December 2018: Bell’s theorem and end-of-year review

Anybody who’s not bothered by Bell’s theorem has to have rocks in his head.

‘A distinguished Princeton physicist’, as told to David Mermin

December has been a bit of an odd month for working – I’ve had a couple of really crappy migraines and lots of pre-Christmas errands to run, and I feel like I’ve got nothing done, but actually I’ve done quite a bit of writing. I finally finished the bat and ball post, and also cross posted to Less Wrong – there are good comments in both places. I also finished a pop book on insight and have started writing a review. And I churned out a fair bit of end-of-year navel-gazing for the newsletter.

I really have got nowhere with physics, though. Though for some reason I wrote this whole thing below on Bell’s theorem anyway.

Went for a walk this morning and got shat on by a pigeon, so 2019 is already going brilliantly for me. Happy new year!

Bell’s theorem

I mentioned last month that I’d got a bit stuck with the van Enk/negative probability stuff, and wanted to think about something different for a while. So I thought I might fill a gap in my knowledge that’s been annoying me for a while. That was the plan, anyway – but as I said above I’ve been crap at doing any physics.

If you learn Bell’s theorem in a physics course, you’re likely to learn a slightly extended version, the Bell-CHSH inequality, with a derivation that looks something like this. Lots of integrals and lambdas and so on. This is more general and usable, but I don’t find it massively enlightening – you don’t come out with a strong gut feeling of NOTHING IN THE WORLD MAKES ANY SENSE ANY MORE, which I would argue is the main thing you want from a derivation of Bell’s theorem.

If you learn Bell’s theorem from a pop science book, you normally get a pared-down version that’s been optimised for the NOTHING IN THE WORLD MAKES ANY SENSE ANY MORE feeling. I think most of these originally derive from Mermin’s excellent explanation in his paper Quantum Mysteries for Anyone. Mermin makes the point that you don’t need to know any quantum physics to see what’s weird about Bell’s theorem. In fact, you don’t need to know any classical physics either:

The impact of this discovery on philosophy may have been blunted by the way in which it is conventionally stated, which leaves it fully accessible only to those with a working knowledge of quantum mechanics. I hope to remove that barrier by describing this remarkable aspect of nature in a way that presupposes no background whatever in the quantum theory or, for that matter, in classical physics either. I shall describe a piece of machinery that presents without any distortion one of the most strikingly peculiar features of the atomic world. No formal training in physics or mathematics is needed to grasp and ponder the extraordinary behavior of the device; it is only necessary to follow a simple counting argument on the level of a newspaper braintwister.

By now I’ve plugged my way through both the physics-course version and the pop-science version, but I was still finding the connection between them somewhat opaque. So my plan was to trace that link. I haven’t got that far, but I did write up Mermin’s version below. This is mostly for my own benefit: I can’t really improve on his one, which is a classic of research distillation, but writing out the argument for myself makes it clearer in my mind. Also, maybe you’ll read my version even if you won’t click on the PDF link.

There’s a 1929 New Yorker cartoon which depicts ordinary people in the street walking around dumbstruck by Einstein’s theory of general relativity. This is a comic idea because the theory was famously abstruse (particularly back then when good secondary explanations were thin on the ground). But Bell’s theorem is accessible to anyone with a very basic knowledge of maths, and is weirder than anything in relativity. I genuinely think that everyone should be walking down the street clutching their heads in shock at Bell’s theorem, and I’ll be happy if I can get one or two more people into this state of confusion.

Of course, you might have rocks in your head, in which case nothing will help you.

Mermin’s machines

Mermin invites you to consider the following set of three machines. I’ve copied the image from the paper:

The machine in the middle is the source. It fires out some kind of particle – photons, electrons, frozen peas, whatever. We don’t really care how it works, we’ll just be looking at why the results are weird.

The two machines on the right and left side are detectors. Each detector has a dial with three settings, labelled 1, 2, and 3, and a red and a green light on the side. When it receives a particle it flashes either red or green. That’s the whole of the setup.

It’s vital to this experiment that the two detectors cannot communicate at all. If they can, there’s nothing weird about the results. So assume that a lot of work has gone into making absolutely sure that the detectors are definitely not sharing information in any way at all.

Now the experiment just consists of firing out pairs of particles, one to each detector, with the dials set to different values, and recording whether the lights flash red or green. So you get a big list of results of the form:

12 GG

13 RG

22 RR

where e.g. ‘12 GG’ indicates that the left hand dial was set to 1, the right hand dial was set to 2, and both detectors flashed green.

