Hi again! Or hi for the first time to new Bristol people 🙂 If you don’t know what you signed up for, this is a sort of experiment to get me into the habit of documenting what I do each month, inspired by this Less Wrong post. Topics vary a bit but it’s normally either physics, something vaguely connected with mathematical intuition, or my attempts at reading bits of continental or postmodern philosophy.
The newsletter’s worked a lot better than I expected it to, and I’ve now been doing this long enough that it’s worth putting the year in the title… when I started it felt too optimistic to specify that it was February 2018.
I’ve been sort of unfocussed again this month. Work is taking up more mental energy than normal, and I’ve had difficulty concentrating on any one thing for long. I still haven’t written up the insight book review, which was my main plan for the blog last month. Wrote chapter summaries and a rough outline, but didn’t get around to filling it in. Hopefully this month, then. (These posts where I’m constrained by, like, actual facts seem to take me ages – the bat and ball one was also really slow. Looking forward to getting back to uninformed bullshit speculation again soon.)
I’m only going to cover a couple of topics this time. One is more chapter summaries from Mathematics and the Roots of Postmodern Thought, an odd book I started going through back in November. Not sure how generally interesting this is, but it’s useful for me. The book is ridiculously dense, and I’m only going to get anything out of it if I think through it in some detail. Writing the summaries forces me to actually do that.
Before that, though, I’m going to talk about the physics society I’m in, the Basic Research Community for Physics. Most of this isn’t specific to this month, but it’s on my mind because I agreed to go on the board of it this year. (This is not a very fancy position and basically means it’s your turn to deal with the email inbox and the bureaucracy of a German Verein. Everyone does it for a year and then finds some new people to fob it off to.)
I first got involved in this through a series of workshops some of the members run, Rethinking Foundations of Physics. I went to the one in 2017 and it was fantastic. I’d left academia a few years before, but they were still happy to have me, and I immediately felt at home with the style of the event. Actually it was almost surreal, like finding a fan club for something bizarrely specific that you’d never quite realised it was possible to be a fan of, and discovering you somehow know all the fan material anyway.
The workshops – Rethinking and various spin-off events – are the main focus of the society. We also mainly get new members through workshops. The organisers have had to walk quite a fine line: if they’re too lax about entry standards or plug the thing too widely, the society will attract every crackpot on the internet. On the other, there’s not much point in creating an institution that exactly reproduces normal academia, and talks about exactly the same topics that everyone else talks about. The workshop method has worked pretty well: everybody who joins already has a reasonable understanding of the culture of the society, because they’ve seen how an event works in practice. It does mean that the society can’t grow very fast, but that’s probably how it has to work to get people who understand what the point is.
In the rest of this section I’m going to describe one of the things I like about it. (It was going to be two things, but I’ve run out of time. Maybe I’ll do the other next month.) Of course, this is just what I get out of it, and other people in the society would probably say something very different.
There’s a style of writing and talking about physics that I really like. It might be easiest to start explaining what it is by throwing out some examples:
- The Mermin paper on Bell’s theorem that I talked about last time.
- These notes on solving classical physics problems with Feynman diagrams. This was probably the oddest bizarrely-specific-fan-club moment of the initial workshop. It’s just some obscure pdf sitting on someone’s website, but on the first evening I talked to someone else who had also read it, and had the same question as me about exactly where the difference between the classical and quantum version is. The diagrams in quantum field theory have loops, and the classical ones don’t: how do the loops get in? (Still haven’t got round to fully figuring that out.)
- Norton’s dome: a Newtonian system that violates determinism. ‘It is a mass that remains at rest in a physical environment that is completely unchanging for an arbitrary amount of time–a day, a month, an eon. Then, without any external intervention or any change in the physical environment, the mass spontaneously moves off in an arbitrary direction, with the theory supplying no probabilities for the time or direction of the motion.’
- Practically everything that John Baez ever writes. One good example would be these notes on the inverse square force law, and weird connections to motion on the surface of a four-dimensional sphere, the hydrogen atom, and other stuff besides.
