Geometry and theology

Boiled the elements down into the axioms
Mistook a fax for remedial tediums
It seems Ezekiel speaks to some
My mind was hazy and numb
And left hand gripped a clump of palladium
Saw the beast with the wings and the talons
The simple answer but it felt out of balance
Bad news like a blue screen of death
Besides the point, but which hue seems best?

I’ll keep conducting these autistic symphonies
These sentences have sentenced me
Like I didn’t have the sense to sense the mere
The presence grows weird
Doesn’t make sense but I don’t fear, not a damn thing
I live like a man who’s already dead
Like I had a motorcycle but my name is Zed
(I’ll be missed), said another clumsy alchemist
Like he just learned predicate calculus
The existential quantifier, a backwards EX
My rap career is a cataloging of defects
Copy edited by Ryan Seacrest
Like he must be new to this
Milo doesn’t exude hubris, chuuch
Like he must be new to this
Milo doesn’t exude hubris

Is this math?

Master of the Die

Look at those lines, that perspective, the symmetry…there is a clear sense of geometry in this picture, and if geometry is math (it is) then there’s a case to be made that this picture is mathematical. Hence the artist is a mathematician. So the picture is math.

Except that’s not quite right, is it? It’s not fair to this piece of art to see it as essentially mathematical. That’s not what the human person who created it was going for. Math doesn’t get to make a claim on any use of parallel lines. Back off, math! Art gets to be art.

I’ve been thinking a lot lately about what counts as math, and what it means to expand what counts as mathematical. Certainly, if you think math is just what’s covered at school, you’re wrong. But then we make these expansions — kids are doing math when they’re playing, they’re doing math when they’re drawing, when they’re braiding hair, when they’re building…

Maybe the issue is in thinking of mathematics as a field or a single subject, as opposed to a mode of thinking that is used by artists, philosophers, lawyers, Wall Street, cashiers, kids, hair-braiders, engravers, everyone. Mathematics isn’t a distinctive activity that you do, it’s a certain way of doing other things.

But then what is that certain way of doing things? Probably not a clear set of criteria, but a kind of family relation: if you’re using numbers, if you’re paying attention to repetition, using the properties of shapes to create something new — you’re not necessarily doing mathematics, but you’re doing whatever it is you are doing in a mathematical way.

But math doesn’t get to claim art.

Some questions about the problem of teachers leaving the classroom

Is it actually a problem for kids? Would schools be more effective places if more teachers on the margins of leaving were to stay in the classroom? How do we know? Is there a correlation between ambitious and teaching skill? What is the correlation?

Do master teacher programs improve learning for a district?

How much of the stress in education about people leaving the classroom could be explained by how uniquely meaningful working with children is? After all, going into management involves a change at work across professions. (Sales managers don’t go on sales calls; you leave the regular police work to get a desk job; you still do rounds occasionally but mostly you don’t see patients, etc.) How much of the problem is that there is a huge emotional gap between teaching and higher-paying work that keeps teachers in the classroom, marginally?

Would people be more effective at their administrative jobs if they were partly in the classroom? Would they be more influential?

William Carlos Williams was a doctor by day, poet by night. No one suggests that there should be more doctor/poet jobs. How do we decide what sorts of jobs their ought to be?

I find this so confusing. What questions do you have? Comments are open.

Would you like to read together at The Old Reader?

Do you miss Google Reader?

Way back when Google Reader closed up shop, my roommate and I decided to migrate ourselves over to The Old Reader. Per the title, it’s a near-clone of Google’s late RSS aggregation tool.

Here are things I love about it:

  • It’s simple.
  • Unlike the rest of the internet, it’s not built around images. It’s built around text.
  • Sharing and commenting is exactly as it was with Google Reader.

My (now former) roommate and I have been continuously using The Old Reader to share and discuss stuff we see on the internet for the past however many years. What I’ve lost, however, is the posse of friends with other interests whose enthusiasms I no longer have access to. They aren’t on The Old Reader; they’re on Facebook, Twitter, elsewhere.

Twitter is, like, I hope I don’t say the wrong thing. The Old Reader is, like, let’s think about this together and we both might be wrong. Very much in the spirit of “rough draft thinking.”

Would you like to join? If so, sign up for The Old Reader (free, but I pay for unlimited subscriptions), and we’ll follow each other and read together.