The second important point, other than the detectors not being able to communicate, is that you have a free choice of setting the dials. You can set them both beforehand, or when the particles are both ‘in flight’, or even set the right hand dial after the left hand detector has already received its particle. It doesn’t matter.

Now you do like a million billion runs of this experiment, enough to convince you that the results are not some weird statistical fluctuation, and analyse the results. You discover two things:

Result 1: If both dials are set to the same number, both lights flash the same colour. So you might get 11 GG, 11 RR, 22 GG etc, but never 11 RG or 33 RG.

This is pretty easy to explain. The detectors can’t communicate, so if they do the same thing it must be something to do with the properties of the particles they are receiving. We can explain it straightforwardly by postulating that each particle has an internal state with three properties, each of which takes two possible values which we label R or G. State RRR will cause the detector to flash red if the dial is in any of the three positions, state RRG will cause it to flash red in positions 1 and 2 and green in position 3 and so on. Then Result 1 can be explained by saying that the detector always sends the same sort of particle to both detectors. For example, if both dials are set to 2 and the particle is RGR, both lights will flash green.  

Mermin says:

This hypothesis is the obvious way to account for what happens in [Result 1]. I cannot prove that it is the only way, but I challenge the reader, given the lack of connections between the devices, to suggest any other.

Now on to the second result.

Result 2: If both dials are set to different numbers, the lights flash the same colour ¼ of the time, and different colours ¾ of the time.

This looks quite innocuous on first sight. It’s only when you start to consider how it meshes with Result 1 that things get weird.

(This is the part of the explanation that requires some thinking, ‘on the level of a newspaper braintwister’. It’s fairly painless and will be over soon.)

Our explanation for result 1 is that particles in each run of the experiment have an underlying state, and both particles have the same state. Let’s go through the implications of this.

Say for instance the underlying state is RGG. I’ve enumerated the various options for the dials in the table below. For example, if the left dial is 1 and the right dial is 2, we know that the left detector will flash red (RGG) and the right will flash green (RGG), so the two lights are different.

Left dial Right dial Lights
1 2 different
1 3 different
2 1 different
1 3 same
3 1 different
3 2 same

Overall there’s a ⅔ chance of being different and a ⅓ chance of being the same. You can convince yourself that this is also true for all the states with two Rs and a G or vice versa: RRG, RGG, RGR, GRR, GRG, GGR.

That leaves RRR and GGG as the other two options. In those cases the lights will flash the same colour no matter what the dial is set to.

So whatever the underlying state is, the chance of the two lights being the same is greater than ⅓. But this is incompatible with Result 2, which says that the probability is ¼.

(The thinky part is now done.)

So Results 1 and 2 together are completely bizarre. But this is exactly what happens in quantum mechanics! You probably can’t do it with frozen peas, though. The particles should be two spin-half particles in the singlet state, the ‘dials’ should be magnets that can be oriented in three states at 120 degree angles from each other, and the lights on the detectors measure spin along and opposite to the field. The particles are actually anticorrelated when the two dials are the same, so if the left detector is wired up to flash red for spin along the field, the right detector needs to have its light wired the other way so it flashes red for spin opposite. (None of the technical details matter if you just want to understand that the results are weird.)

The Mermin paper talks a little bit about how physicists try to explain this result, but I’m not going to go into that. As I’ve mentioned in previous newsletters, I’m personally fascinated by ‘retrocausal’ ideas, where the underlying states of the two particles (RGR, RGG etc) are determined by their initial state and by your final measurement. That might sound weird… but any explanation of what is going on in that machine is going to be weird. The alternatives all look worse to me!

Two quotes

A quick interlude on something completely different: at some point in the month I mashed two quotes together and noticed something interesting.

Empson, Seven Types of Ambiguity:

As a rule, all that you recognise as in your mind is the one final association of meanings which seems sufficiently rewarding to be the answer—‘now I have understood that’; it is only at intervals that the strangeness of the process can be observed. I remember once clearly seeing a word so as to understand it, and, at the same time, hearing myself imagine that I had read its opposite. In the same way, there is a preliminary stage in reading poetry when the grammar is still being settled, and the words have not all been given their due weight; you have a broad impression of what it is all about, but there are various incidental impressions wandering about in your mind; these may not be part of the final meaning arrived at by the judgment, but tend to be fixed in it as part of its colour.

