my name is Dross,
and wen i see
the shiyning text
leap out at me,
i look at wot
it tels my hed -
i read the rules.
i like the red.
my name is Dross,
and wen i see
the shiyning text
leap out at me,
i look at wot
it tels my hed -
i read the rules.
i like the red.
One confusion I wrote down in advance was “I still don’t quite know how to predict that there will not be a simple mathematical apparatus that explains something. Why the motion of the planets, why the game of chance, why not the color of houses in England or the number of hairs on a man’s head?”
I think the main thing I’d look for is an unusual amount of regularity. This comes in two types:
Natural regularity: unusual ‘spherical cow’ type situations like the movement of the planets. Things that are somehow isolated, or where some particular effect strongly dominates, so that only a few variables are needed
Artificial regularity: a lot of the regularity we see around us is there because people engineered it. Dice and coins are good examples. Can’t remember details but I think there’s some interesting stuff on the history of dice, e.g. this link says that ‘Only in the middle of the 15th century did it become standard to use symmetric cubes’. I think it would be hard to invent probability theory when gambling with irregularly shaped lumps.
There doesn’t seem to be any particularly obvious regularity to house colours or number of hairs, they just look like your standard-issue messy situations that don’t tell you much.
Strangely, it can sometimes also go the other way!
One of my most eye-opening teaching experiences occurred when I was helping a six-year-old who was struggling with basic addition – or so it appeared. She was trying to work through a book that helped her to the concept of addition via various examples such as “If Nellie has three apples and is then given two more, how many apples does she have?” The poor little girl didn’t have a clue.
However, after spending a short time with her I discovered that she could do 3+2 with no problem whatsoever. In fact, she had no trouble with addition. She just couldn’t get her head around all these wretched apples, cakes, monkeys etc that were being used to “explain” the concept of addition to her. She needed to work through the book almost “backwards” – I had to help her understand that adding up apples was just an example of an abstract addition she could do perfectly well! Her problem was that all the books for six-year-olds went the other way round.
I think this is unusual though.
I haven’t thought about the bat and ball question specifically very much since writing this post, but I did get a lot of interesting comments and suggestions that have sort of been rolling around my head in background mode ever since. Here’s a few I wanted to highlight:
Is the bat and ball question really different to the others? First off, it was interesting to see how much agreement there was with my intuition that the bat and ball question was interestingly different to the other two questions in the CRT. Reading through the comments I count four other people who explicitly agree with this (1, 2, 3, 4) and three who either explicitly disagree or point out that they find the widget problem hardest (5, 6, 7). I’d be intrigued to know if other people also disagree that the bat and ball feels different to them.
Concrete vs abstract quantities. Out of the people who agreed with that the bat and ball is different, this comment from @awbery does a particularly good job of giving a potential explanation for why:
The problem is a ‘two things’ problem. The first sentence presents two things, a bat and a ball. The language correctly reflects there are two things we should consider. The first sentence is ‘this plus that equals $1.10’. It correctly sounds like a + b; two things. The first sentence presents the state of affairs, not the problem itself. The second sentence presents the problem. The language of the second sentence reinforces the two things idea because there’s still the bat and the ball and they’re compared against each other: ‘there’s this one and it’s more than that one’. The trickiness is that it is a two things problem, but the two things we need to consider are not the most object level single units, but the bat, and the bat-plus-ball. Our brains are pulled toward the object level division of things by the language and the visual nature of the problem. We have to think really hard to understand that the abstract construct of the problem is the same shape as the state of affairs – there are two things to consider in relation to each other – but while the bat and the ball are still involved, they’re reconfigured by a non-intuitive/non-object-like division.
There’s no object level mirror trick in the other two problems, they’re straight forward maths mapping an object level visual representation. The widget problem presents a process which doesn’t change how the machines and widgets relate to each other in its solution. Our brains don’t have to mash up the pond and the lilies to separate the visual presentation to an abstract level. We can see that the pond is the same pond, half covered with lilies then fully covered with lilies at the next step. We don’t suddenly have some new abstract unreal configuration of lilies and pond to contend with.
I think this is why Kyzentun and Ander’s methods help get at the bat and ball problem intuitively – because they bypass the conflict between object level and abstract and translate it into the formal algebra realm. The problem as presented is non-intuitive because the objects visualization it suggests doesn’t reflect the shape of the formal solution.
So I think this is a particular type of problem, one in which visual shape and language of the presentation collude to obfuscate the visualization of the solution at an abstract/formal level. It’s a different type of problem to the other two in this sense, because the objects they present can be used as given in the solution.
Closeness to correct answer. Another interesting possibility is in TheManxLoiner’s comment—that the bat and ball problem is difficult because the incorrect answer is ‘close to the real one’, whereas for the other two problems the incorrect answer is ‘wildly off’. I’ve written a comment in response but I need to think about this more.