The intellectual work that teachers can do but researchers probably can’t

[I’ve written versions of this post many times before. Here here here. Don’t read those, this version is probably better.]

Tomorrow night, I’m going to teach teachers about teaching. I think a legitimate question is, on what grounds am I claiming to know anything about teaching at all?

To be sure, I am pretty confident that I know something very important about my own teaching — in my school, with my students, in my courses, given my personality, etc. I observe my own classrooms (imperfectly) every day. The cumulative evidence of all that observation makes me pretty (not fully) sure that I’ve figured something out.

But the tricky thing about teaching is that this stuff often doesn’t translate to other situations. Just because something works in my classroom (according to me) doesn’t mean that it’ll work in vastly different contexts. To get really specific about this for a second, I teach students who are among the wealthiest children in America. This reality impacts my school in a bajillion ways. Who says that my dumb ideas about feedback will mean anything to the other teachers in my department. Teachers in other schools, and especially high-poverty schools? Forget about it.

(To be fair to myself for a second: I haven’t only taught in my school.)

The point is that there are obvious reasons to doubt that the things I think I know are really truths of teaching. This is even true if we move past the particular practices that I advocate and get behind the thinking and values that support those practices. I think I have a useful way of thinking about teaching, or I think I’ve identified some value that is important for the student experience. Who says that this is anything but my own thinking?

This is the natural place that research on teaching enters the conversation. Whatever you want to say about research, it’s not about my classroom. In general, it’s about forming generalizations in a way that improves upon (e.g.) my ability to make stuff up about my teaching.*

There are lots of interesting edge cases to consider, but I think the generalization about generalizations stands. Researchers might write cases grounded in particulars or engage in a teaching experiment, but the point of those is to contribute to the formation of generalizations that are broadly useful. 

This is getting pretty abstract so let me just get to the point: could researchers ever respect the generalizations that teachers make about teaching as knowledge that stands on par with their own?

The usual way of talking about teacher/researcher parity is to say that researchers excel at making generalizations, while teachers contribute crucial local knowledge. And it’s totally true that teachers do have local knowledge.

But does this really create parity between teachers and researchers? The whole point of broadly useful knowledge about teaching is that it goes beyond local knowledge — it makes a generalization. If what teachers can contribute is local knowledge, then I think we’re just saying that teachers are at best a source of data to the researcher. The teacher inputs local knowledge, the researcher generates broadly useful generalizations.

It’s true that there’s no reason to inherently value general vs. local knowledge, so in a certain sense there can be parity between teachers/researchers. But at the end of the day, what’s broadly useful are generalizations, and teacher knowledge can’t really compete with what researchers contribute.

Or…can we?

I want to speculate a bit about some different ways of sorting out the relationship between what researchers and teachers can contribute. To start, I want to ignore local knowledge for a second and talk about how teachers contribute to generalizations, i.e. researcher turf.

I have a few ideas here, and they’re very rough, so bear with me.

First, researchers are institutionalized while teachers are necessarily amateurs at producing generalizations. The relationship between teachers/researchers can then be folded into the general relationship between amateurs and experts. And, of course, we need experts. But the ecosystem isn’t healthy if it’s entirely populated by experts.

Amateurs play a lot of important roles, even when it comes to forming generalizations that are broadly useful. Here are a few that I’ve read about (too lazy to cite right now):

  • Amateurs can disrespect the boundaries of fields or sub-professions and put together ideas that from an institutional perspective are incongruous
  • Amateurism is in general lower stakes/lower payout than being an expert. If I’m an amateur and my ideas are wrong or useless, my career isn’t on the line. So there’s a way in which amateurs can attend to riskier ideas, or work on lines of thought that are perceived to be less rich in reward or are in general undervalued.
  • Amateurs play an important role in teaching and spreading expert generalizations, but in doing so amateurs often simplify or otherwise improve the results of experts in significant ways.

But this way of framing things — teachers as amateurs, researchers as experts — doesn’t really leave room for teachers to ever get institutional respect from experts as generators of generalizations about teaching, and two further points on that:

  • this is probably true
  • this is much more exciting to me than institutional respect

It’s not good for my $$$, but I am really quite fine accepting the role of an amateur in all this. It’s exciting to try to smash fields together and to not be beholden to conventional wisdom in the field. I can chase ideas about teaching, throw them out there for others and see what resonates for others. We make up the rules as we go. It’s fun!