The thing these have in common is that they’re in a kind of intermediate state between the sort of work you normally see in physics journals and the sort of work you see in philosophy of physics journals. All these papers are rooted in very specific physics problems, normally with actual calculations, rather than more general philosophical topics (‘causality’, ‘the measurement problem’, whatever). On the other hand, they always illuminate some deep conceptual issue. Generally these kind of discussions don’t quite count as ‘research’ in the standard sense (they are often not novel work, more a rethinking of existing knowledge), so for physicists it’s normally expository work in, say, a teaching journal, or unpublished material they stick on their website. It fits a bit better in philosophy of physics, but it’s rather on the specific side.
The Rethinking workshops are full of discussions at this level. Does single particle entanglement make any sense? (This was the big argument at the 2017 one.) Why do we use the Levi-Civita connection in general relativity, and not some other connection? If you like these sort of questions, it turns out that you end up reading pretty much the same things, which leads to the ‘obscure fan club’ feeling which is such an enjoyable feature of the workshops. People are starting to get out papers with the BRCP affiliation themselves, and hopefully we’ll see some publications in this particular style.
Mathematics and the Roots of Postmodern Thought, round 2
I was planning to make notes on all the remaining four chapters, but I ended up writing more than I expected about chapters 7 and 10. I’ve just given a brief description of what’s in 8, and I’ve left out 9 entirely. I might go back to it, or I might just leave things here.
7: The Vanishing Subject. This is where I gave up last time, because the material in this chapter is completely unfamiliar to me. It starts with the philosopher of science Jean Cavaillès, who I’d never even heard of:
Cavaillès was a philosopher of science and mathematics, a critic of Husserl and Kant, and a (twice) decorated hero of the French resistance. The role of his work in the changes that took place in the French philosophical scene after World War II—he was executed in 1944 by a Nazi firing squad—is perhaps unfairly neglected outside of France. I will concentrate on his influence on Foucault, but it extends further than that.
Cavaillès started out in formal logic and apparently had a formalist view of mathematics, influenced by Hilbert and critical of Husserl and Kant. The specifics seem to be very different to Hilbert’s view, though, and I don’t understand very clearly from Tasic exactly what they were:
While Hilbert attacked the romanticism of Brouwer and Weyl… Cavailles was more thorough. He went after the “philosophy of the subject” in general, especially Husserl’s and Kant’s ahistorical intuitions….
Science cannot be reduced to the intentions of individual scientists, but is an entity in itself. Applying this to the particular case of mathematics, we get the following picture. A theorem is not true because someone got an idea and then applied the universal, immutable laws of logic or mathematics, thereby proving the theorem. Rather, the “truth” of the theorem is in its very demonstration, which represents a necessary movement within the structure of science itself. “The true meaning of a theory is not in what is understood by the scientist,” wrote Cavailles in On Logic and the Theory of Science, “but in a conceptual becoming that cannot be halted.”
Following this idea, we come across Cavailles’s line that scientific progress is not a history of accumulation of truths but a perpetual revision through deepening and erasure. On this view, the task of historians of science is to study the constitution of truth as a historical concept within an era, rather than to study what was believed to be true in that era. Foucault says something similar in The Archaeology of Knowledge (1969): “[T]he knowledge of psychiatry in the nineteenth century is not the sum of what was thought to be true, but the whole set of practices, singularities, and deviations of which one could speak in psychiatric discourse.”
That’s all clear as mud to me. It probably doesn’t help that I haven’t read Foucault either. It’s also not obvious to me why this would be ‘formalist’, or inspired by Hilbert, but Tasic does start to explain:
Hilbert was partly inspired by the method of “ideal elements,” fictional entities that mathematicians add to their theories for some strange reason. I have in mind “imaginary” objects like, say, the square root of -1. The formalist idea is that these things are really nothing outside of the context of formulas in which they occur. They are grammatical dummies, produced by grammar itself.
