What is mathematics? If I would have answered this question before I began my teaching career, I would have been provided some cliched, stock answer revolving around its importance to many disciplines like science, engineering, economics, etc. — basically it being some practical workhorse. Now? Well, since I wrote a book about it, it is simply abouthappiness. Finding sporadic moments of bliss learning something new and wonderful about mathematics. Just playing with numbers, puzzles, brain teasers, conundrums, algebraic ideas, etc — mucking around in the sandbox of math. The more I know, the less I know. The less I know, paradoxically, the better I feel about my ideas about mathematics.
Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…
Not that this is necessarily inconsistent with joy. Maybe you get your kicks wandering around anonymous dark mansions — there’s something for everybody. But you’ve got to admit, it sure doesn’t scream out FUN.
So which is it: is mathematics all about happiness or about stubborn frustration? Well, why not both? If, as Singh suggests, we’d best think of mathematics as an “art” then we get to ask, what other art is only about happiness?
I think educators sometimes emphasize feelings of joy and experiences of beauty in math at the expense of all the other things you can feel or see. For a while, I’ve been interested in visualizations of ugliness in math — is such a thing possible? And, as the Wiles quote shows, mathematics often involves far more than happiness.
“Of course math involves more than happiness,” you say, “but the point is that happiness is the goal, the purpose, the carrot that mathematicians are always chasing.”
Here’s an idea though: what if what we’re really in love with is the feeling of being lost in a dark mansion?
There often comes a point in my geometry classes where I ask students to come up with a definition that describes a circle. It’s always interesting to see the ideas they land on. Some students try to capture the perfect curviness of a circle; others talk about the circle’s symmetry. When I share the “textbook” definition (which kids rarely come up with) there are sometimes sighs of relief or yelps of excitement in the room:
Circle: Take all the points that are a certain distance away from a center point — that’s a circle.
It’s a beautifully simple definition, and it’s one that leaves plenty of room for generalization.
If we’re looking to generalize the circle, a great next step is to analogously define an ellipse. I like to imagine the center of a circle splitting in half and turning into the two foci of the ellipse.
(Sorry for the glitchy gif.)
We can define an ellipse in a way that is analogous to the “textbook” definition of the circle:
Ellipse: Take all the set of points that are a certain distance away from two focal points — that’s an ellipse.
Just to clarify, what I mean by “certain distance away” is we’re looking for the sum of the distances to the two focal points. So if our “certain distance” is 10, then a point that is 3 away from one focal point and 7 away from the other would be part of the ellipse.
2. Generalizing Circle Constructions
Sometimes you see geometry problems about goats on a rope. The point being that the rope-stuck goat is restricted by a circle. That goat could trace out a circle, and you can also trace a circle with a pin, string and paper. You can also make a lovely ellipse with two focal points with some string and some pins:
Why stop here, though? The definition of ellipse (let’s call it a 2-ellipse) can be expanded to include 3 focal points, 4 focal points, 176 focal points, really n focal points, no matter what integer n is.
n-ellipse: Take all the points whose distances to each of n focal points sum to a certain number. This is the n-ellipse.
James Clerk Maxwell was sixteen years old when he invented a contraption that generalized the pin and string method for constructing an ellipse. He figured out a clever way to wrap the string around so that you could use pins and string to produce a 3-ellipse.
3. Generalizing Equations
The equations generalize nicely too.
Circle (a.k.a. 1-Ellipse):
(Most students learn a different equation for an ellipse, defined in terms of minor and major axes. It’s not exactly a load of fun to analytically derive that from this equation for the 2-ellipse, I’ve found.)
4. Smallest Ellipses?
Each of the definitions above leave room for a “certain distance.” But are there any restrictions on what that distance can be?
For a circle, that “certain distance” is the radius, and the radius can be whatever magnitude that you like. Given a center, make your radius huge, make it tiny: there’s always a circle of those points just that distance away.
For a 2-ellipse, though, this isn’t so. Imagine one focal point that’s one million miles away from the other, and imagine that we’re interested in the points whose sum of distances from the focal lengths is one inch. That’s simply impossible to achieve. Being one inch away from one focal point would necessarily involve being nearly a million miles away from the other.
At some point, an ellipse might be too small for its focal points.
So: what is that least distance, given two focal points? What is the least amount of distance that you can sum to, given n focal points? What are these critical points of the distance sum function?
For a 2-ellipse, the minimum distance should just be the distance between the two focal points.
But what about for a 3-ellipse? or a 4-ellipse? an n-ellipse?
