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Notation that sings [May. 9th, 2008|05:08 pm]
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Some Actual Maths for a change. Some of it assumes some mathematical sophistication, but the core idea should be accessible to anybody. Have you ever been an arts student? I'm looking at you. Skate over the words you don't know, they're not important - they're just there to explain how the central result connects to more sophisticated and deeper areas of mathematics.

( Go on, click, you know you wanna )
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Fine Structure: The Story So Far [May. 4th, 2008|06:49 pm]
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There's a new Fine Structure story up, called The Story So Far. Go read it. Actually, there's another one that I never got round to posting about, called Failure Mode, which is... rather moving, actually. Great stuff.

Anyway, the missing story ( has now been revealed... )

I'm not going to write too much now, because I have a hangover. But I think there'd be some benefit to thinking harder about the fictional physics - I definitely should have spotted the "superlight" connection. I think there may be more insights to come from that source.

There's now a comments system on the Fine Structure pages themselves, which I suspect will largely replace the discussions here. Please continue to comment on my posts if you want, though! I'm hoping someone else will start blogging in depth about FS. Maybe we need a Fine Structure wiki? An E2 node?
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The White Spider [Apr. 21st, 2008|10:35 am]
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I've just started reading The White Spider, Heinrich Harrer's account of his 1938 first ascent of the fearsome North Face of the Eiger. The climb was one of the pivotal moments in the history of mountaineering, and the book's considered one of the classics of mountain literature. I was particularly struck by the opening paragraphs:
"Writing a book about the North Face of the Eiger? Whatever for?"
The question was put to me by a man of some standing in Alpine circles. I was taken aback and slightly cross, so I gave him a somewhat off-hand answer: "For people to read, of course."
That started him off on a passionate tirade.
"Who's likely to read it? Don't you think the handful of climbers who are really interested in that crazy venture have had quite enough literature on the subject already? Or do you just want to join the sensation-mongers, from whose ranks a serious climber like yourself should keep as remote as possible?"
I answered him: "If all climbers shared your point of view, it wouldn't be surprising to find the newspaper reports overflowing with mis-statements and exaggerations. I believe the public has a right to authoritative information, especially when mountaineering problems become human ones. And I think it is a climber's duty to contribute to the formation of public opinion in such matters."
And with that I dropped the unpleasant argument.
It struck me as I read those words that one could replace "climbing" with "mathematics", "the Eiger" with the name of some serious mathematical problem, etc., and be left with a statement that's equally true, and equally important.

[By the way, there are some excellent, not to mention terrifyingly vertiginous, pictures of Sir Ranulph Fiennes on the Eiger's North Face here. Well worth a look.]
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Too Much Information [Mar. 9th, 2008|12:04 pm]
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There's a new Fine Structure story out, called Too Much Information. In addition, the Crash stories have been incorporated into their own subdirectory, called 1970-. Some analysis of TMI and its implications has already started in the comments threads of my previous FS posts, and I'll have to revise my own thoughts in light of it, but for now:

( Discussion, including spoilers )

By the way, if anyone else is planning on blogging about FS, please leave a link here, so we can all follow what each other are saying :-)
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More Fine Structure badgery [Feb. 29th, 2008|02:40 pm]
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It's become apparent to me that many of the mysteries in Fine Structure depend on the precise order in which the stories occur. Time to get systematic. Obviously, this whole post is full of spoilers.

( Plot summaries )

( Chronology )

I should probably put all this stuff into some sort of Hasse diagram, but right now I can't be bothered :-)
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Thoughts on Sam Hughes' Fine Structure [Feb. 26th, 2008|02:10 pm]
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I've been reading some science fiction written by a guy called Sam Hughes1, and posted on his website Things of Interest. Most recently, I've been reading his Ed stories, about Sam and his (fictional) genius flatmate Ed, who builds battlemechs in their basement and saves the world from annihilation on a regular basis, and to whom Sam plays a sort of bemused Watson. It starts out as a fun bit of wish-fulfilment à la early Sluggy Freelance, and turns into something rather more. Well worth a read.