Ethnomethodology. David Chapman pointed out that these introspective accounts of what people are thinking when they solve maths problems are very unreliable, and that I’d probably be better concentrating strictly on what people do, as in ethnomethodology:
Yes, the fundamental principle of ethnomethodological methodology is “look at what people say and do, and don’t ever speculate about what’s happening in their head, because we can’t know.” At first that seems like a straitjacket, and highly unintuitive; but it forces you to really look, and then you see what is going on.
This sounds promising. I’m only just getting round to reading some ethnomethodology, and I haven’t got my bearings yet.
Cognitive decoupling. There’s a link with cognitive decoupling (in Stanovich’s original sense) that could be worth exploring further. Success in the bat and ball problem seems to involve decoupling from the noisy wrong answer. David Chapman recommended Formal Languages in Logic by Dutilh Novaes for more background on this. So far I’ve read maybe a third of it. I’ve also written a bit more about cognitive decoupling and the history of the term here.
Next steps. I’m not sure where I’m going to take this next. Probably nowhere much for a while, as I have other priorities. But some options are:
Anders came up with a load of similar problems in the comments. These are designed to be cognitively unpleasant in the same way as the bat and ball, so I keep putting them off. I should actually go through them!
I’m going to continue reading Dutilh Novaes and some ethnomethodology.
Connect more specifically to Stanovich’s idea of cognitive decoupling.
Testing theories? Further out, it could be interesting to actually test some theories by trying alternative, disguised versions of the question, on Mechanical Turk or something. Right now I’ve barely considered this, because I haven’t thought through what I’d want carefully enough yet, but it might be interesting to test variations in:
how concrete the things the quantities refer to are (e.g. really concrete like ‘the price of the bat’, or more abstract like ‘the difference between the price of the bat and ball’. Some of Anders’ variant questions might fit the bill
how close in magnitude the intuitive-but-wrong answer is, as in TheManxLoiner’s comment
I’m very ignorant about experiment design, so to do this I’d to get help from someone more knowledgeable. And psych research sounds like a gigantic minefield even if you are knowledgeable, so I’d probably end up wasting my time. But probably I’d learn something from going through the process, and it’s something that could maybe happen in the future.
Update: I’ve done four of these now and have really enjoyed it. It works brilliantly for motivating me to keep a record of what I’m doing, and I’ve had some great followup conversations too. Thanks very much for introducing me to the idea!
Ah, that probably needs clarifying… I was using ‘analysis’ in the sense of ‘opposed to synthesis’ as one of the dichotomies, rather than the mathematical sense of ‘analysis’. I.e. ‘breaking into parts’ as opposed to ‘building up’. That’s pretty confusing when one of the other dichotomies is algebra/geometry!
I agree that algebra and (mathematical) analysis are pretty different and I wouldn’t particularly lump them together. I’d personally probably lump it with geometry over algebra if I had to pick, but that’s likely to be a feature of how I learn and really it’s pretty different to either.
Cool, I like these sorts of lists! Here’s mine:
(Mostly) giving up caffeine. 7 points, ~5 years. Much easier to get up in the morning. I have a single cup of tea maybe once or twice a month if I feel like I need waking up more, and that’s enough to do the job now. Best used in combination with another elite lifehack, highly recommended if you can manage it:
Getting enough sleep. 7 points, ~5 years.
Pomodoros. 8 points, ~9 months. Really excellent and not sure why I resisted the idea so long. Turns out lots of half hour blocks really add up, and it’s significantly changed how I work. This is a relatively recent thing so probably still overexcited about it.
Keeping my desk clear of paper. 6 points, ~2 years. I used to be awful at having stuff piled up everywhere, which would put me off working at home and convince me that I had to go to a library or something. This works by having box files so that the paper never ends up there in the first place.
Lot of calendar reminder email alerts. 4 points, ~3 years. Not exactly life-changing but I have fewer birthday present buying panics.
Todoist. 3 points, ~9 months. It has Gmail integration so I do check it, and it sort of works, but gets clogged with stale stuff too easily. I generally find todo lists hard though so this is good by my standards.
Beeminder. 5 points (but hard to attach a single number to), used for ~4 months 2 years ago and then stopped. Extremely effective way to simulate the stress of having a lot of external deadlines. It worked brilliantly on a time-sensitive project I had, but too stress-inducing for me to want to use permanently. It did do an excellent job of reminding me what being a productive person felt like, and I’d use it again if I really needed to, but mostly it just made me realise I needed to get my internal motivation working better.
