For some parents the logistics of getting a child to and from school present an even bigger challenge than complying with culture demands, reading logs and homework. Every Wednesday, children are dismissed at 12:30 so that staff can attend their professional development sessions. “That’s something else you gotta keep in mind, “Reeder adds. “‘Will I be able to pick up my child on time? Will I be able to have somebody to support me with that?’ School lets out at 3:45 but every Wednesday is “12:30 no matter what.” She hits “12” and “30” hard, hammering the point home. “And we do not have after-school, so you guys have to figure that out.”

The meeting lasts just under an hour, but it opens a portal into the model and culture that explain in no small part the network’s consistent results across its schools. Suddenly it all makes sense: The common criticism leveled at Moskowitz and her schools is that they cherry-pick students, attracting bright children and shedding the poorly behaved and hardest to teach. This misses the mark entirely. Success Academy is cherry-picking parents. Parents who are not put off by uniforms, homework, reading logs and constant demands on their time, but who view those things as evidence that here, at last, is a school that has its act together. Parents who are not upset by tight discipline and suspensions but who are grateful for them, viewing Success Academy as a safe haven from disorderly streets and schools. Charter schools cannot screen parents to ensure culture fit, but the last hour in the auditorium is a close proxy for such an effort, galvanizing disciplines and warning off the indifferent and uncommitted. A the same time, there is something undeniably exclusionary about it. If you don’t have the resources to get your child to school by 7:30 and pick her up at 3:45  — at 12:30 on Wednesdays — Success Academy is not for you. Literally.

Pondiscio makes much of research showing that parental factors make a big difference on the success of low-income students. Parental involvement, two-family homes, strong religious faith are all factors that help. The claim is that Success Academy is essentially selecting for these families in low-income neighborhoods.

Nobody likes talking about it in this book, but also nobody seems to deny it. It’s the unspoken but obvious, glaring fact about Success Academy. It’s a point neatly summed up towards the end of the book:

One former Success Academy school leader was philosophical about all this. “Is it really such a bad thing that this is basically an elite private school that admits by lottery?” he asked. “It’s the first time folks in the inner city have had that chance.”

So it’s not curriculum, not the teaching, not the teachers? Not even the test prep? It’s just selection effects?

Well, mostly selection effects. There are a few other things in the secret sauce. The thing that’s well-known is the intense and systematic behavior management. What I didn’t know about was the entirely sensible division of labor. Principals focus entirely on teachers and students — they have an “ops” person for administrative stuff. The curriculum may not have impressed Pondiscio, but its existence does impress him. The curriculum meets whatever minimal threshold it needs to be to free teachers up to focus on going over student work, calling parents and working with kids. That’s good, in general!

But, yeah, mostly selection effects.

Oh, and my school got a shout out.

The oft-heard refrain at Success to “put the lift on students” and to socialize learning, encouraging children to work collectively in pairs or small groups, is the kind of teaching one would expect to see at Bank Street, Saint Ann’s, or any of the progressive private schools beloved by affluent New Yorkers.

That doesn’t exactly nail the culture of Saint Ann’s but whatever! Not his point.

This is a good interview.

Check it out.

The book was a very good read and very much worth reading. Interesting, provocative, rooted in entirely realistic school observations, it’s definitely worth checking out.

The opening chapter of the novel RED PLENTY is all about mathematical abstraction

RED PLENTY by Francis Spufford was so good. A great deal of the novel is about the frustrated attempt by Soviet economists and mathematicians to reform the Russian economy.