But I don't want to talk about that: I want to talk about his novel-in-progress, Fine Structure, and where I think it's going. It's a collection of loosely-linked short stories whose connections only become apparent later on, somewhat like Trainspotting: I don't want to give away too much of the plot, but let's just say it has elements of Contact, Superman and Strata, in a refreshingly hard-sf style. And it's garnered positive reviews from no less a person than David Brin (search for "Power of Two"). It's probably best to read them in order, but I started with Power of Two and it didn't do me too much harm (and it's one of my favourites).

Edit: further posts on this topic can be found here.

( MAJOR spoilers. Go and read the stories first! )

A word of warning: I have literally lost days to browsing Sam's website. Interesting days, mind :-)

1 Some of you might even know him: he was a maths student at Corpus, Cambridge a few years ago. Or you might have encountered him on some online community or other: he generally goes by the username "sam512".
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Which graduate maths textbook are you? [Dec. 16th, 2007|08:24 pm]
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I ought to be filling in my job application. Needless to say, I'm not: instead I've been reading the archives of my late (and much-missed) friend [info]floor_sitter's blog. I keep finding entries I want to comment on, and then realising that there's no point. Damn, I miss her.

But anyway, a while back, she did a meme called "Which Springer-Verlag Graduate Text in Mathematics would you be?" (which appears not to exist any more, alas).

She came out as Categories for the Working Mathematician. Epic win :-)

Anyway, you're all no doubt dying to know my plans for Christmas and New Year. So's you know:
  • I'll be heading South on Thursday 20th for the USLES panto and cast party.
  • I've made vague plans to meet up with Eugenia Cheng on Friday, then I'm heading to Oxford some time on the Friday evening to see my family.
  • They're going out without me on the Saturday night (the 22nd), so if anyone fancies meeting up in Oxford, that would be excellent.
  • I'll be spending Christmas with my family.
  • [info]wormwood_pearl will be coming down to join me on the 27th.
  • On New Year's Eve, we will be attending [info]elvum's dinner party.
  • We will be heading back North on the 4th.
Meeting up with as many of you as possible would be great, but I don't think we're going to make the VOLES party.
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How I've always interpreted the Sirius/Remus joint birthday present [Nov. 29th, 2007|12:04 pm]
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( Prisoner of Azkhaban spoilers, like you care at this stage. )
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Assyriology update [Nov. 15th, 2007|08:41 am]
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  • Do not try to wget -m the whole of the Electronic Text Corpus of Sumerian Literature. There's rather a lot more there than you'd expect. Why they couldn't just provide a zipfile and/or tarball of the XML they store in the backend database is anyone's guess. I can't be the only geek who's read Snow Crash and wants to contribute.
  • Other than that, I'm really rather impressed with the ETCSL. Check out the mouseover text. Of course, all that stuff needs stripping away for my purposes :-( The background articles on Sumerian language, literature and cuneiform (literally, "wedge-shaped") writing look pretty useful too. Annoyingly, their funding ran out in late 2006, so the site hasn't been updated for a while, and they seem to have made it unnecessarily hard for anyone to take over.
  • The correlation coefficient of a constant signal with anything else, even itself, is always zero, so if I take [info]elvum's suggestion to use autocorrelation then the "short short short short" problem becomes a non-issue. However, I then end up with another signal, whereas what I really need is a single number with higher values representing higher levels of poeticity. Possibly I can limit the number of possibilities I need to check by counting syllables-per-line; or maybe I could just take the maximum value of the autocorrelation? It's been nearly ten years since I did any statistics, so this is all a bit painful. I've tried asking friends in the stats department, and been met with the slightly worried look of an expert challenged on something that's just outside their narrow specialism. I know it well, because it's a look I often use myself.
  • I'm not the first person to apply statistical ideas to analyse the corpus. There's even a book out: Analysing literary Sumerian: corpus-based approaches (or you can buy it from Amazon!) Nothing especially relevant-looking in the chapter headings, but I wonder if I could persuade the library to buy a copy... they don't have it in stock right now, but they do have the intriguing-looking Sumerian or Cryptology? Further investigation reveals that it used to be thought that Sumerian wasn't an actual language, but rather a priestly cryptosystem used for enciphering Semitic texts. More details here.
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Goddess-mother of the Earth [Oct. 29th, 2007|11:05 am]
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I've finally finished reading The Turquoise Mountain, Brian Blessed's account of his first attempt on Everest, and of the filming of the documentary Galahad of Everest. It's taken me ages because in the early stages I skipped about a lot, and it's hard to get excited about reading a bit of the book that you've read before - the time I've taken to read it should not be treated as a poor review! He didn't reach the summit, but he did reach about 25,400 feet (7,750m), before having to turn back due to weather, bureaucratic interference and lack of supplies (the BBC expedition were being supported by the international Peace Climb, who started stripping their tents and gear off the mountain when the BBC were still climbing).

