“A liquid moon”

Winter Trees William Carlos Williams
All the complicated details
of the attiring and
the disattiring are completed!
A liquid moon
moves gently among
the long branches.
Thus having prepared their buds
against a sure winter
the wise trees
stand sleeping in the cold.                                                                           

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?

Chimps are better than humans at some working memory tasks

Amazing. From the research report:

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.

Evolution of working memory

Does this connect to dual process theories? Dual process theories of higher cognition

Mathematical and Scientific Crises

[This post is wildly speculative. File under ‘playing with ideas.’]

On the continuing and fascinating saga of the ABC conjecture and Mochizuki’s befuddling proof:

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.

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.


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.


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.

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


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.





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.”