The book opens on Leonid Vitalevich, about to discover linear programming:

Today he had a request from the Plywood Trust of Leningrad. “Would the comrade professor, etc. etc. grateful for any insight, etc. etc., assurance of cordial greetings, etc. etc.’ It was a work-assignment problem. The Plywood Trust produced umpteen different types of plywood using umpteen different machines, and they wanted to know how to direct their limited stock of raw materials to the different machines so as to get the best use out of it. Leonid Vitalevich had never been to the plywood factory, but he could picture it. It would be like all the other enterprises which had sprung up around the city over the last few years, multiplying like mushrooms after rain, putting chimnies at the end of streets, filing the air with smuts and the river with eddies of chemical dye…

To be honest, he couldn’t quite see what the machines were doing. He had only a vague idea of how plywood was actually manufactured. It somehow involved glue and sawdust, that was all he knew. It didn’t matter: for his purposes , he only needed to think of the machines as abstract propositions, each one effectively an equation in solid form, and immediately he read the letter he understood that the Plywood Trust, in its mathematical innocence, had sent him a classic example of a system of equations that was impossible to solve. There was a reason why factories around the world, capitalist or socialist, didn’t have a handy formula for these situations. It wasn’t just an oversight, something people hadn’t got around to yet. The quick way to deal with the Plywood Trust’s enquiry would have been to write a polite note explaining that the management had just requested the mathematical equivalent of a flying carpet or a genie in a bottle.

But he hadn’t written that note. Instead, casually at first, and then with sudden excitement, with the certainly that the hard light of genesis was shining in his head, brief, inexplicable, not to be resisted or questioned while it lasted, he had started to think. He had thought about ways to distinguish between better answers and worse answers to questions which had no right answer. He had seen a method which could do what the detective work of conventional algebra could not, in situations like the one the Plywood Trust described, and would trick impossibility into disclosing useful knowledge. The method depending on measuring each machine’s output of one plywood in terms of all the other plywoods it could have made. But again, he had no sense of plywood as a scractchy concrete stuff. That had faded into nothing, leaving only the pure pattern of the situation, of all situations in which you had to choose one action over another action. Time passed. The genesis light blinked off. It seemed to be night outside his office window. The grey blur of the winter daylight had vanished. The family would be worrying about him, starting to wonder if he had vanished too. He should go home. But he groped for his pen and began to write, fixing in extended, patient form – as patient as he could manage – what’d come to him first unseparated into stages, still fused into one intricate understanding, as if all its necessary component pieces were faces and angles of one complex polyhedron he’d been permitted to gaze at, while the light lasted; the amazing, ungentle light. He got down the basics, surprised to find as he drove the blue ink onward how rough and incomplete they seemed to be, spelt out, and what a lot of work remained.

It’s the optimism generated by ideas like these that are the true subject of the book, which is the story of the rise and fall of this optimism. The book points out that in a society governed by engineers it was mathematicians and abstract theoreticians that were the main sources of cultural idealism. (In contrast to a place like the US, he says, where lawyers rule the land and writers and artists are the main source of social idealism.)

If you know what happened to the Soviet economy you know the end of this story. The entire book presents itself as a kind of mathematical tragedy, the destruction of the idea of utopian abundance in a planned economy.

Straightedge and Compass

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John Donne’s A Valediction Forbidding Mourning ends with two lovers compared to the arms of a geometric compass over several stanzas:

Our two souls therefore, which are one, 

   Though I must go, endure not yet 

A breach, but an expansion, 

   Like gold to airy thinness beat. 




If they be two, they are two so 

   As stiff twin compasses are two; 

Thy soul, the fixed foot, makes no show 

   To move, but doth, if the other do. 




And though it in the center sit, 

   Yet when the other far doth roam, 

It leans and hearkens after it, 

   And grows erect, as that comes home. 




Such wilt thou be to me, who must, 

   Like th' other foot, obliquely run; 

Thy firmness makes my circle just, 

   And makes me end where I begun.

