My set theory book is getting kind of fascist

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

Fully 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.

…ooh?

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.

Phew. From Set Theory and Its Philosophy.

The piranha problem

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 every going to invent an ethics class that makes people behave ethically.”

“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 MindThe 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.

When I say people, I mean mathematicians

On the wave equation and the beginnings of Fourier analysis:

“Very soon after that, people began…when I say people, I mean mathematicians.”

I only know Rudin from his analysis books, but come on, man! What, are you worried that someone will think mere people would be interested in this?

Another not great moment comes a few minutes later:

“D’Almbert raised a rather strange objection to that. This is a curious objection to my mind, because these were sophisticated people, and why d’Almbert couldn’t see…is hard to believe.”

It’s a lecture on history of mathematics. Rudin warns us at the start that mathematical history is understudied, and that mathematicians aren’t exactly qualified for it. I’m glad he had that awareness, because history starts right here, at that moment of disbelief. The historian has the tools and penchant to turn that disbelief into curiosity.

To be fair, Rudin offers an answer…

“Part of the problem was that the concepts were not well-defined.”

…though this is just another way of saying that they d’Almbert didn’t know modern math. But what was he thinking? Why did the objection seem natural to him? Why did it make sense? (There’s something sort of similar between some of the questions the math teacher and the math historian have to deal with.)

The first 100 pages of “The Righteous Mind”: Reactions

It’s interesting so far! Here are some passages that caught my attention:

Emotion is a bit harder to define. Emotions were long thought to be dumb and visceral, but beginning in the 1980s, scientists increasingly recognized that emotions were filled with cognition. Emotions occur in steps, the first of which is to appraise something that just happened based on whether it advanced or hindered your goals…Emotions are not dumb. Damasio’s patients [i.e. sociopaths, -MP] made terrible decisions because they were deprived of emotional input into their decision making. Emotions are a kind of information processing. Contrasting emotion with cognition is therefore as pointless as contrasting rain with weather, or cars with vehicles.

Yes! Love this thought. Though I wonder why we should think of information processing as more global than emotion. Would it be equally possible (if somewhat radical) to say that all cognition is a kind of emotion? Would that change how we thought of thinking? (I’m thinking about this in light of Alan Jacobs’ post, titled “thinking as delight.”)

The main way that we change our minds on moral issues is by interacting with other people. We are terrible at seeking evidence that challenges our own beliefs, but other people do us this favor, just as we are quite good at finding errors in other people’s beliefs. When discussions are hostile the odds of change are slight…But if there is affection, admiration, or a desire to please the other person, then the elephant leans towards that person and rider tries to find the truth in the other person’s arguments.

And what if there is affection, admiration, and a desire to please the other than doesn’t come in the form of an argument? What if it’s just an internalized, idealized picture of this other person’s moral expectations?

Reminds me of the Talmud (Sotah 36b) which says that when Potifar’s wife demanded that Joseph sleep with her, he looks out the window. He has a sort of vision; he sees his father’s image reflected in the window. This vision tells him: “Your brothers’ names will be inscribed on the stones of the ephod, and you will be included among them. Do you wish for your name to be erased?” And, as the story goes, Joseph says no and gets thrown into prison for it.

What if holding some sort of image of a moral paragon in your head is the way to do and feel the right things?

If people can literally see what they want to see — given a bit of ambiguity — is it any wonder that scientific studies often fail to persuade the general public? Scientists are really good at finding flaws in studies that contradict their own views, but it sometimes happens that evidence accumulates across many studies to the point where scientists must change their minds. I’ve seen this happen in my colleagues (and myself) many times, and it’s part of the accountability system of science — you’d look foolish clinging to discredited theories. But for nonscientists, there is no such thing as a study you must believe. It’s always possible to question the methods, find an alternative interpretation of the data, or, if all else fails, question the honesty or ideology of the researchers.

This is immediately recognizable in others, and therefore should be something that I recognize in myself. I’m sure I do this.

If I am getting better, it’s because I’ve expanded my social circles online to include scientists and research-minded people who would hold me accountable. I would feel embarrassed to believe in a discredited theory. And I’ve incorporated this into my identity, though “identity” just seems like the individual-facing consequence of my social connections.

This is sort of a disturbing thought. Am I only open to research because of my social connections? Do I balance this openness with a skepticism of science as applied to teaching because of the educators who would expect that of me? Where does the individual begin, and the social influence end? Never, I suppose.