This sounds weirdly similar to something else I read recently:

… a change to the input of a combinational logic circuit will cause the circuit to enter a logically inconsistent state until it finally manages to drive all of its outputs to their correct values. A small window of inconsistency opens up between the moment an input changes and the moment the last output assumes its correct value. While this window is open, it is important that nobody trust the output values, which are unreliable as long as the circuit is still settling down.

This is Agre’s Computation and Human Experience, talking about the settling period in an electronic circuit.

Everything I read around the same time tends to look like it’s about the same topic, so I expect to notice similarities like this. The interesting bit is the difference between the two cases. In the circuit case things are designed carefully so that the settling phase is not fixed in the final answer ‘as part of its colour’. You just get the one answer. Whereas the poem needs those traces of the alternate pathways to work.

Review of the year

This part is a load of navel-gazing about me me me, sorry. I feel a bit bad about writing this sort of thing, but I actually like reading these when other people post them, so maybe I shouldn’t.

Blog posts

I managed eight posts this year, or nine if you count writing up yet another version of my ‘two types of mathematicians’ post for Less Wrong. Not a huge number, but not so far off my vague target of about one post a month, either. And I was very happy with the variety of what I produced – it was a good year for exploring and seeing what works.

The most interesting part for me this year was that it felt like I put in very little deliberate effort, either in terms of scheduling time to write or planning out what I’d write. But somehow blog posts came out anyway. I think there were a few reasons for this:

  • I’d finally got into the habit of writing after a year of the blog and a couple of years on the previous tumblr, so sitting down to do it didn’t take much in the way of motivation.
  • Something to do with slowly moving to a more fragmented, task-based way of viewing work, in a surprisingly similar way to how Sarah Constantin describes it in this post and particularly this follow-up comment. I started writing this bullet point and it turned into the outline for a whole post, which I’ll have to write up later. But the relevant bit here is that I now tend to see myself as ‘completing [small task]’ rather than ‘writing a blog post’, and it’s made me much better at using scraps of time for something useful. Fifteen minutes used to look like a totally pointless amount of time, and yeah it’s still pointless if you’re planning to load up a complex argument into your head. But it’s plenty of time for small tasks like pasting some quotes into a file, adding references, fixing a shit paragraph or writing some new shit paragraphs. And it turns out that most of everything can be viewed as a series of small tasks.
  • Starting the newsletter meant that I was producing raw material for it pretty consistently, which I could then polish up/rework for blog posts. (The newsletter has worked really well for me in general; more below.)
  • More people were reading the blog, so I got more responses in the comments and on Twitter. Some motivating social feedback definitely makes it more enjoyable to continue. Very different to the old days of shouting into the void on tumblr.

I think the last one might be the most important, because once you’re part of some kind of conversation there’s natural momentum built in, where you want to respond to comments, or read references people have suggested, which then inspires more writing. Some fun things in the ‘motivating social feedback’ category:

  • The cognitive decoupling elite: This got a fair few comments when I posted it, more than I was expecting, including some really thoughtful ones I still need to go back to. But then a couple of weeks later John Nerst attached the decoupling/contextualising idea to a really controversial topic and sent it flying off around the internet, so it got lots more views and responses. I’d nicked the original idea from Sarah Constantin myself, and it was funny to see it go through a couple more iterations after mine.
  • 20 Fundamentals: A bit later in the year I took up one of John Nerst’s ideas in return, and then got a third list from Christian Hendriks in response to mine.
  • A braindump on Derrida and close reading: this was just a long braindump for the newsletter. David Chapman encouraged me to post it to the blog anyway, and it actually got some interest. And now I’ve finally got it written down it looks like there are a few directions I could take those ideas in next year.
  • The Bat and Ball Problem Revisited: This one grew out of a repost of an old throwaway tumblr post, which was short and crappy but did contain a good question – why is the bat and ball question in the Cognitive Reflection Test different to the other two? The discussion with anders and Kyzentun in the comments ended up being way better than the original post, and I used them as material for the new post. Which still didn’t answer the question. But now I have lots more comments to think about.