I came across this in Stephanie Burt’s book Don’t Read Poetry. She writes:

Each lover “leans and hearkens” after the other, as if Donne and his intimate friend, lover, or wife heard each other across the sea. The balanced eight-syllable lines, with their alternating rhymes, depend on each other too. Their closure seems “just” both mathematically and morally; in their mutual response, one or both of the lovers stands up, or becomes “erect” (yes it’s a penis joke).
If you yourself have ever felt unique or confused or confusing to others, especially in matters of the heart; if you have ever felt that your connection to somebody else–whether or not it is romantic, or exclusive, or recognized by the law–requires some explanation of deserves a passionate defense; if you have friends in a stubborn long-distance relationship; if you have been in any such situation, you might see Donne’s elaborate, challenging metaphors not as barriers to sincerity but as ways to achieve it, ways that take advantage of the tools–metaphor, indirection, complex syntax, rhythm–that we can find in poems. You might even, at least if you are looking for them, see in Donne’s great love poems, this one among them, defenses of what we now call queer relationships, relationships not sanctioned by custom or law, relationships most people in your own society can’t quite understand.

That image at the top, by the way, is a set of compasses held by the British Library from Donne’s time, the 16th century.

Reading Chaim Grade’s “The Yeshiva”

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This is a book that I first read when I was 18 and studying in yeshiva in Israel. But I didn’t really read it very well back then. This is for two reasons. First, I wasn’t sleeping well at all, and I read Grade’s novel only between the hours of 11 and 1. Second, I was a ball of anxieties about secular/religious tensions, and ill-equipped to make sense of the world of ideas Grade was constructing.

In Yiddish the novel is called Tzemach Atlas, and we are quickly introduced to the man himself. Tzemach Atlas is an extreme adherent of the Navarodok branch of the musar movement. He believes that man’s desires need to be entirely uprooted through extreme self-reflection. He can find the flaws and hypocrisies in both common people as well of in great rabbis. He obsesses constantly over his own faults, and has gained a reputation as a brave teacher who is willing to “save” his students for the yeshiva. And yet, on the very first page, we are told that he is haunted by his doubts about the existence of God.

Then a lot of things happen — I won’t burden you with them all! — but we’re eventually introduced to Chaikl Vilner, i.e. Graim Chade, i.e. this is just a stand-in for the author Chaim Grade himself.

Tzemach Atlas eventually takes Chaikl into his upstart musar yeshiva. (Chaim Grade himself spent time studying in a Navarodok musar yeshiva.) Misery ensues for everyone involved, especially Tzemach Atlas and Chaikl.

And then, in the last fifth of the novel, the great gaon and Talmudic scholar Rabbi Avraham-Shaye shows up, vacationing in the small town where the yeshiva is. He takes a liking to Chaikl and invites him to be his personal study partner — this is shocking, as it is wildly unlike a great Torah scholar to take a young study partner.

Amazingly, though, all of this is more or less true. Rabbi Avraham-Shaye is a stand-in for Rabbi Avrohom Yeshaya Karelitz, who published under the pseudonym “Hazon Ish.” Chaim Grade was his study partner for many years.

I’m rereading the book now — I just finished Volume 1, and am ready to start Volume 2 — and the biggest surprise for me has been just how clearly Grade sides with Karelitz. Grade himself left the religious life and fell in with the secular Vilna Circle. Nevertheless, his portrait of Karelitz is almost hagiographical — it’s beaming and admiring.

Though it’s reductive to put it like this, it seems to me as if Grade wanted to write this book simply to take his teacher’s side in an argument about the value of musar.

The real Karelitz was a critic of musar and its ability to improve a person’s character. As Lawrence Kaplan writes in a piece about Karelitz’s views, the musarniks had the whole thing backwards:

The Hazon Ish was of the opinion that the fundamental Musar approach of working on oneself, of turning inward, in order to develop one’s spiritual personality and overcome the obstacles standing in the way of proper observance was fundamentally misguided.

In the novel, Tzemach Atlas, the great man of musarslowly comes apart at the seams. He is a failure in every single aspect of his life. He is unable to keep a “proper” engagement, and runs off with a wealthy secular woman. And though he shows moments of clear moral insight — and impresses his wife with his clarity and moral standards — he alienates himself from an entire town. He then runs off to start his own yeshiva, and finds that he is unable to influence his students in the least.

