November 2018: Brain scans and postmodern mathematics

Hi! I plugged this newsletter again at the start of a blog post this month, and got some new sign ups. So if you’re wondering why you’re getting this, you probably read some kind of odd braindump about Derrida at the start of the month and thought you wanted more of that for some reason. Thanks for your interest in this experiment! If you have second thoughts, let me know 🙂

This one is going to be a bit rushed and undigested. I had a migraine that ate three days of the end of the month, and that’s when I normally write this thing, so I’m scrabbling to pull something together in the last couple of days.

Anyway, some background on what I’m doing might be helpful. I started the newsletter to keep track of what I’m doing on my weird side interests. I’ve got two of these. One is whatever my blog is about, which tends to circle round mathematical intuition and my attempts to make sense of various bits of continental/postmodern philosophy. The other is learning physics, mostly with a tenuous connection to quantum foundations. (This year I’ve had this paper as a focus, and have used it to explore various things to do with quantum physics on phase space, and the brilliantly crackpot sounding topic of negative probability.)

I’m not sure the combination works and I might try to split them out and do something else with the physics next year, but I’m not sure what. So for now you just get both in an incoherent mix.

One last general point, from the first one of these I did back in February:

I’d be very happy to get questions or comments on any of this! I should probably point out though that I’m envisioning this as a place for writing exploratory stuff that isn’t necessarily very polished, or playing around with ideas that are still a bit vague or undeveloped. So, although I definitely want to know if something is confusing or flat out wrong, this isn’t the place where I want to be playing the ‘find as many holes in the argument as possible’ game and I don’t want tons of critical comments. Nothing is ready for that yet!

OK, with that out of the way, let’s start with…

Mathematics and the Roots of Postmodern Thought

I wrote this in my braindump last time:

I really want to understand this parallel story, and how the ‘algebra’/’geometry’ tension played out in the twentieth century. I can figure out who a lot of the main characters are: Bourbaki on the structuralist side, for a start, and Poincare and Weyl on the phenomenology one, in very different ways. Also Brouwer and the intuitionist weirdos must fit in there somewhere. But I’m struggling to find much in the way of good secondary material. About the only thing I’ve found is Mathematics and the Roots of Postmodern Thought by Vladimir Tasic, which has some good material but is kind of all over the place.

I decided to go back to this book and have another look. I’m not sure ‘all over the place’ is really right, more like just ridiculously condensed. The book is only 175 pages long and tries to cover an insane number of topics – each chapter would make an interesting (and much more comprehensible) book in itself. I’ve made chapter summaries of the first half of the book below – mostly to remind myself what’s in it, but it may be interesting to others who want to explore this topic. If nothing else, it gives a sense of how huge the subject is. I’ve only skimmed the second half so far, but I’ll probably include similar summaries in a later newsletter once I get round to reading it in more detail.

1: Introduction. Some background on the Science Wars, and on the aim of the book. This is exactly the thing I’m interested in: a high-level sketch of some kind of alternate-world postmodernism where the all the fights took place in maths rather than the humanities. This would still be a pretty esoteric bunch of fights, but hopefully more accessible to STEM nerds than fights over language:

My primary concern is to demonstrate that mathematics could have been a formative factor in the rise of postmodern theory, and that this possibility stems from the interest in mathematics of its continental “predecessors” and polemical partners. Even on the level of pure possibility, this provides us with a framework for translating parts of the sometimes baffling postmodern rhetoric into a language understandable to the uninitiated, and thus with a context for critically examining various “postmodern” notions.

… it is probably best to think of this book as a story — a speculative reconstruction of a story—and an invitation to a polemic.

2: The Cartesian Circuit. Historical background. Descartes’ circuit:

Perhaps knowledge does not come from pure reason only, in a linear, top-down manner. Something else could be involved, some process of connecting reason with the senses. He seems to have envisioned a kind of feedback loop of mutual justification of theories and facts, where experience and reason cannot be fully separated from one another. I come up with a theory based on observation, and observations in turn justify the theory in a vicious logical circle. This quasi-mystical process, known as the Cartesian Circuit, was for Descartes a proper method.

Then a very fast race through rationalism vs empiricism, Vico, Kant, Fichte, Hegel.

3: Space Oddity and Linguistic Turn. Still mostly background. Kant on Euclidean geometry as intuitively given. The discovery of non-Euclidean geometries making this more complicated. Helmholtz thought spatial intuition was acquired. Some interesting early psychophysics – is ‘visual space’ Euclidean? Hamann and nineteenth-century romanticism: ‘it is only through an “unreasonable,” imaginative individual act of synthesizing experience into a seamless symbolic whole that reason may get a chance to offer its analyses.’