By the way, the name "Brian Blessed" will be immediately familiar to all the Brits, and probably completely unknown to everyone else. He's a much-loved Shakespearean actor, with a huge booming voice and the frame and beard to match, famous to my parents' generation for his role in Z-cars, and to mine for his roles in Flash Gordon, The Black Adder and so on. He's the guy who gets rather anachronistically lynched by the Ku Klux Klan at the beginning of Robin Hood, Prince of Thieves. In real life, he's obsessed with Everest, and particularly with the expedition of Mallory and Irvine.

Some things that struck me:
  • One of the unexpected difficulties of climbing Everest is the heat: even at the North Col, at over 7,000m, they were experiencing temperatures of 40C during the day (at sunset, it apparently drops to -20C in around half an hour).
  • Losing weight on an Everest trip is a given, and losing 20-30lb is not unusual. Brian Blessed, who usually weighs about 16st/220lb/100kg, and dropped to about 14 in the course of his training, was down to nearer 10st/140lb/65kg on his return!
  • You have to be awesomely fit. Blessed claims to have been running "10-14 miles a day, with the odd marathon thrown in for good measure". Even allowing for a bit of dramatic exaggeration (he is an actor), that's pretty impressive.
  • Getting ready to go in the morning at altitude takes ages - ice needs to be melted and boiled for tea, tricky high-altitude gear needs to be put on with fingers made clumsy by cold, and the altitude makes your brain slow and befuddled. On their final day of climbing, it took them nearly five hours to make a start. "Nice to know it's not just me", I thought. But I have a question for the physicist-mountaineers out there: why does water take longer to boil at altitude? Water boils at a lower temperature, so it should be quicker. Is it that it needs to be melted from ice first? Or is it that there's less oxygen around for combustion? If the latter, it should be possible to solve it - the problem of creating high temperatures using chemical reactions at high altitude and with minimal danger and weight penalty has received substantial attention, but I can't find anything about this by Googling. Am I missing something?
  • Shortly before he has to turn back, Blessed sees a hallucination (or vision, if you will) of all the members of the twenties expeditions, sitting on the snow in their shorts, smiling and waving at him. I rather like the idea of the ghosts of all the early mountaineers living together on Everest, able to enjoy the beauty of the mountains but not to be touched by the conditions.

At the less glamorous end of the mountain-activities scale, I climbed my 67th Munro, Chno Dearg, on Saturday. You've seen Trainspotting? The bit where Tommy takes them all for a walk, and they hate it so much they decide to go back onto heroin? That's where we were. Corrour is literally a railway station and a pub (which is now, mirabile dictu, open during the day - the previous owners, having failed utterly to get this whole "capitalism" thing, used to open up only after the last train of the day had left. Unsurprisingly, it's now under new management). It's impossible to reach Corrour by road - even 4x4s have to be brought in by rail. On a good day, it's rather lovely, and I had a great day there last May climbing the two hills on the West side of Loch Treig. Saturday was not a good day. It was rainy, wet, cold and miserable all day. A series of navigation errors (hey, you try following a compass bearing for any significant distance through a sloping bog in the mist) meant we took much longer to reach the summit than expected and came down in the wrong valley, then had to force the pace (with all three of us suffering some form of leg trouble) to get back to the station before the last train came at 1830.