But from the perspective of the novel, this is wholly due to his own faulty ideology and personality. When Chaikl starts spending the night with a local girl (bad Chaikl!) Atlas reprimands him, but succeeds only in inciting Chaikl’s rage.

In contrast is the positive influence of Rabbi Avraham-Shaye. He is even able to influence the child Melechke, who goes about bragging about every little accomplishment for the first 330 pages of the novel. I don’t want to go too hard on him — he’s just 11! — but really he’s a greedy, sniveling braggart who annoys everyone.

But after he visits Avraham-Shaye? He’s a changed boy!:

As soon as the visitor and his escorts withdrew, the younger students surrounded Melechke. Although he had grown and matured somewhat and could already read a chapter of the Talmud, Melechke was still considered a spoiled only child and a show-off who trust himself into the midst of older people so they would test him in Talmud and pet him. His friends couldn’t understand why he hadn’t told them that he had visited Reb Avraham-Shaye.

“I went to see Reb Avraham-Shaye Kosover to discuss Torah with him,” Melechke replied, “and I probably asked him about a dozen questions about the first page of the Tractate Kiddushin. So Reb Avraham-Shaye told me, ‘Come into my house and we’ll open my copy of the Kiddushin and look at that matter together. Then you cn ask what you have to ask.’ So we studied the page together, and all my questions were answered of their own accord. When we left his room, Reb Avraham-Shaye led me to the low window which faced the forest and told me, ‘Climb out. Let’s see if you can climb out of a window as nimbly and quickly as you can ask questions.'” Melechke spread his hands like an old Torah scholar and concluded, “Since I didn’t hear any new interpretations from him, what was there to tel you? How I crawled out the window in the woods, and how he crawled out after me?”

Later, once Chaikl is invited to be the personal study partner of the great gaon, he finds himself unable to remain the night with his landlady’s daughter:

“How could he look Reb Avraham-Shaye straight in the eye and debate lofty matters with him when he did such improper things at night? His shame at deceiving a saintly gaon changed to fear: Perhaps Reb Avraham-Shaye knew? A man like that had divine intuition…”

Later, when the gaon debates Tzemach Atlas in the closing pages of the book, Avraham-Shaye makes the case that through studying the Talmud and its commentaries one essentially gains moral influencers along these lines: “A youngster pores over his Talmud in Vilna and muses that he’s in Babylonia, sitting in the great Talmudic academy of Nahardea, in the beth medresh of Rashi and his scholarly descendants.” So, through Torah study one obtains moral exemplars who effect a profound moral influence. (Or maybe it’s Torah itself that exerts the influence?) Greater than whatever results from the intense moral introspection and trials that the mussarniks recommend.

Two things seem remarkable about this to me. First, that this is essentially the actual view of Karelitz. Second, Chaim Grade seems to have written this novel so as to adopt this perspective wholeheartedly. In the final lines of this book, Tzemach Atlas walks away a broken man, having been absolutely seen-through by the visiting gaon.

There is one place, however, where Grade seems to go beyond his teacher. Late in the book, with nowhere else to go, Chaikl goes to the beautiful wooden shul to sit alone. (Sidenote: the wooden shuls of Poland really are a wonder.) He sits there contemplating — but he is not alone, as Tzemach Atlas has been performing moral introspection up above, in the women’s section. He descends to accost Chaikl.

“Are you studying Musar by yourself?” Tzemach asks.

Chaikl responds: “I came into the shul to look at the carvings. The man who carved the lions, eagles, deer and leopards is a great artist.”

I think the strong suggestion of the book is that Chaikl’s obsession with beauty (and metaphor, as he is the author and poet as a young man) is roughly equivalent to Karelitz’s obsession with Torah. Both are capable of providing a strong moral influence on the subject, something that Tzemach Atlas is unable to comprehend. The book tells us again and again that Tzemach not only doubts God; he also does not comprehend beauty, and (as a consequence) has no respect for humanity — in himself or in others. He is a moral failure because he is a religious and aesthetic failure.