I’m still having trouble with the analytic solution in that paper, but graphically you can just graph a bunch of different distances and try to see where things are headed. Here is a snapshot from this Desmos graph, which graphs a collection of distances for three foci (which you can move around the graph — try it, it’s fun!).
This is a contour plotof the distance-sum function. We could also visualize this as slices of a 3D graph, where the “z” axis is the varying distance-sum.
Here is a 3D graph of a 2-ellipse, showing how it bottoms out at a very narrow ellipse (practically a line segment) at a height of 2 — the distance between the two foci:
And here is a 3D graph of a 3-ellipse, which also bottoms out at a certain distance-sum above 0:
In contrast, here’s the 1-ellipse, a circle. It has no minimum radius:
5. A cool problem
Say that you have three cities that are special to you, for whatever reason. Maybe you grew up in one of them, went to college in another, and have some family in the third.
If your only priority was to be as close as possible to each city, where should you live?
This is really, really similar to the question of where the minimum distance-sum is for an n-ellipse.
(It makes sense that it starts getting circle-ish as we get farther away from the focal points. From a million miles up in the air, those focal points might as well be on top of each other.)
Go to this graph and play around with your cities. You can also add more cities.
7. Now what?
I’m not sure. I tried to write this a few different ways, but in the end all I could figure out was the info dump you see above. Is there a way to rewrite this so that there’s a stronger narrative? Should I come up with problems for each section to give readers a chance to stop and think about stuff?
I think this needs to go on the back-burner, and I just need to write more about math and hope that eventually I figure a few more things out.
“I’m going to tell you that the toilet is going to eat my pee,” he said. It was my son, nearly three, standing with his pants over his ankles. He held himself, umm, firmly.
“I’m going to tell you that the toilet is going to eat my pee,” he repeated.
“The toilet is going to eat my pee.”
“The toilet is going to eat my pee. The toilet is going to eat my pee. Water is going to come out and it’s going to eat my pee.”
“The toilet — it ate my pee.”
This is more or less where my head is these days. The last six months have been ridiculous in our tiny apartment. Minus the random and intense freakouts, our kid is becoming a kid — out of diapers, into a big-kid bed, counting and talking all the time. And our baby is teething, sitting, eating, even maybe sleeping.
But most of all there’s been a lot of poop, pee, yelling, family time at 3 AM, etc.
In the meantime, I’ve been feeling a little bit lost lately. I’m feeling stalled out on a bunch of projects that I was excited about. I’m having trouble getting stuff done. I’m tired, and I feel dumb a lot of the time. This sometimes happens, I’ve learned (even before having little kids) and there’s nothing to do about it, just work it out.
Part of that working out is happening on this blog. I don’t like it, but the only way I know to get out of a rut is to find a low-stakes place to just make an ass of myself, and that’s what I intend to do in this space.
Without getting too whiny, I’ll say that I’m struggling with the same things that I’ve struggled with since the start: what does it mean to be a teacher who writes? what do I know? what can I learn? where can I publish? am I headed anywhere with all this?
I need to figure out if there’s a way to do what I do, but in a way that feels a bit bigger than how I’m currently doing it. I’m not sure what “bigger” means yet but that’s part of what I’m working on.
(Oh god, kid: go to sleep! This bedtime is dragging on and on.)
So: follow my Problems With Teaching blog for things that are a bit more put-together than what I’ll be trying out in this space. There’s an automatic IFTTT trigger so that posts here get posted on Twitter, so you can subscribe to this blog or just follow casually via my @mpershan feed. If I end up blogging a lot here, I’ll try to cut down on my tweets so the internet isn’t over-Pershaned.
What of the moon can be said to be liquid? Not the rock itself, but rather its light, which flows from surface to reflective surface (or from branch to branch).
This reminds me that we only know of the moon as reflected light — save for a few of us who have actually gone and touched the darned thing. The moon can appear huge or small, white or orange, but all this is really a trick of the light, which over time flows into various shapes and colors.
And that reminds me that literally all that we see is only reflected light, and is liquid in that same sense. Some things we touch most things that we know, for most of us, we know through seeing (or our other liquid senses — smelling, hearing). The entire visual world itself is liquid as well, in this sense, and all these images and reflections flow like the moon does. The “wise trees” tell us this truth, about the flowing nature of our perceptive worlds — that’s my read, at least. Your’s?
