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, Intuition and Logic in Mathematics
As I mentioned last month, I wanted to spend my writing energy elsewhere in August, so this one is going to be a lazy sort of links post. Seems appropriate enough for the month of Armpit. There’s your first link.
Derrida, Rousseau, Rameau
Back in May I found this transcript of a talk about Derrida by Christopher Norris. I’ve now read it a couple more times, because it’s about this really general topic that fascinates me that I can’t articulate well, which is something like ‘the tension between meaning and structure’. It’s the one that’s sitting below my ‘two types of mathematician’ recurring obsession, and comes out in this favourite quote of mine by Poincaré that I included at the start, where they become ‘the living sponge’ and its skeleton.
The big example in the Norris talk is actually from music, where Derrida analyses similar tensions in music by picking apart an argument of Rousseau’s against the composer Rameau.
I’ll try and explain myself better soon, but this is a lazy post so I’m just dumping the link here along with an undigested quote:
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. 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. Some readers may well be surprised when they come across that passage, because Derrida is supposed – not without reason – to be highly sceptical about meaning, intention, expressive purport, authorial ‘presence’, and so forth. Yet in these essays what he’s doing is precisely playing off a phenomenological approach, a regard for whatever in the nature of language surpasses a purely structural account, against the structuralist critique of that idea which he sees as being valid, not decisive or definitive, but valid on its own conceptual terms.
That ‘deathly, uninhabited zone of structures’ bit is strikingly similar to Poincaré’s image.
I tried reading some actual Derrida (the Rameau/Rousseau section of Of Grammatology and the ‘Force and Signification’ chapter of Writing and Difference), and the experience was mostly unpleasant. I think what I really want to read right now is ‘Derrida, but as filtered through Norris’s brain and writing style’. Luckily he has a whole book.
Also related: a musicologist talks about Derrida’s Rousseau/Rameau argument.
Bonus musical links: some Rameau instrumental pieces. Also some of Rousseau’s music, but there’s a reason he isn’t remembered as a composer. This one isn’t too bad, but doesn’t really stand up to the Rameau.
I realised at the beginning of the month that I had finally internalised the weirdness of the uncertainty principle. Before, it still felt a bit like an annoying outside condition I was imposing: OK, you can measure momentum or position, fine, I’ll just pick one.
But after spending a decent amount of time thinking about the classical time-frequency analogue, I’m now directly ‘seeing them’ as Fourier pairs. It now feels directly, viscerally wrong to try and impose both, the same way that you wouldn’t try and specify a sine wave and then separately specify the frequency. The sine wave just has a frequency; you can’t try and add a different frequency as an extra condition, because that would be overspecified.
OK, but time and frequency make complete sense as Fourier pairs. They’re the canonical example. Why the hell would position and momentum also be Fourier pairs in quantum mechanics?
I still really don’t get that. So I started thinking about classical waves. But even classical wave momentum turns out to be confusing. (There’s always more physics than you think there is.) See also here. That one’s abstract has a great first line:
Controversies over ‘the momentum’ of waves have repeatedly wasted the time of physicists for over half a century.
I didn’t actually get very far looking at this, so I can’t tell you much about the problem and its solution. The simplest version is: transverse waves are just particles moving up and down, so how can they transfer momentum along the string? But that’s coming out of the linearised wave equation where you’ve made some approximations, and if you don’t make all those approximations there’s a small longitudinal part that comes along for the ride, and that part can carry the momentum. That doesn’t really sound too surprising to me, but I think the question goes deeper than that.
Is Mathematical Truth Time Dependent? I’m not all that interested in the framing question, but really liked the history side. Talks about the transition from ‘anything goes if it works’ eighteenth century creativity as exemplified by Euler, to formalisation in the nineteenth century. Normally this is explained by saying that the nineteenth century mathematicians started turning up weird pathological examples like Weierstrass’s, where intuition fails, but actually Weierstrass and co turn up too late in the day to explain it. Grabiner’s hypothesis: academia expanded and more mathematicians had to teach, so there was a push towards writing textbooks and trying to fit the profusion of results into a coherent formalism.
I also read a section of Lakatos’s Conjectures and Refutations, where he talks through a simple visual proof of Euler’s theorem for polyhedra (vertices – edges + faces = 2). I’d never seen it before, and you can see the very basics of some sophisticated topics in algebraic topology lurking within it, so that gives me a tiny bit more hope of understanding something about cohomology and chain complexes and so on some time in my life. Want to read the rest of the book now.
(The Lakatos extract and the Grabiner piece are collected in this book.)
I finished Winograd and Flores’s Understanding Computers and Cognition. I was planning to write a review but don’t think I have that much to say in the end. The chapters are very short, which is fine for the bits I understood going in but not enough background for the bits I didn’t. Like, there’s one chapter on the vast topic of Heidegger, Gadamer and hermeneutics, and it’s 11 pages long. Followed by a chapter on ‘Cognition as a biological phenomenon’ with some really interesting looking stuff on Maturana and Varela and coupling of living systems to their environment. But, again, only 14 pages for this very dense topic. I guess I can trawl through for references at least.
I did manage to get some stuff finished. I wrote up a piece on my experience with web design, Practical Design for the Idiot Physicist. Also a short follow-up on visual tools vs writing code. Some great comments and links to further reading by ‘anders’ in the comments of that second one.
I added more to the Wigner function post on my new website. Now it has equations and everything.
I plan to spend my physics time next month thinking about interpretations of negative probabilities. Earlier in the month I started rereading one of the few reasonably coherent papers on the subject, and realised that after all the Wigner function stuff I’ve been doing I had a much better background for thinking about the problem, so I want to return to it.
If September’s post is incoherent gibberish, you’ll know that I spent too much time thinking about interpretations of negative probabilities and finally cracked for good.
Hopefully see you this time next month with my sanity intact!