All of these themes are made much more explicit in Grade’s remarkable short story, My War With Hersh Rasseyner. He writes there — and Hersh Rasseyner is an adherent of musar who sounds quite a bit like Tzemach Atlas:

Karelitz believes in simple faith and an obsession devotion to Torah study to guide one’s moral behavior. Grade in The Yeshiva basically agrees — but extends that basic moral perspective to devotion to the humanities, in general.

Modernism in Mathematics

Jeremy Gray makes the case (in here) that modernism applies to mathematics. His modernism consists largely of a move away from representations and towards formal approaches.

So on Lebesgue’s theory of the integral in 1903:

“The axioms specify what the integral is intended to do. They do not start from an idea that the integral is about, say area, or any other primitive concept. It is necessary to show that there is a model of these axioms, but once that is done it is at least possible to prove properties of the integral directly from the axioms and without reference to any model of them. The axioms are sometimes said to define their object implicitly, or to create it. There is no reference to a primitive concept available via abstraction from the natural world.”

And on Kronecker and Riemann:

“Neither man suggested that objects cannot be studied via their representations, but both believed that one must be vigilant to ensure that one establishes properties of the objects themselves and not the properties of merely this or that representation, and to this end it was best to avoid explicit representations whenever possible.”

I didn’t know about the Hausdorff paradox, which feels a lot like Godel. Gray’s summary: “on any plausible definition of the measure of a set there must be non-measurable sets.”

Borel ended up critiquing the use of the axiom of choice to call the paradox into question, but this was another step (apparently) in pushing people to accept that definitions of area are inherently imperfect — pushing us further away from meaning and belief in the representations.

Another interesting point from Gray: you know that thing about the unreasonable effectiveness of math? That wouldn’t have made any sense in the 19th or 18th centuries because math was coextensive with science. Like, there’s nothing surprising at all about the connection between math and the world back then, because math was an attempt to describe the world.

I’m interesting to read more, but I’m feeling as if a question has been answered. Whether we call it modernism or not, this is the time in the history of math when the connection between mathematics and the empirical world was made problematic. If we’re looking for the origins of the idea that math is “useless,” it’s going to be in this movement in mathematics between 1880 and 1920.

Things that I’d like to read: on modernity and mathematics

The world has changed immensely over the past several hundred years. Mathematics has too. Are these changes all related?

Plato’s Ghost looks like a good place to start.

Plato’s Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions.

Building on Gray’s work is this presentation by Susumu Hayashi, which introduces (to me at least) the notion of “mathematical secularization.”

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I also came across The Great Rift.

In their search for truth, contemporary religious believers and modern scientific investigators hold many values in common. But in their approaches, they express two fundamentally different conceptions of how to understand and represent the world. Michael E. Hobart looks for the origin of this difference in the work of Renaissance thinkers who invented a revolutionary mathematical system—relational numeracy. By creating meaning through numbers and abstract symbols rather than words, relational numeracy allowed inquisitive minds to vault beyond the constraints of language and explore the natural world with a fresh interpretive vision.

The focus is on early modernity and the shift to algebra, which is an earlier phenomenon than modernism. But maybe it’s part of the same story?

Also in the category of “is this related? maybe??,” there is a working group of philosophers that call themselves the Mathematics, Mysticism and Secularization working group.

There are all these -isms that I learned about in philosophy of math: empiricism, logicism, formalism, fictionalism (wiki). That’s part of this story too.

What I’m attracted to is the idea that math is as much a part of culture as anything else. Over the last few centuries Western society has gotten less and less comfortable with the abstract, invisible realm of religion and spirits. Wouldn’t that have an impact on how that culture thinks about that other invisible, abstract realm of mathematics?

People used to think of x^2 as referring just to a square’s area, but then it was emptied of that meaning. Is the break of algebra from geometry something like the break of philosophy from theology?

People used to think that mathematics was a search for ultimate truths, not just conditional ones. Are we living in a mathematically relativistic world?

