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.

YouCubed, Reviewed

This exponents activity is neither original nor at all an interesting version of the idea. It’s no better than what most teachers would make on their own, if they wanted to teach exponent rules inductively.

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Better versions of this are readily available in practically any textbook, but Illustrative Math has a totally free and online unit on exponents that does this activity better. It’s less tedious and repetitive and it asks questions to push students towards generalizations, rather than asking kids to churn out rows and notice the structure at the very end (“discovery”).

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Yes, it’s at a Grade 8 level, but this lesson is pretty much there too. And if you can wait a few months, you’ll have the high school version available too.

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.

Continuous induction is a thing!

Great question:

Great answer: Yes!

From the paper:

Consider “conventional” mathematical induction. To use it, one thinks in terms of predicates — i.e., statements P(n) indexed by the natural numbers — but the cleanest statement is in terms of subsets of \mathbb{N}. The same goes for real induction.

I think of induction as being about dominoes falling, but the focus on sets is a different way of talking about induction — even for the natural numbers.

To be OK with this paper’s explanation of continuous induction, we have to be OK with the idea that its sets that the inductive property — it’s not a process metaphor, it’s a property metaphor, or something like that. Maybe a useful metaphor is flammability, rather than dominoes. Some sets are made of stone. Others you set a spark and the whole thing goes up in flames. Those are the inductive sets.

Here’s what it means for a set of real numbers to be inductive:

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The three conditions say what it means to be inductive, and the theorem states that if a set is inductive then it is fire: something that’s true for part of that inductive set has to be true for the whole thing.

(Something I’m wondering about real induction: can’t the spark can be planted either at the front or back of the set? I think with slightly different conditions for inductivity you could start this at b.)

The paper goes on to use this continuous induction to prove some of the classic theorems of calculus and analysis, including the Intermediate Value Theorem. Here’s my attempt to restate the proof in the paper.

So, I’m going to tell you three things about a graph. First, it starts like this and ends at 10:

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Second, the function is continuous.

Third, the function does not ever cross zero.

Question: Are you guaranteed that this graph stays positive the whole time?

Answer: Yes, duh. But as with a lot of this foundational stuff, the challenge is to say why.

So let’s make the following inductive argument, by first collecting all the inputs 0 through 10 of my function that yield positive outputs into a set. So far, just from the graph, you know that this set contains all the real numbers from 0 to about 2. We’d love to show that actually all the real numbers from 0 through 10 are part of this set.

Now, let’s tee up the conditions for this set being inductive.

  1. 0 is in this set, because duh I showed you that 0 is in this set.
  2. Pick any number that’s already in this set — it looks like 2.2 is in it — then there has to be some larger number in the set i.e. also yields a positive output. That’s because this function is continuous, and so approaching 2.2 from the right, eventually the outputs need to get really, really, REALLY close to the true output when 2.2 is the input to f. That necessarily means that it’s going to have a positive output, since the output of 2.2 is positive. So there needs to be numbers larger than 2.2 that are in this set if the function is continuous.
  3. And suppose that you knew some range from 0 up to but not necessarily including 8 where was positive, i.e. still hanging out above that zero line. Well, even though we “but not necessarily including”-ed 8, 8 has got to be positive anyway. Because go backwards at all from 8 and you should hit positive outputs, but because this is continuous eventually all those positive values need to be (eventually) super-duper-duper close to f(8). So 8 must have a positive output.

And by induction, tada, there’s nothing stopping you from extending this to the entire domain of 0 to 10, it’s positive the whole way through. And since this is just a restatement of the Intermediate Value Theorem, you’ve proven that too, by induction.

I still have questions, but this is very cool.

Motivation

My interest in any particular piece of mathematical content varies. There are some things that I think are just absolutely fascinating, others…nah. So I think if I was, at bottom, motivated by a love of mathematics then my teaching would ultimately suffer for it.

Working with children, the future looms: I want these kids to have skills, to pass the tests, to get into schools, to get through college, to find meaningful work, to be able to see the world differently in the (hopefully) long, long time they’ll spend outside of schools. But when you take a look at what is genuinely preparatory out of what we’re supposed to teach, it’s hardly everything. People say that school math is mostly useless, and I do see what they mean.

So the answer for me is in the present, and I don’t worry too much about everything else that’s going on. The situation is that four or five times a day, a bunch of people get into a room and are supposed to study mathematics. I honestly don’t know why they’re supposed to study mathematics, but that’s just it: they are.

They are in this room, and I am also in this room and am supposed to help them learn it. If I quit tomorrow, someone else would do it, but I haven’t quit and I am in the room.

So given all this, the question for me is always, how can I do my job without making anybody feel dumb or miserable? And that gets me pretty far.

Is this math?

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.

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.