Fortunately, as we staggered onto the platform at 1825, we were met by a large group celebrating the fact that two of their number had bagged their final Munro, who took one look at us and thrust a bottle of whisky into our hands :-)
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The nature of Monkey was irrepressible! [Oct. 19th, 2007|08:10 pm]
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[music |Godiego - Monkey Magic]

Two works of Oriental literature of which I have recently become aware:

The Epic of King Geser: I learned about this while reading Sergei Lukyanenko's excellent (excellent) novel Night Watch. Geser is a Tibetan/Mongolian epic poem that tells the story of King Geser of Ling, his semi-divine origins, his battle against the kingdom of Hor (Mongolia) and the Enemies of the Four Directions, before he briefly descends to hell and finally ascends to heaven. It's around a thousand years old, one of the very few epic poems still surviving in oral tradition, and is considered to be the longest work of literature in the world - if it were written down in its entirety (presumably, including all the local variations), it would run to over 20 million words. The Wikipedia article's a bit confused - the talk page might shed a bit more light. Or perhaps not.

[Did I mention how good Night Watch is, by the way?]

Journey to the West: better known in the West as Monkey, the title of the translation by Arthur Waley and the 1970s TV series (originally Japanese, dubbed into English by the BBC), both of which my flatmates have on DVD :-). You can see the highly earwormy opening theme here (or here with Japanese introduction). It could be described, very inaccurately, as a Buddhist/Taoist Pilgrim's Progress with animal spirits and kung fu. It tells the story of Prince Tripitaka's pilgrimage from China to India to fetch lost Buddhist sutras, accompanied by three immortal helpers: the invulnerable, irrepressible, battle-loving Monkey, the disgraced angel turned pig spirit Pigsy, and the disgraced angel turned cannibal water-monster Sandy. The translations of the character names, by the way, seem to be due to Waley - "Tripitaka" is actually a translation back into Sanskrit of the nickname used for the character throughout most of the book. The TV series at least is a wonderful mix of adventure, silliness, fighting, and Buddhist thought and allegory, and my understanding is that the novel is much the same, and also a satire on Ming dynasty politics. Tripitaka (or Xuanzang to give him his Sunday name) was a real person, and he made a pilgrimage to India through modern-day Uzbekistan and Afghanistan from 626-645AD, and spent much of the rest of his life translating the texts he brought back into Chinese. The first account of his journey is his own Journey to the West in the Great Tang Dynasty of 646. Legends around his journey grew up almost from the moment he returned, and theatrical versions have apparently been performed continuously since the 13th century; the novel usually called Journey to the West was written in the 1590s, and is usually ascribed to Wú Chéng'ēn. It's considered one of the Four Great Classical Novels of Chinese literature. Me, I didn't even know that there were Four Great Classical Novels of Chinese literature until a few days ago.1

I can't help but feel that one or other of these must have been the inspiration for the Great Circling Poets of Arium...

1 That is, I'd assumed there were at least four classic novels in the Chinese literary canon, but I didn't know there was a standard choice of the four Greatest :-)
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Tactical mathematics [Oct. 15th, 2007|12:07 pm]
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Term started back a couple of weeks ago, and I've been teaching undergraduates (in addition to the private tutoring/undermining of the principle of open-access education that I do to bring in beer-and-climbing-wall money1). I've noticed that I'm spending an awful lot of time not teaching the material, but rather teaching what I think of as the "tactics" of mathematical problem-solving. Things like "simplify whenever you can", "use symmetry (of whatever form)", "Replace terms you don't understand with their definitions", "give names to things", and "if the problem starts with 'show that for all wombats ...', then start your proof with 'Let x be a wombat', then show that x has the desired property." It's not that they don't know how to do the things they need to do; they don't know what to do. They lack guiding principles to allow them to follow their noses through problems. As a result, they get stuck, or waste time on lengthy calculations that they should have been able to see were unnecessary.