Mathematicians sometimes talk — with pride! — of the uselessness of their work. Is that the end result of the sorts of processes described by these authors?

I have no clue, and I have no idea when I’ll be able to read those books. But the questions seem interesting and confusing.

Stop Making Sense

Van Morrison in Rolling Stone, 1978:

But a lot of your most danceable songs have very profound lyrics.
I don’t know, this thing about lyrics – I’m just catching on to this. If you get some of the facts together . . . I mean, I sell records in places where they don’t speak English. And I’ve experienced listening to Greek singers, for instance, and not knowing what the words are, but I get a story and a feeling from it even though I haven’t a clue to what’s being said. So if English-language songs can sell in non-English-speaking countries and people can be touched by them, then we can see how irrelevant the words are.

Sometimes it seems that you let the words dance when you sing them; you release them, and they take off in their own directions.
The only time I actually work with words is when I’m writing a song. After it’s written, I release the words; and every time I’m singing, I’m singing syllables. I’m just singing signs and phrases.

But this can’t be the whole story. Music with incomprehensible lyrics isn’t the same as listening to instrumental music. It’s also not the same as listening to scat or wordless yammering. Somehow the presence of words — maybe it’s just the possibility of meaning — changes the way we listen to a song.

Mayyyybe

Beyond that, though, we’ve grown ever more aware of the problems with centralizing the internet. Traditional blogs might have swung out of favor, as we all discovered the benefits of social media and aggregating platforms, but we think they’re about to swing back in style, as we all discover the real costs and problems brought by such centralization.

From this, explaining why Signal v Noise is leaving Medium. Look, don’t get me wrong, I would love a return to blogging. I would personally benefit from a return to blogging. But I really see no reason to expect a return to blogging.

Every blog is either (a) a legacy blog, created during the heyday or (b) the blog of someone with a significant social media presence. Blogs don’t even bother with the blogroll of old — blogging is entirely dependent on already having the potential to draw a readership. If you’re starting out blogging now, how could social media not be part of the puzzle?

Maybe blogs are ready to “swing back in style” in the same retro sense that vinyl sales are growing, I don’t know. A small minority defined against the dominant culture — that still seems to me the future of blogging.

 

 

Some choice quotes from Jeff Tweedy’s memoir

I think I want to use this blog to share more quotes from things that I’m reading. Right now that’s Jeff Tweedy’s very funny and touching memoir Let’s Go (So We Can Get Back).

I thought this was perceptive about the changing function of music reviews:

I will say that today, reviews aren’t quite the same as they were back in the early ages of rock journalism. Reviews back then devoted way more ink to trying to describe what music sounded like. That was their main purpose. It’s why people read them, because it was the only way to decide if you wanted to spend your money on a particular record.

I’m already looking forward to explaining those weird CD sampling stations that they used to have at Barnes & Noble to my kids.

From rehab:

I’d sit in group sessions and listen to other patients talk about their lives, and what they’d endured was beyond anything I could imagine … One guy told us about seeing his father murder his mother when he was nine and that he had his first taste of alcohol that night because his father forced him to drink whiskey, thinking it would make him forget what he’d seen. Hearing a story like that made me ashamed of how little I had had to survive and how much pain I’d derived from so much less actual trauma. What was I gonna say when the group got to me? “Um … I cry a lot. I get scared sometimes. I have headaches, and it makes it hard to make music.” That was the worst of it. I was out of my league.

One time, after a group session, a few of us were in the smoking room and I confided to them, “I feel like I shouldn’t even open my mouth. I don’t want anyone to get the idea that I think my situation compares.”

“Listen to me, motherfucker, listen.” Getting right up in my face. “Mine ain’t about yours. And yours ain’t about mine. We all suffer the same. You don’t get to decide what hurts you. You just hurt. Let me say my shit, and you say your shit, and I’ll be there for you. Okay?”

Pain is pain is pain is pain. To elevate someone else’s pain is ultimately to make it strange — to alienate that other person from you.