This works brilliantly and I’m so glad I tried it. I got the idea from moridinamael on LessWrong and was expecting it to at least be a fun way of keeping track of what I do in a month and doing a bit of reviewing. But it turns out to have a surprising number of other benefits. Quoting some bullet points straight from his post, because I found the exact same things:  

  • It’s a cure for the goodism that leads you to have fifteen unfinished drafts of blog posts that never see the light of day. You’ve got a month. Wrap it up.
  • You’re forced to organize your random notes and copy-pastes and unlabeled links into something readable enough for others that it ends up passing as powerful reference material for your future self.
  • Your thinking on a given topic will “advance” in a way that it otherwise wouldn’t. You’re forced to finish and polish the stubs of thought processes that you may have thrown in. (A lot of this finishing and processing is happening subconsciously throughout the month. If you know you’re going to have to share it at the end of the month, your brain will give you something worth sharing. Without that pressure, your thoughts just tend to continue on, going in circles for years without ever resulting in anything useful to even yourself. Or at least, mine do.)

That last one is particularly powerful, and I’ve noticed that my thoughts have advanced on a lot of topics where they were just wheeling around pointlessly before. (Blogging also does this, but the monthly deadline makes the effect stronger.) This is mostly a very good thing, but it does mean that I’m ending up with way too many topics to think about, because writing solidifies ideas that I’d otherwise have just dropped if they’d stayed in my head. I don’t have enough time for all of them, and I need to do some kind of idea triage (see below).

I’m not planning to make any changes to the newsletter at the moment – I had considered doing something else with the physics part, but I’m not sure what so for now it will stay as one incoherent mess.

As always, thanks for reading!


I’m having a bit of a slump at the moment where I’m not getting much done, so it’s not a great time to do a review. I might come back to this in the spring, but here’s a few thoughts.

  • I’m very happy with my idea of sticking with one paper throughout the year, and also with my choice of that paper. It’s rich enough to connect to quite a few interesting ideas (epistemic toy models, QM in phase space, negative probability) but constrained enough to stop me just dabbling superficially in too many things, which is what normally happens.
  • My original aim for the year was the following (from my February newsletter):

My plan for 2018 is to go beyond just learning some physics in my spare time and to do ‘something novel’, interpreted broadly. ‘Novel’ in this case doesn’t have to mean original research (though that would definitely count) – I’m thinking of a wider conception of what counts as a novel contribution, in the style of Chris Olah and Shan Carter’s Research debt essay (I wrote some comments on it here).

A good online explanation of something that doesn’t currently have a good online explanation would definitely count by my standards. Just following a lecture course or doing some exercises from a textbook wouldn’t.

This didn’t really happen. I suppose some of the stuff I wrote here about the Wigner function would just about count as a ‘good online explanation’, but it’s less than what I was aiming for. I kind of dropped the thread towards the end of the year, which didn’t help. I think there is some more worthwhile stuff to write up, and it’s worth just carrying on for now with the same aim – I’m not going to set a new one for 2019 at the moment.

  • I haven’t got to the ‘self-sustaining’ point that I have with the blog, where work just happens without feeling very effortful. Partly that’s because physics is harder work than writing rambly bullshit. But I also don’t have the social context I do with the blog, where other people are trying to do similar things, so I’m mostly just plugging along on my own. That was better in 2017, because I went to more workshops connected with with the BRCP, which has like-minded physicists within academia. This year I went to one workshop in Vienna, which was good, but I wasn’t presenting there and the topic was tangential to my main interests so there wasn’t a big incentive to get anything done. I’ll probably be more involved with the BRCP next year, so that might help, but I should also think about what else I can do.
  • I don’t quite know what I’m doing with that website I set up, and haven’t got into a great routine with posting there. I’ve got a couple of half-written bits to push to it some time, but it’s been a slow start.

Hm that all looks a bit negative, and probably more negative than it should be. I was more focussed than I was in 2017, and I picked a good topic, and I’m in a good position to keep thinking about it next year.


I read a few books off of David Chapman’s list:

  • The Reflective Practitioner (I reviewed this one)
  • Agre, Computation and Human Experience
  • Winograd and Flores, Understanding Computers and Cognition

The Agre book is still percolating through my brain and inspiring a few thoughts. Winograd and Flores covered too much too quickly and didn’t quite work for me.

I also dipped into:

  • Merleau-Ponty, The Phenomenology of Perception
  • Dreyfus, Being-In-The-World

but didn’t make any serious effort to read the whole thing in either case.