Anyone who values truth should stop worshiping reason. We all need to take a cold hard look at the evidence and see reasoning for what it is. The French cognitive scientists Hugo Mercier and Dan Sperber recently reviewed the vast literature on motivated reasoning (in social psychology) and on the biases and errors of reasoning (in cognitive psychology). They concluded that most of the bizarre and depressing research findings make perfect sense once you see reasoning as having evolved not to help us find the truth but to help us engage in arguments, persuasion, and manipulation in the context of discussions with other people. As they put it, “skilled arguers…are not after the truth, but after arguments supporting their views.” This explains why the confirmation bias is so powerful, and so ineradicable. How hard could it be to teach students to look on the other side, to look for evidence against their favored view? Yet, in fact, it’s very hard, and nobody has found a way to do it.

So the Humean, emotional, non-rational view of judgement leads to a social perspective on learning. It’s almost never rational argument that leads to someone changing their views. It’s rational argument used in the service of social signalling that helps people change their minds.

Or is this not his view? I have 247 more pages in which to find out!

Jewish Internment Camps in Canada, 1940 -1943

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European refugees who had managed to escape the Nazis and made it to Britain, were rounded up as “enemy aliens” in 1940. Many were interned on the Isle of Man, and 2,300 were sent to Canada, mostly Jews. They were transported on the same boats as German and Italian POWs. They were sent to camps in New Brunswick, Ontario and Quebec provinces where they were mixed in with Canadian fascists and other political prisoners, Nazi POWs, etc.

From wikipedia. From the Vancouver Holocaust Education Centre:

Upon arrival in Canada, the refugees were spread out in makeshift prisoner of war camps in New Brunswick, Quebec and Ontario. While some commandants and guards displayed tolerance – if not sympathy – for their prisoners, others combined anti-German and anti-Jewish attitudes when dealing with them. After a visit to Camp N in Sherbrooke, a military observer noted “strictness arbitrarily applied,…rude and appalling language and indulgence in antisemitic remarks [which] are particularly objectionable.”

Meanwhile, refugees interned in England were quickly gaining release and most were soon engaged in the war effort. The British, admitting their error, informed Canada that the refugees could be returned to freedom in Britain, although made it clear that they preferred that they be released into the safety of Canada. But Canada had resisted pressures in the past to grant admission to Jewish refugees, and officials were determined not to let Jews gain entry through the “back door” of internment.

Those who wished to join the British Pioneer Corps (a non-fighting unit) were soon able to return to Britain. Also released were scientists who had been working on top-secret military intelligence technology, and a few others needed for war-related work. The rest languished behind barbed wire in Canadian camps; some would stay there for as long as three years. They called themselves the “camp boys.”

The Vancouver Holocaust Education Centre makes it seem like the camps were decent enough places — decent enough, considering that Jewish refugees were living side-by-side with Nazi POWs. Internees had access to art studios, lectures, a kind of “university.” Much more here.

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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.”

Gutierrez on the tensions of teaching

Although teachers must recognize they are teaching more than just mathematics, they also have to reconcile that fact with the idea that, ultimately,  they are responsible for helping students learn mathematics. Teachers who are committed to equity cannot concern themselves with their students’ self-esteem and negotiated identities to the exclusion of the mathematics that the students will be held responsible for in later years. Yet preparation for the next level of mathematics must also not be the overriding feature of a teacher’s practice. In answer to which of the two foci are important (teaching students or teaching mathematics), I would answer “neither and both.” It is in embracing the tension…”

Rochelle Gutierrez, Embracing the Inherent Tensions in Teaching Mathematics from an Equity Stance

What does it mean to live in tension? I suppose it’s an emotional feeling, of not knowing at any given moment which way you should lean. I like the idea that teaching involves tensions that can’t be resolved, though I admit I’m still fuzzy on what it means to teach in a way that reflects such a tension.

“1” and Russell’s Paradox

I’m falling in love with Classic Set Theory, an introductory set theory text. The introduction to Russell’s paradox motivates it in a way that I’d never seen before.

When I’ve thought about Russell’s paradox in the past I’ve sometimes thought, ok, this makes sense, but how did he come up with this? Or, even, what line of thought could even have led to conceiving of something so clever?