That’s the spirit in which I’m going to teach this class tomorrow night. I’ve got this stuff I’ve figured out about teaching. I don’t want to make myself sound like a tin-hat Alex Jones-type, but I do think that what I’ve learned about teaching goes against a certain conventional, institutional, expert way of thinking. And it is the result of mashing up a bunch of things — trial-and-error in the classroom, reading research, experimenting by giving presentations to teachers. And if it’s not broadly useful as a generalization about teaching? Hey, that’s OK too. There’s very little at stake here.

So, I’m not an expert, and neither are you: maybe these ideas are useful to you? Let’s find out. That’s the way I approach this stuff right now, as an amateur.

Talking Holocaust With Your Kids


I was picking up the baby and Yosef, my three-year old son, at the synagogue their daycare rents space from when we wandered into the setup for an event. There were rows of chairs facing a projection screen and six enormous white candles at the front of the room — clearly a Holocaust memorial service, on the occasion of Yom HaShoah.

We walk through the room and Yosef is all intrigued by whatever it is that intrigues three-year olds about everything. I’m trying to get him to put on his coat with the baby strapped to my chest, clawing at my neck. Nearby is a friend of mine, also trying to shove her daughter’s arms into the jacket.

Yosef wants to know what all the stuff in the room is for.

“It’s for the Holocaust,” I say.

“Really it’s against the Holocaust,” says my friend.

“That’s right,” I say. “Really it’s anti-Holocaust.”

“What’s the Holocaust?” Yosef asks.

“Well, we’re out!” my friend says. “Have fun!” She strollers her daughter away, and we follow behind them. They go home, and we start walking up the big hill up to our apartment. I point out a car, a street sign, a crack in the sidewalk, a clump of dirt.

“What’s the Holocaust?” asks Yosef, a kid who apparently doesn’t know how to take a hint.

“Well,” I say, “well.”

“Well. Well, well well. Yup. Well. So,” I say, “this is a sad story. Yeah, let’s start with that.”

I’m the sort of parent that takes my kid’s questions as a challenge. “Why does it get dark during winter, Daddy?” he asks. “That’s right, do you even know, Dad? Or are you some sort of chump that can’t explain how seasons work to a kid who sometimes pees on the wall?”

So of course I can’t help myself. The Holocaust will be my biggest parenting win yet.

“It’s a sad story,” I say. “Do you know Irene from shul? Right, so she wasn’t born in New York City. She was born in a place called Germany, and she grew up there and her whole family did too. All of her friends and family. And then one day, the Germans told the Jews that they had to get out. We don’t want you any more! Go, and if you don’t go then we’re going to hurt you.”

“Did they get better?”

“What do you mean.”

“After they got hurt?”

“Ah. Umm, no. They didn’t get better.”

“Why not?”

“Because they hurt them in a way that wouldn’t get better.”

“Anyway,” I continue, “Irene ran away. She went to a different country, and then she came to New York City. But most of her friends and family stayed in Germany, and they got hurt.”

“How does it end?” he asked.

“That’s it. They got hurt, Irene had to run away. That’s why it’s a sad story.”

“But it’s happy for the people who ran away.”

“Well, yeah, except they were still sad about everyone who they loved who didn’t run away and got hurt.”

We thoughtfully pause. A leaf falls from a tree. We hear the sounds of children playing. I get ready for a poignant moment with my son.

“Tell another sad story!” he says. He’s laughing. “Tell it again! Tell another sad one.”

“Another one? Alright. OK, so here’s another sad one: this wasn’t the first time this has happened, and it’s happened other times, and it’s probably going to keep on happening. That’s sad.”

“Ha ha! Tell another sad story!”

My poignant moment is backfiring. Somehow I’ve managed to warm my son’s heart with the story of the Holocaust. I put on my serious face.

“Well, I’m too sad from telling all these sad stories. That’s it for now.”

“Tell another!”

“You know how Pharaoh had slaves? Well there were slaves in America too, and it wasn’t that long ago either.”

“OK,” he says. “Hey, there are three sidewalks here, but they turn into five! There are three here, and then they’re five. And NOW THERE ARE THREE AGAIN! DADDY LOOK. WHY? WHY IS IT THREE ACROSS?”

