Homotopy Type Theory

I’ve had the Univalent Foundations Program’s book on Homotopy Type Theory on my to-read list for quite a while after reading about the project on Michael Harris’s blog. For some reason, recovering from a nasty fever was the exact right time to skim-read the bits I would know about, viz, the Introduction, and the chapters on Set Theory, Category Theory and real numbers.

Call me a rude mechanical but I’ve always thought that people who go in for type theories have missed a number of points. Yes I do know that Homotopy Type Theory is currently the subject of active research by people who are cleverer in their sleep on a bad day than I am awake on a good day, but as someone once remarked about the "highly motivated individuals" that were popular in recruitment in the 1970’s, the catch is that they can all highly motivate themselves up a gum tree.

Type theory was an ugly kludge invented by Bertrand Russell to get round the fact that, in its unrestricted form, the Axiom of Comprehension leads immediately to inconsistencies of which Russell’s Paradox is the most famous. The "set of all sets that are not members of themselves” looks like a well-formed definition, but now consider that very set. If it does belong to itself, it doesn’t, and if it doesn’t, it does. Russell’s kludge was to stratify logical formulae into “types”, and impose the rule that a set could only belong to a set of higher type than itself. It worked well enough for him to finish the project of showing that mathematics could be developed from “purely logical foundations” that was the aim of his massive Principia Mathematica.

A few years later, however, Zermelo, Fraenkel and Skolem devised the current axioms of mainstream set theory, traditionally called ZFC “Zermelo-Fraenkel with (the Axiom of) Choice” (for some reason Skolem’s name always gets left out). The foundationalist programme became “mathematics is derivable from the axioms of ZFC and mathematical logic” instead of “mathematics is derivable from the Axiom of Comprehension, type theory and mathematical logic”. A lot of people are very happy with that, including me. The point of the foundational programme was to show that such a derivation was possible, not to argue that fractions were really ordered pairs of massively nested copies of the empty set. Once we have ZF, we don’t need the kludge that is types.

And there it should have died. Along with the biplane, the TOG tank, the Sinclair C5, airships and programmable calculators. All may have been wonderful and useful once, but the world has moved on. And the same goes for mathematical theories, which are developed to solve problems. Cantor’s set theory was not an attempt to fabricate a “new language for mathematics” but an attempt to understand the limit points of Fourier Series. It so happened it let other mathematicians re-state other theories in a clearer and more systematic way, which was why it was adopted so quickly. We still use it because it’s still the best way of stating many definitions and theorems. But as a subject on its own?

In its own right, set theory is an interesting for a) large cardinal theory, or b) Cohen forcing constructions for independence proofs and proving the existence of weird objects. These are not going to make your e-mails any safer or your pictures any less fuzzy any time soon. People work on set theory as they work on model theory, both of which John Bell drummed both into me back in the day, but I’m not going to sell you on the commercial benefits of saturated structures (generalisations of the idea of algebraically-closed fields). It’s interesting to some people, but it’s a creek off the main river of mathematics. The same goes for any foundational subject.

Category theory is foundational in that sense. It was devised to formalise proofs and constructions that occurred in multiple branches of mathematics, and to formalise the “X and Y have different fidget groups, but fidget groups are preserved under twiddles, so X and Y are not twiddle-equivalent” arguments that were appearing in algebraic geometry at the time. There’s some quiet satisfaction from the moment when you realise that an SQL inner join is really a pullback in disguise, but that knowledge does not make you a designer of more stylish queries. In the same way that, just because you can show that a folklore Haskell programming trick actually illustrates Kan Extensions, it doesn’t meant that knowing anything about Kan extensions will make you a slicker programmer. Academic computer scientists love their Haskell and their category theory, but if either was a pre-requisite for a job at Google, Business Insider would have run an article called “Here’s the far-out math theory you need for a job at Google" a long time ago.

And there’s one more thing. Type theories have, in general, a non-classical logic! (Except at the “-1” level, where you can do classical logic.) Would you have guessed? I have nothing against the study of multi-valued and modal logics, though again, I’m not sure I want my taxes to pay for it. I get (or did at the time I read about it) why it appears naturally as the “natural logic” of certain categories of sheaves, but that’s no more profound than saying that the “natural logic” of non-complemented lattices is non-classical, and nobody thought to do that. For some reason failure of the law of the excluded middle is seen as some kind of abstract virtue and I can’t help hearing alarm bells when it is so presented. It’s something to do with hair-shirts, I think. Maths goes better with the occasional non-constructive proof by contradiction.