[I should mention to the Oxbridge types reading this that the Scottish university system is very different to the English one. Students arrive a year younger, and don't specialise until their second or third year. As a result, we get lots of undergraduates who are only doing maths because it allows them to do some other course in some other faculty, and have no intention of specialising in mathematics. Plus, the permanent members of staff tend to keep the best students to themselves.]

At first, this was getting me really annoyed. With the possible exception of the last one mentioned, these tactics are so basic (and so independent of context) that they should have been drilled into everyone from primary school onwards. Why should I have to do the job that their school maths teachers should have been doing? I want to give these students the red pill, and show them just how deep the rabbit hole goes, but I'm having to stop and say "take the pill from my hand. Now, put it in your mouth. Now, swallow. OK, if you can't swallow right away, then take a drink of water. *Sigh*. Pick up the glass..."2

Now, though, I'm not so sure. Precisely because these tactics are so basic, and so generally applicable, they're likely to be much more use to my students than, say, partial differentiation. They should have been taught them before, but they haven't, and here's my opportunity to make a difference. Plus, from a certain viewpoint, this kind of problem-solving technique is the beating heart of mathematics. If you haven't read Timothy Gowers' essay The Two Cultures of Mathematics, you might like to do so now: in it, he discusses the split between "theory-builders" and "problem-solvers". I come very much from the theory-building side myself, but I can see and appreciate the other side.

But then again, maybe this is why all maths teachers (or at least, the good ones) are a bit crazy. In order to retain their sanity, they have to convince themselves that the basic, tactical, bread-and-butter stuff they have to teach is the real point of the subject. Like Mephistopheles, they have tasted the eternal joys of heaven, and are now tormented with ten thousand hells in being deprived of everlasting bliss. Why this is hell, nor are they out of it.

I've been doing some reading: specifically, I've been reading Polya's How to Solve It, which is about precisely this kind of stuff (and about how to teach it effectively). I've also been reading Stephen Fry's autobiography Moab is my Washpot, which, though not featuring much mathematics, does contain the scene where Stephen's terrifying engineer father and self-appointed O-Level mathematics tutor finally breaks through to Stephen and shows him what mathematics is really all about.
When he grasped the completeness of my ignorance and my incompetence he did not gulp or gasp, I'll give him that. He stuck by his own beliefs and went right back to the beginning. He taught me something that I did not understand: the equals sign.

I knew what 2 + 2 = 4 meant. I did not understand however even the rudimentary possibilities that flowed from that. The very thought of an equals sign approximating a pair of scales had never penetrated my skull. That you could do anything to an equation, so long as you did the same to each side, was a revelation to me. My father, never once flinching at such staggering ignorance, moved on.

Then came the second revelation, even more beautiful than the first.

Algebra.

Algebra, I suddenly saw, is what Shakespeare did. It is metonym and metaphor, substitution, transferral, analogy, allegory: it is poetry. I had thought its a's and b's were nothing more than fruitless (if you'll forgive me) apples and bananas.

Suddenly I could do simultaneous equations.
This has also made me realise quite how lucky I was with my school mathematical education. As I've mentioned before, I was happily multiplying matrices at the age of 11 or 12, and inverting matrices of arbitrary size at 15: most of the people in my office didn't encounter them until they were 17 or 18. We had a whole year from 13 to 14 doing random interesting maths while we waited for the syllabus to catch up with us. And solving unseen, difficult problems was always a big component, whether for public-school scholarship exams, Olympiads (I never made the national team, but I made it through a couple of rounds of selection) or just as part of ordinary teaching.