And it was true. The sidewalk started with three little lanes but then expanded into five, back to three again. And who knows why they made it like that, really. How are you supposed to make sense of something like that?

A Jew thinks out loud about Farrakhan

I don’t know a ton about Nation of Islam. A few years ago I read Manning Marable’s biography of Malcolm X, and I left that book with an enormous admiration for Malcolm X, feeling like we’d lost a really great American. And if only for that, I’m prejudiced against Louis Farrakhan.

The other piece of context I have for Farrakhan is Freddie de Boer’s excellent essay in Harper’s, titled “The Charmer.” Here are a few of the choice bits of that essay that pertain to his anti-Semitism:

To speak of what Farrakhan gets right about racism is to risk lending credence to all he gets wrong. For just as he has been correct in his indictment of white supremacy, so too have his critics been correct in their indictments of him. Farrakhan’s ample critical gifts have always come packaged with abundant bigotries: hatred of Jews, mockery and fear of homosexuals, denigration of women. I have heard occasional attempts to deny Farrakhan’s anti-Semitism, but these defenses approach lunacy. Farrakhan’s distrust of and anger toward Jews are as central and constant as any other aspect of his philosophy; anyone reading over his speeches for examples of anti-Semitic rhetoric will quickly find herself spoiled for choice. Jews control the banks, they control the media, they control the government, Israel knew 9/11 was coming.

Why then does Farrakhan fear Jewish reprisals? For no good reason, it seems, other than his palpable anti-Jewish paranoia, along with the anti-Semite’s tendency to see one of the world’s most oppressed peoples as the shadowy driver of all events.

So let’s take for granted that Farrakhan’s anti-Semitism is not subtle or interesting in and of itself. It’s not a phenomenon that demands nuance. It just is: he’s an anti-Semite.

OK, but Tamika Mallory, national co-chair of the Women’s March, was in attendance at Farrakhan’s most recent speech. And, apparently, it’s not an isolated thing, as she’s posted about Farrakhan before on social media:

Here is her defense, it seems:

If I understand, she’s saying that whatever Farrakhan’s faults, he is an important leader. She doesn’t share his anti-Semitism (or, presumably, his deep homophobia and anti-feminism!) but she won’t refuse to associate with him or to praise him.

OK, so what’s the generalizable principle here? I think it’s: Don’t refuse to associate with or listen to or offer praise to someone who plays an important, positive, communal role, even if they have deep, disgusting faults. 

Now, is her invoking this principle hypocritical? I don’t know much about Mallory, so I can’t say. It’s certainly true that e.g. Aziz Ansari didn’t get this sort of benefit of the doubt from the left, but then again that might be entirely consistent with the principle above. Ansari is, at the end of the day, an artist. Farrakhan’s faults — his paranoia and hate — may run deeper than whatever it was that Ansari was guilty of, but then again maybe it gets overridden by the Million Man March? I don’t know.

And, besides, I actually agree with that principle to a point, and wish it was more widely followed.

Is it anti-Semitism for that principle to be selectively invoked for Farrakhan but not for others? I’m not sure. Remember, Farrakhan isn’t just a hater of Jews. He’s homophobic, among other things. So the claim that Mallory and other leftists wouldn’t defend Farrakhan if he had issued hate speech about some other group…Farrakhan hates a lot of people. I’m not sure this argument really flies.

(If it does fly, it’s because of the intensity and persistence of his anti-Semitic speech, even compared to his homophobia or anti-feminism.)

So, where does that leave us? What’s at stake?

I think 95% of the emotion at play here is about whether Jews get to be counted as the oppressed or oppressors in the US today. Of course the answer could be both, and intersectionality would call for that, but in practice intersectionality is a cherry-on-top nuance over a baseline judgement: oppressed or oppressor?

Lots of Jews generally want to be seen as oppressed because of, you know, all of Jewish history.

Leftists want Jews to be seen as oppressors because they’re mostly white, and especially because of anti-Zionism.

Mallory could easily apologize for Farrakhan’s anti-Semitism, but doesn’t want to, because Jews (as oppressors) have it coming for them, just as white people do in Farrakhan’s rhetoric.

Jewish critics want to hold Mallory accountable to leftists and progressives to force a reckoning, and to extract a concession that Jews are victims of oppression.