I have no problem with what consenting mathematicians choose to talk about in the privacy of their conferences, though if it was me, I wouldn’t use a lot of taxpayer’s money to fund research into Univalent Foundations. Voevodsky is selling it as a theorem-prover, and that will aways get some attention, but you and I know that it wouldn’t help much even if it did produce an effective theorem-prover. Types can only capture a certain class of errors, not something subtle everyone has so far missed about about (say) Cohen-Macauly rings over finite fields.

So do you read the book and follow the work? Look, some people still swear by the λ-calculus for dealing with functions. I know it works. But anyone who actually used a λ-function in actual production code would find their code re-factored to get rid of it at the first opportunity. Ditto types: I’m sure the maths is impeccable. It’s the project that’s a little pointless.

The whole foundations thing was done in the early twentieth-century. The point of mathematics is to solve problems, and while the majority of those problems mostly still come from physics or gambling probability theory, some now come from cryptography, computing, biology and other sciences. Most are, in the end, to do with solving differential or difference equations. I’m going out on a limb here, but I’m pretty sure no-one is going to improve image-enhancement techniques with higher homotopy types.

And next up… a health report

I spent eighteen (18) hours asleep on Thursday. My Fitbit said so. I would sleep for maybe seventy or eighty minutes, sit on the side of my bed, get up by leaning against the wall, walk very carefully to the loo where I got rid of some water from one end and took some more on at the other. A very little bit more. I didn’t eat a thing. I had no appetite. I did not even want to get out of bed.

That was after taking Wednesday off because I had a fever and I couldn’t actually walk upstairs, but had to crawl on my hands and feet, and then grasp the balustrade as I got to the top. Still had to do that this morning, but managed to get out to Richmond for some lunch and to stock up on cough syrup and Lemsip. You would be amazed how few cough syrups are actually alcohol- or ethanol-free. I made it back just in time to lay on the couch for a rest. When my blood pressure drops so much that if I sit up too quickly I get very dizzy. I haven’t even eaten any chocolate, though I did have a Cote Creme Caramel.

(You don't eat this if your digestion is gippy)

If I spend an entire day in bed, it’s almost always because I’m recovering from food poisoning, but it wasn’t that. All I was doing on Tuesday afternoon was coughing slightly. I went to the gym and everything. Then at midnight I woke up and, looked at the clock and realised I wasn’t going anywhere for at least a couple of days. Hot, headachy and fever-dreams, you know, obsessional and meaningless.

It’s half-term. There were children on the train. It was also (effing) cold, so my immune system could have been operating at less than 100%. I blame the parents: they should have had the little darlings back home, eating tea and watching Blue Peter by the time I got anywhere near a train. And the railway companies should spray all their evening trains with some nasty chemical that kills all known germs and viruses before we grown-ups get on. I have no idea what Vladimir Putin has infected your kids with this winter, but it’s a winner. Spray it over NATO forces, wait two days and he can take over Europe before lunch with a couple of divisions of marines.

When Donald Trump is President, there will be no more children on commuter trains. I bet Ivanka didn’t travel by train anywhere, anyday.

Would younger viewers look at Godard and say “Well, I guess you had to be there at the time”?

A friend sent me this link to an edit of early Godard movies.


Godard in Fragments from Criterion Collection on Vimeo.

Inspired by this, and if it wasn’t a school night, I would stay up and watch La Chinoise and Two or Three Things I Know About Her. But it is, and I have the last two episodes of S7 of Sons of Anarchy to watch.

Watching this made me wonder: would younger viewers look at it and say “Well, I guess you had to be there at the time”? Or is the sheer style and innovation still clear today? Or does it come across as “Yeah, they did that avant-garde stuff in the 60’s. Have you seen Cremaster?"

Jack Donovan’s The Way of Men

Jack Donovan is famous for the idea that men should operate in gangs, and for being gay while saying so. He should be famous for the distinction between "being a good man” and “being good at being a man”, because that is worthy of J L Austin himself, and for his identification of the four tactical virtues: strength, courage, mastery and honour.

Being a good man has to do with ideas about morality, ethics, religion, and behaving productively within a given civilisational structure… Being good at being a man is about…showing other men that you are the kind of guy they’d want on their team if the shit hits the fan… [demonstrating] strength, courage, mastery [and] commitment… A man who is more concerned with being a good man than with being good at being a man makes a very well-behaved slave.