1 My STEP student surprised me the other day by showing me a result he'd discovered for himself. Specifically, given that ab = c for some fixed c, he'd found how to minimise the product ab (I'll let you work it out for yourselves). I had to explain to him that (a) it was very, very cool that he's inventing and solving his own problems at the age of 17, (b) no, this result almost certainly isn't new. The difference between "new" and "original" can be a subtle one at times. But hey, my student did a cool thing! :-)
2 Or rather, because I'm trying to get them to think for themselves, "I've got this pill in my hand, and we want to get it into your stomach. Now, how can we get things into your stomach? With a hypodermic needle, yes, but I don't have one of those to hand. Maybe there's some kind of tube to your stomach we could slide it down? No, no, pull your trousers back up. Is there, perhaps, another tube we could use? And where does it come out? Well, I suppose technically your nose is connected to your oesophagus, but that wasn't really what I was thinking of..."
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Book meme [Oct. 11th, 2007|12:27 am]
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Taken from [info]michiexile, [info]neoanjou and [info]half_of_monty, and probably others.

These are the top 106 books most often marked as "unread" by LibraryThing's users (whenever someone started this meme). Bold what you have read, italicize those you started but couldn't finish, and strike through what you couldn't stand. Add an asterisk to those you've read more than once. Underline those on your to-read list.

( Read more... )

26 completed, 7 given up on, one in progress. There are some excellent books there - I'd strongly recommend almost all of those that I've bolded and not struck out. I'm genuinely surprised that so many people seem to have had such trouble with The Name of the Rose - it's slow going until about 2/3 of the way through, but the final 200 pages or so grab on to you and don't let go. Maybe I found it easier because I'd read enough philosophy to appreciate the joke of them all speaking in a parody of Aristotelian logic?

I try not to re-read books too much: I could very easily slip into re-reading the same few books over and over. And life is short, and unread books are legion. To be honest, I could have underlined nearly everything on the list.
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SICP [Sep. 24th, 2007|12:49 pm]
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I got back from Wales on Tuesday, to find my copy of Structure and Interpretation of Computer Programs (sometimes known as the Wizard book, or SICP, or "sickup", as I've taken to calling it) waiting for me. It's the first-year computer science textbook used at MIT, and early signs are that it's going to be worth its weight in rather good coffee.

( Chapter summary )

I've been working through the exercises, and I'm up to about page 70 (most of the way through Chapter 1). So far, there hasn't been anything really difficult, though a couple of the exercises require you to do Actual Experiments on a computer, and since, what with one thing and another, I don't have a working one at the moment (I'm typing this on [info]wormwood_pearl's laptop), I haven't been able to do these. Some of the algorithms are new to me, and I've particularly enjoyed doing some programming with loop invariants (but how do you find them in the first place? Presumably there are techniques...), but it's hard to shake off the feeling that a lot of it is old hat; I've been programming computers for nearly twenty years, I'm no longer surprised by the concept of a data structure :-) I've enjoyed the philosophical and historical asides - if an algorithm dates back to ancient Egypt, they'll tell you which papyrus fragment it was found on - and the generally thoughtful and informed tone of the book. The progression from procedural to data-driven to metalinguistic programming reminds me of Eric Raymond's The Art of Unix Programming, but this is surely not coincidence - like many expert Unix hackers, Eric was a Lisper first. [info]totherme may be amused to learn that his old tutor, Joe Stoy, is a frequently-cited contributor :-)

1Including some probabilistic algorithms - the ease with which one can roll a die in Lisp is not such a trivial matter :-)
2Plus ways of making change from a dollar or pound - I was amused to note that they haven't updated it to reflect the disappearance of the halfpenny :-)
3They're quite fond of the prefix "meta": for instance, they call an interpreter for language X written in language X a "metacircular interpreter", rather than just a circular one, for reasons which I can't quite fathom.
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Long-term project SITREP [Sep. 10th, 2007|12:39 pm]
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Current status on some of my long-term projects:

Thesis: Up to ( 36 pages )

Birdshot!: I had ( some good ideas for this )

Moving in with [info]wormwood_pearl: ( Nominally moved in at the beginning of August. )

Learning Haskell: I'm going to ( officially give up on this one for now. )

Diet: This is going ( really well. )