Leechblock type browser extensions. 4 points, used them on and off for ~4 years up to about two years ago. I think the one I liked most was called Crackbook, which added a delay to the page load time instead of outright blocking it. They tend to work OK until they don’t. There’s no particular reason I stopped using them, except that the problem doesn’t seem so urgent now I have a normal full time job and value my free time a bit more.
Thanks! I have been meaning to add a ‘start here’ page for a while, so that’s good to have the extra push :) Seems particularly worthwhile in my case because a) there’s no one clear theme and b) I’ve been trying a lot of low-quality experimental posts this year bc pandemic trashed motivation, so recent posts are not really reflective of my normal output.
For now some of my better posts in the last couple of years might be Cognitive decoupling and banana phones (tracing back the original precursor of Stanovich’s idea), The middle distance (a writeup of a useful and somewhat obscure idea from Brian Cantwell Smith’s On the Origin of Objects), and the negative probability post and its followup.
This is the most compelling argument I’ve been able to think of too when I’ve tried before. Feynman has a nice analogue of it within physics in The Character of Physical Law:
… it would have been no use if Newton had simply said, ‘I now understand the planets’, and for later men to try to compare it with the earth’s pull on the moon, and for later men to say ‘Maybe what holds the galaxies together is gravitation’. We must try that. You could say ‘When you get to the size of the galaxies, since you know nothing about it, anything can happen’. I know, but there is no science in accepting this type of limitation.
I don’t think it goes through well in this case, for the reasons ricraz outlines in their reply. Group B already has plenty of energy to move forward, from taking our current qualitative understanding and trying to build more compelling explanatory models and find new experimental tests. It’s Group A that seems rather mired in equations that don’t easily connect.
Edit: I see I wrote about something similar before, in a rather rambling way.
Thanks for writing this, it’s a very concise summary of the parts of LW I’ve never been able to make sense of, and I’d love to have a better understanding of what makes the ideas in your bullet-pointed list appealing to those who tend towards ‘rationality realism’. (It’s sort of a background assumption in most LW stuff, so it’s hard to find places where it’s explicitly justified.)
Also:
What CFAR calls “purple”.
Is there any online reference explaining this?
I’m not a massive fan of the ‘postrationality’ label but I do like some of the content, so I thought I’d try and explain why I’m attracted to it. I hope this comment is not too long. I’m not deeply involved but I have spent a lot of time recently reading my way through David Chapman’s Meaningness site and commenting there a bit (as ‘lk’).
One of my minor obsessions is thinking and reading about the role of intuition in maths. (Probably the best example of what I’m thinking of is Thurston’s wonderful Proof and Progress in Mathematics.) As Thurston’s essay describes, mathematicians make progress using a range of human faculties including not just logical deduction but also spatial and geometric intuition, language, metaphors and associations, and processes occurring in time. Chapman is good on this, whereas a lot of the original Less Wrong content seems to have rather a narrow focus on logic and probabilistic inference. (I think this is less true now.)
Mathematical intuition is how I normally approach this subject, but I think this is generally applicable to how we reason about all kinds of topics and come to useful conclusions. There should be a really wide variety of literature to raid for insights here. I’d expect useful contributions from fields such as phenomenology and meditation practice (and some of the ‘instrumental rationality’ folk wisdom) where there’s a focus on introspection of private mental phenomena, and also looking at the same thing from the outside and trying to study how people in a specific field think about problems (apparently this is called ‘ethnomethodology’.) There’s probably also a fair bit to extract more widely from continental philosophy and pomo literature, which I know little about (I’m aware there’s also lots of rubbish).
There’s another side to the postrationality thing that seems to involve a strong interest in various ‘social technologies’ and ritual practices, which often shades into what I’ll kind-of-uncharitably call LARPing various religious/traditional beliefs. I think the idea is that you have to be involved pretty deeply in some version of Buddhism/Catholicism/paganism/whatever to gain any kind of visceral understanding of what’s useful there. From the outside, though, it still looks like a lot of rather uncritical acceptance of the usual sort of traditional rubbish humans believe, and getting involved with one particular type of this seems kind of arbitrary to me. (I exclude Chapman from this criticism, he is very forthright about what he think is bad/useless in Buddhism and what he thinks is worth preserving.) It’s probably obvious at this point that I don’t at all understand the appeal of this myself, though I’m open to learning more about it.
Not quite knitting, but close—you may like this piece by Sarah Perry explaining a spinning metaphor of Wittgenstein’s:
And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres.
Oops, I fixed that in my blog version and then accidentally posted the old draft here. Edited now, thank you!
I also identify more with the elephant, which I (probably unhelpfully) think of as the one that ‘actually does maths and physics’, in the sense of gaining insights into problems and building intuitive understanding.