The view that “the object is the product of the method,” coupled with a distrust toward the individual anthropomorphic creatures who apply the method, is lurking at the background of formalism. Cavailles simply radicalized a built-in feature of it.
… A symbolic “object” such as the square root of -1 was originally not something that represented a ready-made idea. There was no idea present to begin with, so it could not be represented. This object made its appearance in the process of finding a formula for the solutions of certain kinds of equations. Today, we would simply say that the square root of -1 is a solution. However, the sixteenth-century Italian algebraists who were looking to solve these equations—Girolamo Cardano, Niccolo Tartaglia, Scipione del Ferro, Rafael Bombelli—had no concept of square roots of negative numbers as objects. They applied the method purely formally, and these pseudo-objects started showing up.
I don’t really get formalism, and this complex number example annoys me. It’s true that they were originally found in this formal way. But complex numbers are so clearly about something – rotations in two dimensions – that it just looks like an accident of history that mathematicians happened to find them before they knew what they were.
On the other hand, there are other parts of algebra that are just plain weird, and resistant to a satisfying intuitive explanation. Finite groups seem a bit like that. There are 5 groups of order 8, and 14 groups of order 16. Why? Or worse, that horrible list of differential structures on spheres. Look at that table of numbers. Where the hell did they all come from? Maybe I’d have more sympathy to a formalist argument for stuff like that.
Then there is lots more about Foucault, which also doesn’t make much sense to me. I think I just need to read about Foucault some time.
8: Say Hello To The Structure Bubble. This is about structuralism and Sassure’s structural linguistics. I don’t know a whole lot about that, but more that the big fat zero I know about Foucault, so I had more success with this chapter. It’s mostly background, and I’ll lump it in with my discussion of chapter 10.
9: Don’t Think, Look. This is something about Wittgenstein and language games. I haven’t read it.
10: Postmodern Enigmas. I found this chapter really interesting. It returns to Derrida, and ties in several earlier points in the book. I think I now have slightly more idea about what Derrida is trying to do and it’s maybe different to what I was imagining.
I’ve been talking a lot recently about this talk by Christopher Norris, where he situates Derrida at this kind of meeting point of structuralism and phenomenology:
Structuralism basically takes one side of the chicken/egg dilemma I mentioned at the start of this paper, putting its chief emphasis on system, code, convention, the arbitrary nature of the sign, and all those elements of language that must be in place before we can even begin to communicate. Whereas phenomenology in Husserl’s conception, and as Merleau-Ponty conceived it later on, was about the strictly irreducible surplus of expressive meaning over anything that could possibly be articulated in terms of a structural account.
In mathematics, the structural side corresponds to the set of symbols we use to formalise a problem, and the rules of inference we use to transform those symbols and get to the solution. The phenomenological side is harder to describe, but it involves… whatever those symbols and inference rules actually mean, so that a maths problem is about rotations of a shape or the motion of a point or something. Something that isn’t just an exercise in pushing symbols across the page according to set rules.
Structuralism tried to get the meaning just out of the network of symbols and rules. Not individual meanings of the symbols, because it’s clear that the specific symbols are arbitrary, but in their relations, the ‘structural differentiation’ between symbols.
Small nitpick: this is true for later versions of structuralism, at least. Tasic points out that Saussure probably didn’t believe this:
Let me introduce a terminological convention to avoid unnecessary confusions between Saussure and his alleged followers. Let us reserve the word “structuralism” for something that may be closer to Saussure’s line of thought, namely, the view that structural differentiation is a necessary but not a sufficient condition for the occurrence of meaning. As for the other kind of “structuralism,” where meaning is supposed to spring out of the structural differentiation alone, I propose that it be denoted by the term “functionalism.”