Munro-bagging: up to ( 61 out of 284 )

Becoming a l33t martial artist: I've been doing ( some capoeira. )

Learning to juggle 5 balls: I'm getting ( 8 catches pretty consistently. )

Reading Ulysses: Haven't looked at it since I reached ( page 500. ) I seem to have so many other things competing for my attention :-)
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[Jul. 18th, 2007|05:11 pm]
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I've been tutoring a high-school student for his Cambridge entrance exams recently. Yesterday, we had the following conversation1:
Him: I've been reading Fraleigh's book this week.
Me: Sorry, never heard of it. Is that one of the set books or something?
Him: It's called "Introduction to Abstract Algebra" or something like that.
Me: Doesn't narrow it down much, I'm afraid. Have you got it with you?
[He hands it over]
Me: Hmmm. Sets, [flickflickflick] relations, partial orders, equivalence relations, looks fine... [flickflickflick] groups, permutations, cosets, all very standard stuff... [flickflickflick] vector spaces, rings, OK, so far so good... [flickflickflick] Sylow's theorems? That's a bit advanced for an introductory book, surely? [flickflickflick] Simplicial homology? WTF? [flickflickflick] Galois theory?
Him: Do you think that would be useful for my first year, then?
Me: I think that will be useful up to and including your fourth year.
Him: So if I can work my way through it this year...
Me: If you can work your way through it this year, I'll start taking classes from you. So, how are you getting on with it?
[Warning: Actual Maths starts here]
Him: Well, I'm up to the bit about groups, but I need to go over it again.
Me: Yeah, that's par for the course when you're learning from books, I'm afraid. Anyway, groups are interesting because they capture the notion of symmetry in a very general way. You can take any object of any type, and take the transformations that leave it essentially the same, and they will form a group.
Him: What do you mean, "transformations"?
Me: Well, that depends on the type of object. Have you got as far as group homomorphisms?
Him: I've got as far as isomorphisms...
Me: [thinks: wtf?] OK, well a homomorphism is [scribbles definition], and an isomorphism is a homomorphism f such that there's another homomorphism g which is its inverse. [scribbles down, accompanied by "Moral Definition"]
Him: Why have you written "Moral Definition"?
Me: That's because there's an immoral definition, which you sometimes see, which works for groups and rings and things but not more generally. [scribbles down "Immoral Definition: an isomorphism is a bijective homomorphism".] Hmmm, I wonder what Fraleigh uses... [flickflickflick] Aha. "An isomorphism is a bijective homomorphism." Sod. [horrible suspicion starts to dawn...] Let's look at the index... [flickflickflick] "cartesian product, cases: argument by, centre" - oh God, he's written about homology without using categories. In [flickflickflick] 2002! Why would anyone do that? Arararargh!
Anyway, we spent most of the rest of the class talking about field extensions and Galois theory (as examples of places where equivalence relations and groups were useful): I don't know if he understood any of it, but I had fun :-)

1 Lightly edited: we may not have used precisely these words, and the conversation didn't exactly happen in this order, but everything above was said at some point :-)
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The private life of the gerund [May. 16th, 2007|09:32 am]
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The relationship between human languages and programming languages is interesting to me. In general, programming languages are a lot simpler (which is great, because it makes it easier to learn lots of them): they also have rather less vocabulary (but that's OK, because you're actively encouraged to make up your own). We talk about programming "idiomatically": while it's often possible to use one language's idioms in another, they can look almost as odd as, say, Japanese idioms used in English. In the evolution of programming languages, you can see a super-compressed version of the evolution of human languages: ideas, grammatical forms, and vocabulary transfer from one language to another where there's contact between the two communities, and languages split into mutually-unintelligible descendants when the communities fracture. There's at least one example of creolization that I'm aware of: the programming language Perl can be considered a creole of C, shell, sed and awk (and has since absorbed features from many other languages). Perl's also been heavily influenced by that other notable creole, English, and incorporates several features from human languages (like topicalization). It's no coincidence that Larry Wall, the original creator of Perl, studied linguistics at graduate school. In the opposite direction, you'll sometimes hear hackers describing human languages in terms derived from programming languages: Japanese is sometimes said to use reverse Polish notation.