(Russell’s paradox is one of the great stories of math: a letter to Frege, just as he’s completing his master work, announcing a fatal contradiction in his system, effectively ending Frege’s life’s project. Wow! The above panel is from Logicomix, a book that I left feeling disappointed by. I felt it didn’t live up how awesome the story is, but maybe my expectations were just too high.)

Anyway, here’s what I think I currently understand:

Frege figured out a way to define “1” without talking about oneness. Instead, he made it all depend on uniqueness.

Definition: A set C is a singleton if there is an x \in C and if anything else is in C, that thing is equal to x.

Frege doesn’t say anything about the number “1” in defining singletons; it all depends on equality. So Frege is free to use the notion of singleton-ness to define the number “1”.

Definition: There is a set of all the sets that are singletons. Call that set [1].

It’s a pretty nifty trick, and it shows that lurking within our notion of equality is some notion of oneness.

So far, that’s Frege: “1” is a set, [1].

But if [1] is a set, we can start playing around with that set. For example, can make a new set that contains just one element:

\{ [1] \} is a singleton!

Which leads to the weird notion that { [1] } is a member of [1]. And, of course, [1] is in ${ [1] }$ itself, so we get a weird series of inclusions:

[1] \in \{ [1] \} \in [1]

Not paradoxical yet, but weird. And close enough to Russell’s paradox that you can see where he might have started playing around with these ideas. OK, so there’s a weird way that [1] as defined by Frege is a member of itself. How could we mess around with that idea some more?

Definition: V is the set of all sets.

And then clearly, V is also a member of itself.

And now we’re so, so, so close to Russell’s paradox itself: the set of all sets that doesn’t include itself.

Part of what I find so exciting about all of this is the connection to Frege, but also the sense that there’s a natural progression from the definition of “1” to the paradox. What it is is that the definition of “1” does open the door to weird inclusions, since it’s a set of everything that is a singleton. And once that door is open, everything can march on through.

The only way to close that door is to restrict what is allowed to become a set.

In response to Heidi’s post

https://tooteeter.blogspot.com/2017/12/8-12-by-12-inch-paper-cardinal-vs.html

How did you get into my 3rd Grade classroom?? I was dealing with all of this just a few weeks ago.

My situation was slightly different, and because it informs my thoughts about your questions, I”ll share it here.

I decided to move to measurement out of frustration with my students’ ability to subtract. In particular, I was asking them to find the distance between two numbers on the number line, and it felt like we weren’t going anywhere.

Based on previous years memories of students who start measuring at the 1′ mark, I began class with a picture of a ruler and a pencil on the board. I placed the pencil so one end was at the 1′ mark, the other at the 6′ mark. How long is this pencil, I asked the class? The first view was, as you might expect, 6′, and I don’t remember if another kid or I was the one to disagree, but I drew the class’ attention to the alternate view, which was based on two things:

a) Counting the spaces, not the lines
b) Our ability to slide the pencil to anywhere that we want on the ruler (i.e. it wouldn’t make sense for the length to change)

Then I gave out rulers and asked them to measure that piece of paper, that same Investigations activity.

OK, and one last thing: when kids end up with different measurements, one thing I’ve learned to do is to turn it into a statistics problem. At the end of their measurements I tallied their measurements on the board in a list. I point out that variation in measurement is totally normal — there’s no way to make a perfect measurement, there is always error. So, I asked, looking at our measurements, what do we figure the TRUE measurements of the paper to be?

What does all this mean? My first guess is that a lot of number line strategies don’t make a ton of sense to kids yet, but that measuring helps. When we want kids to realize that 20 + 30 = 15 + 25 because you can slide both down the number line? That seems like a strategy best built on something like experiences with a ruler.

(It also reminds me of this research paper that we talked about, where teachers who focused more on measurement seemed to help struggling students more than those teacher who dug in on arithmetic.)

Second, an alternative to asking kids to remeasure until they get the correct measurement might be to record their various measurements and then to ask the statistical question about them: what do we think the true measurement is, given the spread of measurements we’ve been making? can we remeasure to get closer to that true measurement? Because, unlike addition or subtraction, there isn’t a way to measure the true length — every measurement is an approximation.

All this said, I’m not really sure why it makes sense to kids to start at 1 instead of 0. I like your idea that moving between tiles and lengths might get at the true relationship, but I find myself still puzzled about how kids see the ruler.