I (also probably unhelpfully) think of the rider as a more of a sort of dull bean counter who verifies the steps in my reasoning are correct afterwards, and ruins my fun for some of my wilder flights of fancy.
I’m slowly learning to like the rider more—it’s doing more than I give it credit for.
Probably some of the issue is trying to fit everything into these two categories. I think Sarah Constantin has convinced me that there are at least three things in the world—flow states, formal step-by-step reasoning and insight. I’ve been unthinkingly lumping flow state in with insight as the good stuff, and leaving the rider with just formal verification. Someone else might lump insight differently.
I like this, but I don’t think mimesis is always a bad thing, at all. It’s often a useful stage on the route to deeper understanding. You see this in teaching sometimes: you’re trying to teach the cross product, but they’re learning that they need to underline their vectors, and that they should put some punctuation and explanatory words between their equations so another person can follow the argument. Eventually they will definitely need to learn both sets of things, but if you just get back vector salad with explanatory words interpolated between it they’ve still learned something that will be useful in their mathematical career.
I’ve never been a teaching assistant for a proofs course, but I imagine you have to mark a lot of epsilon salad, because I’m pretty sure I produced a lot of epsilon salad as a student in the course of internalising the language.
These days as a relatively noob programmer it’s normally me doing the babbling, and I’ve used this strategy consciously: offer up some network protocol salad or version control salad and gauge from my boss’s face how much it sounds like the real thing. I find that when I’ve filled in an outline like that and know roughly how to talk a language, it’s much easier to fill in the detailed steps.
Happened to look this post up again this morning and apparently it’s review season, so here goes...
This post inspired me to play around with some very basic visualisation exercises last year. I didn’t spend that long on it, but I think of myself as having a very weak visual imagination and this pushed me in the direction of thinking that I could improve this a good deal if I put the work in. It was also fascinating to surface some old visual memories.
I’d be intrigued to know if you’ve kept using these techniques since writing the post.
No, I also definitely wouldn’t lump mathematical analysis in with algebra… I’ve edited the post now as that was confusing, also see this reply.
Your ‘how much we know about the objects’ distinction is a good one and I’ll think about it.
Also vim over emacs for me, though I’m not actually great at either. I’ve never used Lisp or Haskell so can’t say. Objects aren’t distasteful for me in themselves, and I find Javascript-style prototypal inheritance fits my head well (it’s concrete-to-abstract, ‘examples first’), but I find Java-style object-oriented programming annoying to get my head around.
I like this and agree that this thing deserves its own name. In my own head (you may not agree) this view often also includes ideas like ‘explicit formal metrics often get Goodhart-ed into useless cargo cults, top-down rational plans often erase illegible local wisdom’, etc. The kind of cluster people seem to get from Seeing Like A State, The Great Transformation, etc. (I’ve never read either of those myself though.)
To my mind this cluster is something like ‘pomo ideas grafted on analytic rootstock’, rather than the normal continental rootstock. And I think the main influence it misses because of this is phenomenology (gworley I think may be pointing somewhere similar). Thinking seriously about subjective internal experience often pulls people towards a more thoroughgoing rejection of modernism than the ‘skeptical modernism’ one.
I don’t understand any of this well myself, though, and I’d struggle to unpack any of this into a compelling argument for someone who didn’t basically already agree with me.
I’m enjoying this whole series, but this one’s extra relevant because I’m doing something similar right now in a less structured way, so I can pick up some ideas.
I’m working on being able to notice, when talking to people, that I can expand my awareness out to include my surroundings. (The idea is that this gives me space to notice what I feel and have curiosity about what they feel, rather than being locked into ‘polite conversation bot’ mode, but right now just noticing is the important bit.)
I like the noticing timeline, it fits very well with what I’m experiencing. I’m sort of between stages 2 and 3 depending on the situation, and normally either remember just after or notice some time during.
I think I’m going to experiment with an explicit marking action and a tally, and probably toy situations as well. Thanks for all the ideas!
This is only tangentially relevant, but adding it here as some of you might find it interesting:
Venkatesh Rao has an excellent Twitter thread on why most independent research only reaches this kind of initial exploratory level (he tried it for a bit before moving to consulting). It’s pretty pessimistic, but there is a somewhat more optimistic follow-up thread on potential new funding models. Key point is that the later stages are just really effortful and time-consuming, in a way that keeps out a lot of people trying to do this as a side project alongside a separate main job (which I think is the case for a lot of LW contributors?)
Quote from that thread:
Also just wanted to say good luck! I’m a relative outsider here with pretty different interests to LW core topics but I do appreciate people trying to do serious work outside academia, have been trying to do this myself, and have thought a fair bit about what’s currently missing (I wrote that in a kind of jokey style but I’m serious about the topic).