Chimpanzee memory has been extensively studied. The general assumption is that, as with many other cognitive functions, it is inferior to that of humans; some data, however, suggest that, in some circumstances, chimpanzee memory may indeed be superior to human memory. Here we report that young chimpanzees have an extraordinary working memory capability for numerical recollection — better even than that of human adults tested in the same apparatus following the same procedure
Watch that video for details of the test. The biologists think this might be connected to eidetic imagery:
These data showed that the chimpanzee subjects can memorize at a glance the Arabic numerals scattered on the touch screen monitor and Ayumu outperformed all of the human subjects both in speed and accuracy. Our results may be reminiscent of the phenomenon known as ‘eidetic imagery’ found by Jaensch. Eidetic imagery has been defined as the memory capability to retain an accurate, detailed image of a complex scene or pattern. It is known to be present in a relatively high percentage of normal children, and then the ability declines with age…The results fit well with what we know about the eidetic imagery in humans.
To me, this raises the tantalizing possibility that our working memory limitations are a core part of what it means to be human. We’ve been designed to hold fewer things in our head at once.
Other things I want to read, in relation to all this. I’ll add more if I find (or if people find for me!) other related reads.
Five years ago, Cathy O’Neil laid out a perfectly cogent case for why the (at that point recent) claims by Shinichi Mochizuki should not (yet) be regarded as constituting a proof of the ABC conjecture. I have nothing further to add on the sociological aspects of mathematics discussed in that post, but I just wanted to report on how the situation looks to professional number theorists today. The answer? It is a complete disaster.
From Andrew Gelman’s post on the ongoing replication crisis facing psychology and many, many other fields:
At this point, certain earlier work was seen to fit into this larger pattern, that certain methodological flaws in standard statistical practice were not merely isolated mistakes or even patterns of mistakes, but that they could be doing serious damage to the scientific process. Some relevant documents here are John Ioannidis’s 2005 paper, “Why most published research findings are false,” and Nicholas Christakis’s and James Fowler’s paper from 2007 claiming that obesity is contagious. Ioannidis’s paper is now a classic, but when it came out I don’t think most of us thought through its larger implications; the paper by Christakis and Fowler is no longer being taken seriously but back in the day it was a big deal. My point is, these events from 2005 and 1007 fit into our storyline but were not fully recognized as such at the time. It was Bem, perhaps, who kicked us all into the realization that bad work could be the rule, not the exception.
Two entirely different scientific controversies occurring in entirely different fields and of an entirely different nature. They have nothing in common. Right?
Right? I don’t know if there’s anything here, especially as a non-expert in this (and any) field.
(Sigh. I do love teaching, but sometimes I wish I had a knowledge base and expertise that I could really rest on. All this stuff of controversy makes me wish that I could really speak with authority on any of this fascinating stuff. A different lament for a different time…)
OK, so let’s toss one more scientific controversy into the mix: Yann LeCun’s post about AlphaZero-style machine learning, and the opaqueness of AI’s solutions to human observers:
The main message was, in essence, that the current practice in machine learning is akin to “alchemy” (his word). It’s insulting, yes. But never mind that: It’s wrong! Ali complained about the lack of (theoretical) understanding of many methods that are currently used in ML, particularly in deep learning. Understanding (theoretical or otherwise) is a good thing. It’s the very purpose of many of us in the NIPS community.
But another important goal is inventing new methods, new techniques, and yes, new tricks. In the history of science and technology, the engineering artifacts have almost always preceded the theoretical understanding: the lens and the telescope preceded optics theory, the steam engine preceded thermodynamics, the airplane preceded flight aerodynamics, radio and data communication preceded information theory, the computer preceded computer science. Why? Because theorists will spontaneously study “simple” phenomena, and will not be enticed to study a complex one until there a practical importance to it.
And maybe this is the tension that unites all three of these scientific controversies. We aren’t living in an moment marked by intense industrial or even informational explosion. Rather we are living in an age of immense theoretical constructions that outstrip any single person’s ability to understand them. On the corporate side this includes the algorithms of Google or Facebook. The controversy over machine learning is of the value of algorithms whose methods are unknown to the operators — e.g. facial recognition software whose methods for identifying faces would itself need to be studied in order to be comprehended.
Now, the crisis facing psychology and other sciences isn’t exactly new and its causes include non-open data practices as well as publication bias and a commitment to conceptual replication. (More here.) But it seems that a lot of scientists were treating statistics itself like a black box. You collect some data, and then toss it into the statistics pot and, boom, out comes an effect. Science was using statistics as a big, mysterious theoretical edifice, and now we’re grappling with the aftermath.