Other relevant books:

  • The Eureka Factor – this is a pop book on insight that I just finished and am going to review
  • Brooks, The Well Wrought Urn – some New Criticism
  • Empson, Seven Kinds of Ambiguity – more New Criticism (or maybe a precursor to it)
  • Tasic, Mathematics and the Roots of Postmodern Thought – this covers way too much way too quickly but I find the topic interesting enough to get a lot out of it anyway. Will finish making chapter summaries at some point.

And then, of course, I’ve read a massive pile of stuff off the internet, the same as I do every year. I can’t be bothered to go through this at the moment. Except for this talk by Norris on Derrida, which inspired the braindump and has probably been the most useful single thing I read all year. Which is weird, because I don’t know how I found it or even why I cared about Derrida in the first place.

Next year

  • I definitely want to read some ethnomethodology. Suchman’s Plans and Situated Action for sure, and then see what I feel like.
  • Lakatos, Proofs and Refutations. Read a couple of extracts this year and now want to read the rest.
  • Maybe Wittgenstein’s Philosophical Investigations?
  • My stretch goal for next year is to read ten consecutive pages of Derrida in one go without giving up.

Idea triage

I’ve never done anything like this before, and I’m very sceptical of my ability to actually change my focus – generally it feels like I’m interested in whatever I’m interested in, and there’s not much I can do about it. But I’ve picked up too many threads this year, and some of them are more promising than others, and it would be good if I could prioritise.

Anyway the threads I can remember now are:

  • Negative probability, the van Enk toy model, quantum mechanics in phase space, etc.
  • Something to do with research distillation – explaining bits of physics I can’t find good online explanations of. What I’m planning with the website but haven’t actually done much of.
  • I spent a really ridiculous amount of time this year making that website, which has basically nothing on it. I wrote up my experiences in Practical Design for the Idiot Physicist, which was my own favourite post to write, but didn’t get a whole lot of interest and mostly doesn’t lead anywhere in particular…
  • … though I did get some good links on live programming from anders to follow up.
  • The Derrida braindump leads in at least two directions I find promising. The first is this ‘alternative universe where postmodernism hit the maths department’ idea that’s been rolling round my head. At the least I’ll finish making notes on the Tasic book.
  • The second thing is the bit on how human music has an extra ‘engineered’ quality that’s missing from birdsong (defined intervals between notes, lengths of notes, etc) and how people go about engineering their surroundings to have these kind of discrete, repeatable components. This is why I need to read some ethnomethodology. Also going further into speculative territory on ‘algebra’ and ‘geometry’ used in a weird way as correlates of Derrida’s weird usage of ‘writing’ and ‘speech’. ‘Algebra’/’writing’ as the engineered side, ‘geometry’/’speech’ as the pretheoretic background.
  • Decoupling. I think there is something interesting there if I keep digging. The insight book I’ve been reading is also weirdly relevant.
  • The inevitable ‘two types of mathematician’ question.
  • That stupid bat and ball, and related questions like psychology survey methodology.
  • ‘The Cluster Structure of Thingspace’. I spent a lot of time at the beginning of the year arguing in my head with this Yudkowsky post and now I have a long half-written post that didn’t quite work and I don’t know what to do with it.
  • Not been much of this this year, but rambling about academia, what’s wrong with physics, etc. Also the ‘crackpot time’ posts (didn’t do one last year and should really do an update).

Whoa that’s a lot of threads. I notice that some of these topics are things that are ‘in the air’: live programming, research distillation, Academia Is Broken, survey methodology. I find them interesting but don’t have any specific expertise (well maybe a little bit on Academia Is Broken), and somebody else will write about them if I don’t. I’m not going to stop myself thinking about them, but they don’t seem so high priority compared to the weird crap that nobody else is going to do. That would be:

  • Negative probability and the van Enk model
  • Derrida thread 1: ‘postmodernist maths department’
  • Derrida thread 2: ‘algebra’/’writing’ vs ‘geometry’/’speech’
  • Decoupling

Of course, there’s probably a good reason why nobody is thinking about those things… they are probably terrible ideas! But I’m only going to find out by trying, and I feel like the whole point of independent work outside of academia is to think about high-variance weird stuff that nobody gets paid to think about. (Also it’s the most fun.)

The bat and ball question is maybe also in this category, but… I’m sick of it at the moment. So I think those four bullet points above are the winners of this triage thing.

Next month

I’ll write up the insight book review. Also hopefully I’ll actually do some physics.

Cheers, and hope you have a good 2019, ideally with less pigeon shit than my one,