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


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”


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.”

Screenshot 2019-02-13 at 11.38.10 AM.png

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.


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.

Reading “A Root Cause of the Teacher-Diversity Problem”

Check it out here, it’s great.

So far the main effect of having a second kid in the house has been to make me, in pretty much every way, a little bit dumber. You know what I mean by dumber: not as clever, etc.

Anyway, one of the ways in which I’m dumber is that I’m a worse reader with worse impulse control. So last night I found myself reading an article right before bed (another mistake) and having trouble understanding a perfectly clear article by Melinda D. Anderson (the one I mentioned). And then I had the questionable decision to tweet the author my questions.

I really feel for journalists on Twitter. How do you know when to engage, when someone is worth talking to? Melinda was very generous in helping me understand what she wrote, as was Grace Chen, so I figured I’d write a bit explaining what I understood, out of thanks.


About 15% of public school students are Black:

Screenshot 2018-01-25 at 8.23.56 PM.png

About 16% of public school students were Black in 2012, but only 7% of teachers were Black.

Screenshot 2018-01-25 at 8.26.23 PM.png

(Figures and sources from this NCES report.)

So even as Black people make up about 12% of the general population in the US and even as that share of the population is modestly rising, the percentage of Black teachers is not increasing. A disproportionate percentage of the teaching population is not Black.

The question is, why? There are roughly two sorts of answers.

Supply-side: The issue is in the pipeline. This is mostly what people talk about, and it takes myriad forms: Black people graduate high school and college at lower rates, and hence are less available for teaching positions. Black people are disproportionately uninterested in becoming teachers, for whatever reasons.

Indeed, that NCES report that I linked to focuses on the pipeline:

Screenshot 2018-01-25 at 8.46.32 PM.png

But there’s another set of factors to look at: what if the teachers are there, but schools just won’t hire Black teachers?

Demand-side: Districts are biased, either implicitly or explicitly, against Black teachers. For reasons rooted in stereotype or false-associations, Black candidates are less appealing to schools, so they don’t get hired.

That’s what Melinda wrote about in the Atlantic piece. She wrote about districts that were sued for being biased against Black teachers, and the court agreed. (Or, rather, one district was issued a decision by a court to stop discriminating against Black teachers, and another took action after they got spooked by a suit that was in the works.)

One thing I was wondering is whether it’s really possible to disentangle supply and demand considerations. If Black teachers experience bias in school hirings then they’ll be less likely to pursue careers in schooling. If you start with a dearth of Black teachers in the pipeline, biases will creep up. These factors might not be possible to truly separate.

My own school has disproportionately few non-White teachers (and students). Is the issue supply-side or demand-side? Both, probably, though I suppose it only matters to the extent that this helps a school decide what to do. As Grace points out, most in education prefer to frame this as a “won’t somebody think about the children” issue, i.e. it helps kids to have Black teachers. I’m not saying I’m skeptical of that, but as Grace notes this could be a way for schools to avoid responsibility for bias — the dearth of Black teachers is then not unlike having not enough computers or too few pencils.

But how do you tell a school to stop being biased against Black job candidates? The answer for both of these schools was affirmative-action policies. Is there a stronger case for affirmative action in the presence of bias than if the problem is in the pipeline? I think so, though I feel fuzzy about how to think about this.

Many people I talk to who are otherwise strong advocates for diversity in the teaching force are uncomfortable with affirmative action. But maybe there’s an important difference to folks between affirmative action to correct a present bias vs. affirmative action to address a historical bias?

I’m confused. Melinda said at one point in our Twitter chat that she wrote the piece to provoke thinking and discussion, and it’s definitely done that for me.