Now I certainly have the intuition that structural differentiation alone isn’t enough to ‘put the meaning in’. And I think this is common. I’ve quoted both of these before, but here are two similar passages by Derrida and Poincaré. I got to Derrida’s one by following this lead from the Norris talk:
Derrida has a very striking and evocative passage in one of his early essays, ‘Force and Signification’, where he says that once you have completed a structuralist analysis of a literary text – here one might think of Roman Jakobson’s exhaustive (and exhausting) analysis of a Shakespeare sonnet – what’s left is something like a city that’s been laid waste by some man-made or natural catastrophe. He makes it sound like a neutron bomb, you know, those bombs that do no damage to buildings and infrastructure but kill all living creatures within miles around, so you have this kind of deathly, uninhabited zone of structures that survive but the life has gone out of them.
Derrida’s own version is:
Thus, the relief and design of structures appears more clearly when content, which is the living energy of meaning, is neutralized. Somewhat like the architecture of an uninhabited or deserted city, reduced to its skeleton by some catastrophe of nature or art. A city no longer inhabited, not simply left behind, but haunted by meaning and culture.
Now compare this to Poincaré:
Perhaps you think I use too many comparisons; yet pardon still another. You have doubtless seen those delicate assemblages of silicious needles which form the skeleton of certain sponges. When the organic matter has disappeared, there remains only a frail and elegant lace-work. True, nothing is there except silica, but what is interesting is the form this silica has taken, and we could not understand it if we did not know the living sponge which has given it precisely this form. Thus it is that the old intuitive notions of our fathers, even when we have abandoned them, still imprint their form upon the logical constructions we have put in their place.
Poincaré’s quote suggests that you need to go to the pretheoretic background, ‘the old intuitive notions of our fathers’, to get the meaning into your symbols and understand them as being about rotation or velocity or whatever. This makes sense to me and was also what I assumed Derrida wanted to do. But according to Tasic it’s something a bit weirder than that, and more concerned with getting the meaning out of the structure itself.
(There’s a lot of ‘according to Tasic’ here, and I know I should just read Derrida. Every secondary source adapts him for whatever story they are trying to tell themselves, and the stories don’t really match up, so I’ve been getting buffeted around by Tasic’s Derrida and Norris’s Derrida and Agre’s Derrida. I still hate Actual Derrida’s writing style, though, so I never get very far. Anyway in the rest of this, when I say ‘Derrida’ it’s Tasic’s Derrida.)
Derrida’s point is that although a structure is just a set of symbols and rules, it’s a generative one: the rules of inference allow you to produce a vast number of possible statements, most of which you don’t know. And the insight he takes from structuralism is that because the meaning of a given symbol depends on its relation to other symbols, and because we’ve never uncovered all these relations, the meaning we do have access to will shift over time. And this seems to be where he’s looking to get the missing meaning into structuralism:
What Derrida wants, apparently, is not a destruction of structuralism as such, but its reconstruction in a manner that would include some equivalent of intuitionist mathematics. The result would be a new structural “science” that he calls grammatology.
I’m not sure how much I like this. I don’t understand it very well, and it’s hard to articulate, but… it seems to be at the wrong level, or something? Like it’s assuming you’ve already dug right down to the bedrock, and got the right symbols to express everything with, and then you’re talking about what they generate. But normally you’re instead trying to formalise, to get some symbols that are useful in the first place. And I think this still needs the pretheoretic background, some stuff that resists being described as a bunch of symbols at all. But I find all of this to be horribly confusing, and I’ll stop here.
I’ve been learning a lot from this book, but it’s pretty frustrating overall. There’s a huge amount of material, it goes incredibly fast, and there’s very little logic to the ordering of topics – I feel like most of the middle chapters could be shuffled into a completely different order without losing much. The last chapter worked much better for me because it linked back to earlier ones and made connections between topics. I’d have appreciated more of a guiding thread throughout.
Still, I’m very happy that this book exists, because I don’t think there’s anything else like it. It’s possible to find individual bits of the story elsewhere – e.g. the connection between structuralism and Bourbaki – but nothing else I’ve found so far tries to bring it together into a high level picture. Even a rushed, confusing high level picture like this one.
Maybe I’ll finish the insight book review. Maybe I’ll concentrate on the same thing for more than three days at a time. Who knows.