What Donovan gets is that men want the good opinion and recognition of other men because only other men know what was involved, intellectually, personally and physically, in our achievement. Women’s opinions don’t count because women don’t care about the things that men care about: strength, courage, mastery and honour. Men don’t really care about their supervisor’s opinion of them because their supervisor doesn’t give a damn about anything but how well we fit in to the machine. (Tactically the opinions of supervisors and women matter, because life is easier if neither are bitching all the time, but one wouldn’t want to base one’s entire life around keeping the wife and boss happy. Would you?)

What makes Donovan’s book refreshing is that he doesn’t blame feminism. Women only appear as sirens inviting men to wreck themselves on the rocks of all things soft and compromised. There’s a reason for that we’ve already mentioned.

His villain is modern technology and the post-industrial economy and society that it creates. This has removed most of the chances that men have to acquire and exercise the four tactical virtues: strength, courage, mastery and honour. Processes and mechanisation has removed the need for Mastery from all but a handful of mostly intellectual roles. Power steering means women can drive buses, so that Strength isn’t needed. An astonishing level of public safety and policing means that we can withdraw large sums of money from machines in the middle of the night on busy roads without a thought of being mugged - there goes Courage. The whole idea of Honour from one’s fellow man is a joke when equal opportunity legislation means he turns out to be a woman.

However, while this is the right criminal, it’s the wrong crime. Being good at being a man can’t depend on a particular mode of the economy or the exact arrangements in society. A man can have strength, courage, mastery and honour in most economies and societies - but it will look different in each one, and perhaps each may regard the others’ as un-manly.

Men aren’t compromised by feminism, or post-industrial society, or the Health and Safety at Work Act. Some are compromised by marriage, children and the need to earn a living. They may not always have been physically soft. Modern entertainment technology, plus the commuting that scatters workers to the four quarters at the end of the day, has probably lead to fewer chances for men to spend time with each other socially. But some men have always skived and tried to cheat each other: that’s why we have all the commercial law we have now. If bakers had never adulterated flour with chalk, I might have believed in a Golden Age of Manly Virtue, but they did and I don’t. There is no “crisis of manhood”, it’s always been this way, only the costumes change.

Acquiring and living the four tactical virtues is a personal project that a man pursues despite the economy he works in, the society he lives in, and the men he knows. He cannot suppose that the men around him want to be virtuous, nor that the society values virtue, nor that it will be rewarded. A man chooses to work towards being good at being a man because he cannot live as a compromised person. And it has always been this way.

Donovan does not think that men in post-modern Capitalism will ever act as a united political force. I agree. That leaves personal action, which always feels a little bathetic after some high-grade, wide-ranging social analysis.

What does a man do when he wants to become better at being a man? Donovan’s answer is to “start a gang”. Not as in the Sons of Anarchy, but as in a bunch of like-minded guys to do stuff with.

You need to learn how to read each other and work together as a group. Go to the shooting range. Go hunting. Play paintball. Go to the gym. Take martial arts classes. Join a sports team. Take a workshop. Learn a useful skill. Get off your asses and do something. In harder times, the men that you do these kinds of things with are going to be the first men you call. They will be your gang. They will be your us.

Errr, no. That’s not quite enough.

When you are on the end of a wrongful dismissal, you need a union or an employment lawyer, preferably one who knows some journalists. You need a criminal lawyer with a flair for publicity to handle that false rape accusation. When the pipes burst, you need a plumber, and when you break a bone, you need a surgeon. Your gang is unlikely to include one, and certainly won’t include all, of the specialists you need. You need huge amounts of personal fortitude to sustain the campaign, and your buddies can’t help you with that, beyond a few platitudes. Because you’re going to need a lot of cash or the ability to live on very little - and your buddies won’t help you with that. They are regular guys with regular jobs like you. Dealing with misfortune in post-modern capitalism isn’t like defending the village from raiders or saving the animals from a flood. When bad things happen in post-modern capitalism, you have to get to Krasnoyarsk with the kindness of strangers. And once you get back from Krasnoyarsk, you might hang out with the guys again, but they won’t be your gang. They will be a bunch of guys you shoot the breeze (or the paintball) with, but who, when the shit hit your fan, were no more help than some guy on the pavement last week.