And where do I stand? I think this is some of the danger of thinking too much about identity and status. Is there anti-Semitism in the US that we want to fight against? Is there an actual plan on how to protect Jews from hate crimes?

I want to avoid the sort of metaphysical issues that talk of anti-Semitism usually devolves into. I’m interested in protecting Jews from the sort of things I’m afraid of: shul shootings, terrorism, street harassment. I’m not interested in fighting anti-Semites, I’m interested in defeating anti-Semitism, and it’s not clear to me that the two are the same project.

So Farakkhan is a loony, but I don’t see much good coming from trying to take down Mallory for her association with him. The Women’s March isn’t about to add an anti-anti-Semitism plank to their work anyway.

The set of all singletons doesn’t exist

From Classic Set Theory:

Use appropriate axioms out of Zermelo-Franekel axioms 1-6 to show that \{x: x \textup{ is a singleton} \} is not a set.

I’m fascinated by sets that go wrong, and I was surprised that set of all singletons leads to trouble. Here’s my best attempt at explaining what goes wrong.

The issue with “is a singleton” as the defining property of a set is that any set can be turned into a singleton via the Axiom of Pairing in ZF set theory. The Axiom of Pairing is like a machine: give it two sets, it’ll smoosh them together into one set. Give it the same set twice, and it produces a singleton. That way, there’s a singleton for every set.

Slightly more rigorously, let X be any set at all:

“Hi, my name is X and I’m a set.”

Feed two copies of X into the Axiom of Pairing; the axiom then spits out the set whose elements are just the elements of X and, well, X:

“Yo. The Axiom of Pairings made me. I’m the set whose elements are the same as those of X and … err, ok, so just X. I’m \{ X \}.

So, for any set X there exists a singleton set whose only element is X. Great! So what’s wrong with the set of all those singleton sets?

The problem is another axiom, the Union Axiom. The Union Axiom is another little set theoretic machine. Feed the Union Axiom a set (chomp chomp) and it spits out another set. This new set contains all the elements that are in subsets of the original.

An example is probably useful. Say that your set is the set of all the basketball teams in the NBA: {Bulls, Pacers, Knicks, etc.}. The Union Axiom is the set of all the players in the NBA. The Union Axiom bites through the husk of one collection and produces a new one out of the things living just one level down.

Here’s where we start breaking things: what if you feed the set of all singletons to the Union Axiom? (Get ready, because here comes everything.)

  1. The Axiom of Pairings can turn every set into a singleton.
  2. The set of all singletons collects all of these singletons into a set.
  3. The Union Axiom would create a collection of all the elements of elements of the set of all singletons, i.e. literally every set.
  4. So the set of all sets exists.

And that’s great, if you love contradictions, because now you can make any set you want, including Russell’s famous one. Because in this version of set theory the way that you block Russell’s Paradox is with the Axiom of Separation, which says that you can define a set using any property, as long as it’s a subset of some currently existing set. Now, though, we have a set of all sets. Everything can exist, bats explode out of the belfry, Pandora breaks the seal, boom!

“Let there be a set that’s a subset of that set of all sets, containing all of the sets that are not members of themselves.”

And that’s Russell’s paradox, and it leads to contradiction.

All of which is a long way of saying, the set of all singletons does not exist. Which is weird, because I had written previous that this was the way Frege defined “1” within his system, as the set of all singletons. Is this set itself one way that the problems with Frege’s system manifested itself? Or was I misunderstanding Frege’s definition?

One subtlety I’m missing out on right now is how the Zermelo-Fraenkel set theory system compares to other set theories. This bit from Wikipedia seems to summarize what I’ve just started to understand:

Gottlob Frege and Bertrand Russell each proposed defining a natural number n as the collection of all sets with n elements. More formally, a natural number is an equivalence class of finite sets under the equivalence relation of equinumerosity. This definition may appear circular, but it is not, because equinumerosity can be defined in alternate ways, such as by Hume’s principle.

This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theoryZFC and related systems, because in such systems the equivalence classes under equinumerosity are proper classes rather than sets.

I’d like to understand some of the other sets that lead to trouble in ZF set theory. What sort of properties, in general, lead to contradiction? And I’d also like to understand what sort of choices you make in type theory in order to permit this sort of definition of “1.”