In parallel to this romanticist story, formalism was also cranking up, trying to make the foundations of mathematics more rigorous after the confusion caused by non-Euclidean geometry:

Mathematics was making its own “linguistic turn.” With the inertia of a luxurious ocean liner, it was veering toward the discrete, formal-computational approach that reflected a growing concern with language.

4 – Wound of Language

Starts with romanticism in art:

There is more to art, infinitely more, so to speak, than meets the eye. It is an invitation to a dialog, an opening toward the unseen, unforeseen, and unforeseeable.

This invisible “remainder”—or rather the gap between the finite physicality of artwork and the infinity of possible interpretations, the gap between looking and seeing-as—opens up the playing field to irreducible individual experiences and self-realizations of the observer (reader). It is the interpretive process that associates art with life and vitality and liberates the mind from the despotism of the eye.

Bergson, Schlegel. In a somewhat similar way, some mathematicians (it’s vague on who, at this point in the text, but Weyl and Poincaré crop up in the next chapter) objected to the reduction of mathematics to symbolic logic:

They objected to the notion that the total meaning of a (mathematical) text is somehow encoded in the text itself and is subsequently recovered by a quasi-mechanical process of “decoding.”

The rest of the chapter is something to do with Brouwer and his weird phenomenological view of the continuum. I skated over this as I don’t know anything about Brouwer and couldn’t be bothered to make the detour, but I should go back and read it properly.

5 – Beyond The Code

Weyl on the continuum and his correspondence with Husserl:

“Finally a mathematician,” he wrote in a letter to Weyl, commenting on Das Kontinuum, “who understands the necessity of phenomenological thinking [and of] finding a way toward the primal ground of logical-mathematical intuition [. . .].”

Then it goes into some complicated stuff with Derrida, who also studied Husserl’s Origin of Geometry. Again, need to read properly.

Next up is Poincaré, who really hated the logicisation of mathematics. I’ve read some of his rants before and they are entertaining. For instance, there’s this one in Science and Method:

… we might imagine a machine where we should put in axioms at one end and take out theorems at the other, like that legendary machine in Chicago where pigs go in alive and come out transformed into hams and sausages. It is no more necessary for the mathematician than it is for these machines to know what he is doing.

Interesting quote on Poincaré’s view of spatial intuition as grounded in the human body:

For Poincare, this intuition is due to the body as much as it may be due to the mind. In a similar vein, Bergson wrote of the “instinct,” Husserl wrote of the “living body” moving through the life-world, while the twentieth-century French philosopher Maurice Merleau-Ponty held that the body’s knowledge of the world is older than that of the intellect. Poincare explicitly places this preintellectual knowledge in the context of evolution, and thereby—like, say, Heidegger—into history.

6 – The Expired Subject

This is a long chapter about Hilbert’s program for formalising all of mathematics from basic axioms. Surprisingly (to me), he didn’t completely want to get rid of intuition – he wanted to keep some kind of pretheoretic intuition of discrete objects. He wrote:

[A]s a precondition for the application of logical inferences and for the activation of logical operations, something must already be given in representation: certain extralogical discrete objects, which exist intuitively as immediate experience prior to all thought. [,..] The solid philosophical attitude that I think is required for the grounding of pure mathematics—as well as for all scientific thought, understanding and communication—is this: In the beginning there was the sign.”

Tasić points out that this last slogan ‘quite openly challenges the romanticist notion that action in some sense precedes knowledge… Even more interestingly, Hilbert’s assertion can be construed as an ironic play on a phrase from Goethe’s Faust, “In the beginning there was the act”’.

Possible interesting reference – ‘Husserl’s The Crisis of European Sciences (1936), where the approach is indicted as the loss of mathematical meaning through “mechanization” or “technization.”’

Then a lot more about Church, Turing, Chaitin which I haven’t read very carefully. I can’t make myself care about formalisation, even though it’s clearly important to the story.

The second half of the book throws in a whole lot more names: Cavaillès, Foucault, Saussure, Bourbaki, Wittgenstein, and more on Derrida. And it looks pretty coherent, rather than just randomly chucking some postmodernists into an otherwise unrelated story. But I’ll have to get through that another time.

The Radon Transform

Physics learning has been going pretty slowly recently, so I’m not going to say much. But I did learn this nice bit of applied maths that I hadn’t seen before.

Say you’re doing a CAT scan of a 2D slice of brain, as in the 100% accurate image below:

You fire X-rays through this slice – those are the wiggly red lines – and see how much radiation comes out the other side. (Or something. This is the cleaned up theorists’ version, which might not bear much resemblance at all to what actually happens.) For every line through the brain slice you get a number out, which is something like the line integral of the density along that line.