And maybe this is a way to think about the Mochizuki controversy. True, the edifice he’s created is impossible to penetrate and basically no one understands it. But that puts it in good company, along with machine learning engines and the statistical packages that psychologists run their data through. Maybe there was a time when science was about understanding, but now we’re trying to take science beyond mere understanding, and struggling to find the rules of engagement.
Definition: A set is said to be pure if no individuals belong to its transitive closure.
Well there are always individuals, right?
Consider now the following axiom candidate. Axiom of purity: Every set is pure
The main reason for the [axiom of purity] is that, as was discovered fairly early, it is is not necessary to assume the existence of individuals in order that set theory should act as a foundation of mathematics, while if we rule them out from the outset, we can simplify the theory.
A TOTAL BAN ON INDIVIDUALS
After 1963, not even set theorists had any use for individuals. Worse, there are proofs in set theory that do not work if we have to allow for them. So it is unsurprising that in the last 40 years individual have largely disappeared from view.
FULL PURITY FOR MATHEMATICS. NO INDIVIDUALS. ALL ARE SETS. THIS WILL ENSURE THAT EVERYTHING IS WELL-FOUNDED AND REGULATED.
However, we shall not follow this trend here.
The reason is that to do so would cut our theory off from at least one of its intended applications. It is by no means obvious what justifies the applicability of mathematics in general to what lies outside it…the most natural, if not only way, way to ensure that that calculus is available to be applied to counting non-mathematical things — chairs, electrons, thoughts, angels — is to allow such things into the theory as individuals.
A fundamental tenet of social psychology, behavioral economics, at least how it is presented in the news media, and taught and practiced in many business schools, is that small “nudges,” often the sorts of things that we might not think would affect us at all, can have big effects on behavior. Thus the claims that elections are decided by college football games and shark attacks, or that the subliminal flash of a smiley face can cause huge changes in attitudes toward immigration, or that single women were 20% more likely to vote for Barack Obama, or three times more likely to wear red clothing, during certain times of the month, or that standing in a certain position for two minutes can increase your power, or that being subliminally primed with certain words can make you walk faster or slower, etc.
The model of the world underlying these claims is not just the “butterfly effect” that small changes can have big effects; rather, it’s that small changes can have big and predictable effects. It’s what I sometimes call the “button-pushing” model of social science, the idea that if you do X, you can expect to see Y. Indeed, we sometimes see the attitude that the treatment should work every time, so much so that any variation is explained away with its own story.
In response to this attitude, I sometimes present the “piranha argument,” which goes as follows: There can be some large and predictable effects on behavior, but not a lot, because, if there were, then these different effects would interfere with each other, and as a result it would be hard to see any consistent effects of anything in observational data.
The analogy is to a fish tank full of piranhas: it won’t take long before they eat each other.
From Andrew Gelman’s blog. Read the whole thing, which I think has applications for education and education research.
I sometimes think that there are two fundamentally opposed world-views that are currently competing for dominance:
The most important factors that impact culture, education, behavior are incredibly un-obvious and invisible. They are subliminal, subconscious, invisible, linguistic. Therefore, to improve the world, sweat the details.
The most important factors are big, obvious things. They are economic, material, visible, sitting right in front of our noses. To improve the world, improve these big things.
I don’t think these categories break down cleanly along any of the left/right lines, but I do think the more educated you are the more you’re (we’re) asked to notice the little things. Maybe there’s a correspondence there?
“Nobody is ever going to invent an ethics class that makes people behave ethically after they step out of the classroom. Classes are for riders, and riders are just going to use their knowledge to serve their elephants more effectively. If you want to make people behave more ethically, there are two ways you can go. You can change the elephant, which takes a long time and is hard to do. Or…you can change the path that the elephant and rider find themselves traveling on. You can make minor and inexpensive tweaks to the environment, which can produce big increases in ethical behavior. You can hire Glaucon as a consultant and ask him how to design institutions in which real human beings, always concerned about their reputations, will behave more ethically.”
From Haidt’s The Righteous Mind. The elephant/rider metaphor is sort of clumsy, but his point is that rationality usually chases our moral intuitions; it’s rare for reason to override those intuitions.
Could an ethics class ever change the moral intuitions? Could a math class? What does ever change those moral intuitions?
Haidt’s answer seems to be that most changes to our intuitions come from our interactions with other people who we admire and want to admire us. I don’t know what that could mean for the possibility of moral education.