That’s the difference between the mythical life of the savannah tribe and post-modern capitalism. In post-modern capitalism, each man needs his own Rolodex of useful contacts, from plumbers to employment lawyers, electricians to employment agents. He needs the time and skills to keep these contacts fresh, and any businessman will tell you that doing that can take a week or more from your month. He needs to have something useful to trade with these contacts: he needs to be in their Rolodexes. The principle is the same - he needs to demonstrate he can be useful to those other men - but the camaraderie will be missing.

Few people have those social skills. Which is why we had the Yellow Pages, and have Google now. Building that kind of Rolodex takes a long time the moment you live in a town much larger than about five thousand people. So let’s add “the social skills needed to build and maintain a rolodex” (I’m trying to avoid the n-word) to the list of tactical virtues, though it’s not one Aristotle would have recognised. His towns were small enough that “everyone” knew each other.

The Way of Men is one of the better books on masculinity today. But it still doesn’t understand just how corrosive post-modern capitalism is, and how it turns everything to its advantage.

Heck. Even Jack’s selling merchandise.

£15 for a Korean Martial Arts Film at the Curzon Bloomsbury? My Inner Pricing Manager is Offended

I went to see The Assassin at the Curzon Bloomsbury (or “The Renoir” if you’re old-school) the other Saturday. The 11:00 AM showing. In the Minema screening room.


It cost £15.

It’s a good film. There were quite a few people there - for a Korean martial arts film showing at the Renoir. And get they must have spent a LOT of money on the Takero Shimazaki-designed overhaul of the cinema, of which this is a sample.

But £15?

When the DVD, which will come out in six months at the most, will cost about £8 in the Covent Garden branch of Fopp?

I paid £20 to see the Star Wars movie at the Odeon Leicester Square, and I was happy to do that because some films need to be seen on a BIG screen. The Curzon Bloomsbury does not have a big screen, though it does have more comfortable seats. I think the Minema screening room is not a lot bigger than my back room, and I live in a small terraced house.

My inner pricing manager is offended. I’m not sure what I think would be a good price. The Prince Charles charges £11. That feels about right, as they have refurbished the Prince Charles as well. I think I paid about £12 to see The Big Short at the Cineworld Haymarket the Saturday before that. I know the Curzon want me to sign up as a member, when I will get two free tickets and “reduced” rates. Except those “reduced” rates are the standard rates, and the walk-in price I paid was a premium rate. The annual membership (for me) is £45, less £30 for the free tickets, which is £15, and I think the discount isn’t as much as £3. I would need to see six films a year before I broke even on that one, which is just about possible.

It ain’t cheap to go to the movies no more. And I’m not sure the movies they’re showing are worth that kind of money. Maybe having Rohmer, Robbe-Grillet and Godard on DVD does spoil one. 

Maybe I need to go to the ICA more. After all, they show more or less the same films as the Curzon chain. But this week it’s only showing The Assassin at 16:00, and that’s when I leave the office over in the City. So maybe not the ICA then. While I’m still working.

John Bodi - Death by a Thousand Sluts

While I’m talking about books you should read, this is the first book for quite a while that I read in one sitting. Do not read this if you are female - you simply won’t understand it. Do not read it if you have never had a bad week in your life, because you won’t understand it either.

And when you get to the end, you might realise that Bodi’s real achievement is that, throughout the depressions, crises and breakdowns, not once does he turn to the doctor and ask for some anti-depressants. He never doubts that his troubles and inner chaos can be sorted out without drugs and by action. I’ve had some bad moments, but none as bad as some of the ones Bodi describes. And his insights into people and his descriptions of the London Day Game scene are worth every penny.

Just not if you’re a girl of either sex.

The Scarfolk book really is weird, and it’s the illustrations that are most weird.

I am alive really, it’s just that I’m not taking any photographs and the stuff I’m thinking about is stuff I’ve thought about before and I don’t want to go on about the same things in public.

So instead… read this...

I saw it a while ago in Foyles, found it intriguing, bought it and only now have been in the mood to read it.

It took me a long time to realise that you should always buy a book you want to read, even if you can’t read it right away, because it might not be available in six month’s time when you’re in the mood again. This is especially applies to art books that accompany exhibitions, and art books generally. The really good ones don’t get remaindered, though you may be able to get them cheaper on Amazon. Support your local bookstore however (in my case, that’s Foyles).

The Scarfolk book really is weird, and it’s the illustrations that are most weird. Someone went to a ton of trouble to do those and there are lot of them.