# Continuous induction is a thing!

Great question:

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:

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:

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

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.

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.

# Save Big Money at Menards

Do you know the Menards jingle? Two options:

1. What?
2. Of course I do!

In other words, you’re either from the Chicago area or you’re not. I am, and as we’ve been visiting my parents this week I’ve had many chances to revisit that particular aspect of my childhood.

More importantly, though: here are a few of the CDs and tapes that are hanging around in my childhood bedroom.

Bridges to Babylon, The Rolling Stones – I don’t know how old I was, it could have been anywhere from 8 to 12 years old, so let’s say that I was 10. I told my mom that I wanted to get a CD, so she drove me to the local Blockbusters — one of those Blockbusters that carried music. She trotted me in front of some salesperson who I was totally intimidated by. Then the Blockbusters guy asked, “So, what sort of music do you like?”

How didn’t I understand that I would need to prepare an answer to that question? I had no plan. None. I thought that I would walk into that store, ask for Music, and then be granted Music.

“Uhh…I like music where you can hear the words,” is what I said, which is stupid because there is lots of good music where you can’t hear the words, and in fact some of the best music exists in that sweet spot where you are 60% sure of what the singer is saying.

(Until a week ago, I definitely thought the Magic School Bus theme had a line that went like this: raft a river of blood!)

So Blockbusters guy handed me two CDs, and we bought them both. The first was The Rolling Stones’ Bridges to Babylon.

The second was freaking the Beatles’ Sgt. Pepper’s Lonely Hearts Club Band, which means that Blockbuster guy saw this little kid asking for music where you can hear the words and decided that he either wanted (a) one of the best albums from the best band of all time or (b) an irrelevant late-period album from one of the best bands of all time. What was that thinking? I wish I could talk to that guy, but I can’t because that Blockbusters isn’t there any more, so that guy probably had to get a new job.

Everclear, So Much For the Afterglow

This is from a few years later, and I was definitely getting closer to finding my musical tastes. I have no idea where I would have heard of Everclear, though my main sources of access to music were MTV, VH1 or the radio. Actually I have vague memories of the “Father of Mine” video on VH1, so that’s probably where this came from.

I remember spending whatever pre-teen money I had on the CD, bringing it home and putting it in the boombox. At first I really liked it. But after a few tracks (yeah) I realized (yeah) that every song (yeah) eventually turns into a chorus of yeahs (ye-ah). To the record with only one pop idea!

Various Artists, Wild Wild West Soundtrack

I swear, I can still do it! Let’s see how far I get…

Wild Wild West! Jim West! Desparado! (something), no you don’t want na-doe(?)…

that wasn’t nearly as much as I thought I could do. Damn.

This CD was huge when I was at summer camp, which I think was when I was going into 8th Grade. I remember this one was on heavy rotation on our bunk stereo.

And it’s not bad, honestly. BLACKstreet, Dr. Dre and Eminem, Enrique Iglesias, let the rhythm take you over. 95% of the time when a bunch of 8th Graders get into a group they make worse decisions than any single individual would, but I’d have to say that this is the exception. Ridiculous as it is, this was my route into rap/R&B.

David Gray, A Century Ends, Lost Songs, White Ladder, A New Day at Midnight

My David Gray fandom as a teen is…not flattering. Here’s how it happened: starting around 8th Grade I started playing music with friends, i.e. in bands. I played keyboard/piano — at first a dinky one, and then a proper instrument. And this was terrific — I met a lot of people who liked music and played music with them.

This, though, I don’t know. Here’s what I’ve found: just because you are excellent at playing an instrument does not mean you have good taste in music. In fact, some of the most insufferable listeners of music I know are musicians because they’ll like a band that makes “interesting” but terrible music.

Anyway, David Gray is not “interesting” in any significant way, though he was recommended to me by a guitar player friend with terrible taste. (At least he did as a kid. Maybe he is reformed from his jammy, hippie ways.)