 

 

I had a great conversation with Albert Brooks once. When I met him for the first time, I was kind of stammering. I said, you make movies, they live on forever. I just do these late-night shows, they get lost, they’re never seen again and who cares? And he looked at me and he said, [Albert Brooks voice] “What are you talking about? None of it matters.” None of it matters? “No, that’s the secret. In 1940, people said Clark Gable is the face of the 20th Century. Who [expletive] thinks about Clark Gable? It doesn’t matter. You’ll be forgotten. I’ll be forgotten. We’ll all be forgotten.” It’s so funny because you’d think that would depress me. I was walking on air after that.

Conan O’Brien, who gets it. Who says the present doesn’t matter?
Posted in Fun

“Equity” is dead, long live equity

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By the time organizations — even organizations whose work I really like — start using the language of equity to advertise their work, it’s a sign that we’ve overtaxed the latest bit of edu lingo. “Equity” is at that point in the edu fad life cycle; it’s beginning to mean just about anything.

I don’t know if there’s anything to do about this. I think this is less about education and more about the corporate world — business lingo isn’t much better than edu lingo. People want to signal that they get it, without getting too bogged down in what exactly “getting it” entails.

The thing I try to remind myself is to be specific and to use familiar, boring words whenever possible. In place of stuffing meaning into abstract terms, I try to put it into sentences. And instead of “equity” I try to talk about the particulars: unsafe classrooms, hot schools, bad water, inexperienced teachers, and so on. This is my personal resistance to the educational world’s endless desire for catchy language, as I think it’s really all we’ve got.

Scattered thoughts about: Springsteen on Broadway

Here are some scattered thoughts:

  • Springsteen always structures his shows into little mini-arcs, acts, cycles of songs, whatever you want to call it, and he does exactly the same thing here. So “Growin’ Up,” “My Hometown,” “My Father’s House” and “The Wish” are all about his childhood, birthplace, father and mother respectively. Then we get stuff about the road and how it opened up his vision of America: “Thunder Road,” “The Promised Land,” “Born in the USA.” You get the marriage songs with Patty, his case for national despair and national hope, and then “Born to Run” ties it all up in a bow. In other words, this is super highly structured into suites, which is what he generally does though not in this precise way.
  • The songs that I felt were most transformed in this setting were “My Hometown” and “Tenth Avenue Freezeout.” “My Hometown” always sounded corny and sad-sacky to me in the context of arena rock; it makes a lot more sense whispered and confessed. And “Tenth Avenue” became a sort of wake for Clarence Clemons, Danny Federici, all others who had been lost. It was touching and sad.
  • The song that brought me to tears was “My Father’s House.” Part of what happened was I get confused between some of the tracks on Nebraska because the instrumentation and phrasing are similar, so I thought he had launched into “Reason to Believe,” a comparatively upbeat song…when I realized it was quite the opposite, it felt crushing. Part of what makes it work so well is the story of Bruce, as a child, entering the bar to retrieve his father perfectly parallels the vision of running into his father’s home. And Bruce’s perspective that we emulate those whose love we seek but don’t receive is insightful, as is his confession that the voice and stage persona he constructed is an idealized vision of his father. In his memoir Born To Run it’s clear that all of these revelations come from his years in therapy.
  • But what exactly is going on with Patty? What does it mean to sing “Brilliant Disguise” with your wife of 27 years? Just to review, “Brilliant Disguise” is a song Bruce wrote in the early years of his first marriage — that was a failed marriage — and it is stunning to me that this first marriage was able to survive this song for a second. I mean, how do you hear this song from your spouse (or your own mouth) and not immediately know that you’re headed for divorce? Now you play the loving woman / I’ll play the faithful man // But just don’t look too close / Into the palm of my hand, I mean come on! I have no idea what it means that Patty is singing that song with him? Are they staring down this fear together? That she supports him in this moment of doubt? I don’t quite get it.
  • Here is what I was thinking for most of the film: Bruce looks old, he sounds old. He is 69 years old, and he looks it and sounds it. This is to my mind the most artistically interesting thing about the entire enterprise. Pop music in general (rock in particular) is a genre in and about youth. It’s not that Bruce is the only aging rocker around, but I think this is the only time I’ve seen aging performed. If you go see The Rolling Stones or Paul McCartney or Elton John, you can see aging denied; you can see old people act and perform young, and giving the audience a chance to experience a kind of eternal youthfulness. That’s good, that’s a natural thing for aging pop stars to do. There is also a mini-genre of dying performed — this is Johnny Cash’s last few albums, Warren Zevon playing “Keep Me in Your Heart for a While.” This is interesting too. But I don’t think I’ve ever seen anyone in pop music do quite what Bruce does in this show, which is reflect on regrets, deaths, relationships, and try to summarize a legacy in a way that only makes sense from the perspective of someone who is decidedly looking back. He looks smaller than life, which is an artistic choice; the second he goes on tour with the E Street Band he’ll be back playing that other role. But what’s special about this isn’t the setting or the music but the choice to perform as a 69 year old.
  • One day Jay-Z will do a nice stint doing something similar. This particular thing might only work if you have a well-known (to your fans) legacy to deconstruct. Not very many pop artists have the longevity and penchant for myth-making to have that sort of narrative. Mr. Carter does, I’m not sure who else, though I’m sure there are others.