But as I said, programming languages tend to be a lot simpler than human languages. In fact, so-called object-oriented languages (like, say, Java) have been described (perhaps "derided" is a better term) as languages in which everything is a noun: dually, so-called functional languages (like Haskell) could be described as languages in which everything is a verb. Actually, verbs and nouns cover pretty much everything in most programming languages. Perl has pronouns and conjunctions, and pronouns can be added to Lisp with sufficient cleverness; a couple of languages have something vaguely resembling adverbs.

So you can imagine my pleasure at learning yesterday that the language called J has gerunds :-)

*downloads J interpreter*


Fig. 1: The gerund attacks some peaceful pronouns (image courtesy n. molesworth)
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Bifurcation and the logistic mapping [May. 9th, 2007|12:27 pm]
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I've been reading Ian Stewart's book Does God Play Dice? The New Mathematics of Chaos, and I got inspired to play about with some of the stuff described therein. You've probably all seen this image before:



The formula that produces it is almost shockingly simple, and yet the behaviour it produces is really quite surprising: yet another reminder that "complicated" is not the same as "deep".

( The maths behind the picture - generally accessible )

Cool, eh?
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A series of unfortunate occurrences [Feb. 16th, 2007|12:19 pm]
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I got back from the high that was the Scottish Juggling Convention to find the following waiting for me:
  • A big stack of letters from the library, who had decided to cut out the customary "Your book is due for renewal" stage, and proceed directly to the "Return your book NOW or pay the replacement cost (or £20, whichever is higher), and pay a £5 fine. Per book. For eight books" stage. So, I was looking at at least a £40 fine, with a strong possibility of £200, and possibly more. Damn those six-month library loans. Fortunately, I could still find all the books, and subsequent investigation showed that the reason I hadn't got any warnings was because they'd reset my email address from the one I'd specified to one I never checked. With a combination of outrage, politeness, and having Right on my side, I managed to get let off the lot (plus about 80p of fines that I did actually owe them, but I'm not complaining). Hurrah!
  • A lovely email from [info]benparker saying that while Birdshot! showed great promise, and deserved to be finished, Two Shades of Blue would instead be taking The Matrix: The Pantomime to Edinburgh. Which was pretty gutting, but not entirely unexpected: TMTP is a cracking show, it's already finished, and it'll be a lot easier to market than a Dick Barton/film noir spoof. As evidence of this, [info]benparker and [info]r_e_mercia apparently couldn't decide if Birdshot! was a James Bond spoof or a Thirties spoof, even after reading the script. [Hint, guys: it's both, and neither. Try a decade later, and read up on Dick Barton, Special Agent].

    This was doubly annoying, as I'd just missed the deadline for STaG's own Fringe submission process. I should really read those STaG emails more thoroughly...

Anyway, I still intend to finish Birdshot! - now I know its fate for certain, I should be able to do some writing on it again :-). I don't suppose OULES/CULES/BULES/ALES would be interested in doing it as a summer show? Camp spoofery, stage fighting, tormented Private Eyes, beautiful dames, deadly deathtraps, and a Plot To Hold The World To Ransom? The cast can be pretty much any size - there's plenty of scope for doubling-up, and we can add as many henchpersons as we like. I've got about 35 minutes written at the moment, and a fair idea of where the rest is going.
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Pride and Prejudice [Feb. 1st, 2007|12:20 am]
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I've just been watching the film of Pride and Prejudice (the one with Kiera thrice-Knightley as Lizzie), and had rather a shock when I realised that Mr Bingley was being played by someone I went to school with! I think he may even have been in my English class when I read the book for GCSE...

It's surprisingly weird watching him on screen - partly not having seen him for years, but mostly seeing someone I know in an actual film. Not at all like watching a friend on stage.

Anyway, he's a nice chap, so it's good to see he's doing well...
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