The only thing embarrassing about liking David Gray as much as I did was that, for a few years, I was a child whose favorite artist was charting only on Adult Contemporary. But, look, the guy has some good songs.

Most importantly, though, David Gray was probably the first artist who I liked partly because I knew his story. His first few records underperformed commercially (presumably critically also given the songs he wrote). He then, as a sort of last ditch effort, married his extremely acoustic balladeer sound with some simple synthy things and had a surprise hit (“Please Forgive Me”). The video involved a piano crashing, I think. Then he released Lost Songs, which gave his new fans a chance to hear all of his unpopular music — I dug it.

I remember an extremely embarrassing conversation with my father, on our way to a hockey game. Usually on our way to my hockey games we listened to one of the following dad-approved artists: Billy Joel, Bruce Springsteen, Meatloaf, Rod Stewart, Bob Dylan.

(All first-round inductees to the Dad Rock Hall of Fame, and to my heart. Except for Rod Stewart. Forget that guy.)

Anyway, the embarrassing thing I said was “You know Bob Dylan sort of sounds like David Gray.” And my dad sort of huffed and said, “No, that guy sounds like Dylan.”

Which is mostly embarrassing because, no? They don’t? The only thing they have in common vocally is that it’s sometimes hard to make out the words that they’re singing. As established, this was a major concern of mine as a young listener. Presumably that’s what I was getting on.

Do you remember at big music stores (which is what I was mostly frequenting) they had those CDs with headphones dangling and you could preview a CD but only while standing next to a pole that was also a CD player?

I do. I bought this at Barnes and Noble after hearing “Bent” on VH1.

Papa Roach, Infest [non-explicit version]

Man, so this is precisely the kind of kid I was. I was watching MTV and I saw the video for “Last Resort,” the hit single from Papa Roach about suicide. Now, I didn’t care about the lyrics — surprise, Blockbusters guy! — and I just knew that I didn’t have riffs like that in my life yet. Plus, it’s catchy, so sue me.

The pickle I was in was this: there is a lot of cursing on this album, and that was a no-no in my parents’ home. So…the intensely non-cool compromise was to order the non-explicit version from the internet, maybe the first purchase I ever made from the internet, come to think of it.

What’s sort of funny is they reword the lyrics insert so that there are no curses there either. Which means the lyrics page reads like this:

Cut my life into pieces / this is my last resort

Suffocation, no breathing / don’t give a (bleeped) if I leave my arm bleeding

This was the way my childhood was. I did all these things that should have been cool, and they even sort of sound cool if we speak in general terms. Once we get into the details, the details are never cool.

So for instance it’s true, I did play in bands all through high school, and we even got gigs. Cool!, you say. But then, ruining a perfectly good thing, I go on: yeah, we were a Jewish band, mostly playing on Purim or Channukah, mostly dancing music for classmates and our rabbis. We played a bar mitzvah once. We were a pretty big deal.

And also I played keyboards, the single un-coolest instrument you can play in a band. With conventional rock instruments, here is the ranking:

1. Guitar
2. Drums
3. Vocals (controversial ranking, but this is my experience)
4. Bass
5. Saxophone
6. Are there any other instruments in the band?
7. Piano keyboard

The White Stripes, Get Behind Me Satan

Here is where I start to get my act together.

I remember where I was when I first heard “Seven Nation Army,” which is not on this album (this is Jack White’s piano album, which I liked for obvious reasons). I first heard it in the dorms at my yeshiva high school (it had a dorm for the dormers) and I was with my friend Shmuel in Ariel’s room. Ariel wasn’t there, but he had a radio.

The radio was on and we heard Meg White’s drums come in. Boom boom boom boom. Steady, icky thumping, one after the other, relentless.

I just want to be clear: I never danced, but I swear to it — we were dancing around Ariel’s room. In the scene in my memory, we were just drumming on things, yes, yes! This is the stuff. Mainline it, please.

Goodbye, Everclear. So much for the afterglow.

***

And much more. That’s what I called music. This is volume 1. There is much more.