So you have a function that takes in lines and spits out numbers. (There’s some choice in exactly how you implement that: you could parametrise the lines by gradient and y-intercept, or something else.) But what you’re probably really interested in for a brain scan is a function that takes in points instead of lines, telling you what the density of the brain is at each point in the slice. As in, a picture like these ones.

The Radon transform is the transform that takes you from the function of points to the function of lines. In this case you want the inverse Radon transform, taking you from lines to points. I’m not going to write out the maths – the Wikipedia article isn’t bad.

Anyway, the same setup turns up in quantum physics on phase space! In this case it’s called quantum tomography and it’s used to reconstruct the quantum state from measurements which are line integrals over phase space. And some of the weird equations I’d seen before that used some unintuitive-looking ‘phase point operators’ are just related to this Radon transform. I haven’t completely fitted the pieces together yet, but it’s all slowly starting to make sense.

More than two types of mathematician

David Chapman sent me a link to this essay by Timothy Gowers this month. I’ve read it before, but it’s worth another look – as Gowers points out, it splits mathematicians into two groups in a way that’s interestingly different to the ways I normally emphasise. I started writing a reply but never finished the draft for some reason, so I thought I’d at least write up the half-finished version here. The argument is a bit weak because I need to think about it more.

Gowers’s main distinction is between problem-solving and theory-building. When I collected lots of these sorts of essays and bucketed everything into two rough groups, I put problem-solving with algebra and theory-building with geometry:

The two clusters vary a bit, but there’s some pattern to what goes in each – it tends to be roughly ‘algebra/problem-solving/analysis/logic/step-by-step/precision/explicit’ vs. ‘geometry/theorising/synthesis/intuition/all-at-once/hand-waving/implicit’.

(Edit to add: ‘analysis’ in the first cluster is meant to be analysis as opposed to ‘synthesis’ in the second cluster, i.e. ‘breaking down’ as opposed to ‘building up’. It’s not referring to the mathematical subject of analysis, which is hard to place!)

Problem-solving/theory-building is possibly the weakest link in that bucketing and gives some indication of where to look to unbucket them. I personally strongly identify with both the geometry and the theory-building sides of these dichotomies, so it’s easy for me to lazily assume that that’s standard and want to keep them together. I think I can justify this as the more natural choice but it might look a bit forced, I don’t know.

Gowers says this about the problem-solving side early on:

At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better.

This is interesting to me because I’m strongly drawn to subjects that are not like this. I always get sucked in by pretty much the same thing, which is a problem where there’s some formalism that works well enough but I don’t know what it means. Currently for me that’s the phase space formulation of quantum mechanics, and negative probability more generally.

This is kind of a reasonable strategic choice, because I don’t have the raw mental horsepower to compete in subjects where the basic objects are nailed down unambiguously at the start and the task is to apply technical brilliance to them. But I’m not doing this deliberately, I just can’t stop thinking about these sort of questions whether it’s a good idea or not.

Problem-solving requires having a problem in the first place – something that’s been sharpened up enough to be stated clearly. So it makes sense that people with an inclination for problem solving would gravitate to fields where you don’t have to spend a lot of time figuring out what the basic objects of study even are.

This doesn’t explain why you’d have to bucket problem-solving with algebra, and theory-building with geometry. And in general, I don’t think you do. But geometry does seem to be a very good source of these sort of big nebulous messes. Torsion in differential geometry is a classic one where it’s famously hard to develop intuition and figure out the big picture – I’ve got badly nerdsniped by this one in the past. On the other side, algebra tends to emphasise the syntactic component, which is more amenable to producing clean problems to solve.

So I think I have a vague explanation for the bucketing I chose. It’s possible to have a mix of preferences from the two lists, though. You could definitely be a theory builder who resolves big nebulous algebraic messes, for instance. I’m still not sure how I’d choose to 2×2 it if I was going to split out a second dimension, but I’m at least getting clearer on exactly what I’ve thrown in the two buckets.

Next month

I should actually finish my blog post on the Cognitive Reflection Test that I said I’d post in November. (It’s not really that long or complicated, I’m just getting bored of the topic. Anyway it’s mostly written now.) Then for physics I probably need to learn something different for a while to get unstuck. I’ve got a couple of ideas.

I’m also having a break from the sort of hard-work tomes I keep ending up slogging through at the moment, and reading some easy pop psych. I’m currently reading this book on insight, which I’ve been meaning to read for ages, and it’s pretty good so far. Not sure what I’ll read after that, but maybe this weird sixties cybernetics self-help book would be interesting. Other recommendations welcome!

At the end of the month I’ll probably do a bit of an end-of-year review and evaluate how well the blog/newsletter/studying thing is working. I’ve never done one of those before but it sounds like a good idea. So the next one of these might contain an abnormal amount of navel gazing.

Anyway thanks for reading! If you decide you no longer want this thing, let me know and I’ll unsubscribe you 🙂