Scribble, scribble, scribble

sing

My choir is putting on a "Come and Sing" on the afternoon of Saturday 2008-09-20. Vaughan Williams and (alas) Rutter. Details at the far end of the link in the first sentence. Please come and, er, sing.

While I'm plugging the New Cambridge Singers: we'll be singing Bach's Magnificat in Trinity Chapel on Saturday 2008-11-22. (In the original E-flat version, for those who care.) It'll be good. Please come but preferably don't sing unless you join the choir first.

More consistent notifications of our concerts than you'll get from me are available by email. Let me know if you want to be signed up.

news

I don't normally read the Sunday newspapers, but I was staying with my parents last Sunday and took a look at the papers they'd bought: the Sunday Times and the Sunday Telegraph. Both had about 15 pages of news. The Times had a front-page announcement about Obama/Biden and a short article on some Obama-related subject. Apart from that, every single story in the news section of either paper was UK-specific in some way: mostly things happening in the UK, and a few stories about Brits abroad.

Perhaps there was a separate section called "International News" or something. Even if so: what a parochial notion of "news" these people are working with.

(Probably this is old news to people who read newspapers more than I do. And those wouldn't have been my Sunday papers of choice; maybe some others are different.)

car

I can haz car.

For those who care about such things: It's a Nissan Almera, it's three years old and has about 11k miles on it, and it has lots of bells and whistles that I don't care about.

For those who see the world as I do: It's a car. It appears to get me successfully enough from A to B. I haven't smashed anything or killed anyone with it yet. Yay.

tested

I have been weighed in the balance and not found wanting. Be very afraid.

(In other words: I somehow managed to pass my driving test. This is a Good Thing because my theory test would have expired in three days, and a Bad Thing because I am clearly far from being an expert driver. On the other hand, I do know I'm not an expert driver, which apparently distinguishes me from the majority of people who have just passed their test.)

I have absolutely no idea how one goes about buying a car without spending far too long on it or ending up with something deeply suboptimal. I suspect the answer is that most people spend far too long on it or end up with something deeply suboptimal.

derision

Paul Graham's two latest essays are on Ambition and Distraction. I look forward to his thoughts on Uglification and Derision. (I'd bet a modest sum that the appearance of those topics next to one another is deliberate on PG's part.)

job

The new job is good so far. Really cool technology, interesting problems to solve, good people, still small enough to be fun.

We converted Heather's cot into a bed today. She's currently asleep on the floor of her room. Of course.

Excellent application of MD5-breaking technology.

Further to my earlier desultory comments on zombies, a contrary view. Lanier's main point is, I think, originally due to Hilary Putnam, though quite possibly Jaron Lanier thought of it independently.

I've always had a soft spot for Knuth's "literate programming", but it seems that I'm in a tiny minority. I'm pleased to find that the practice of LP is not entirely dead; for instance, there's a little community of people doing it, apparently for fun, in a wiki at literateprograms.org. But I think the right way to bring wikis and LP together is to make each section be its own page. Perhaps (though I rather doubt it) worthwhile literate programs could then be created in the same sort of informal ways as wikis are. And wrecked in the same sort of informal ways, too.

random

Random state-of-the-world report:

• I have a new job (starting on 2008-06-09): I'll be doing mathsy things for Light Blue Optics. They make holographic video projectors, which is less cool than it sounds (the images are only 2-dimensional) but still very cool.
• I'm learning to drive, finally. It seems to be going quite well so far (deaths: 0; cars written off by insurers: 0).
• Heather is continuing to grow and learn, as small children generally do. It seems very likely that she'll follow in her parents' footsteps and be an early reader.

zombies

About a year ago I quoted an extremely silly philosophical argument about consciousness. It's somewhat related to a more famous but almost equally silly argument in the same field, the one about zombies. ("Quaaaaalia!")

Since it's that time of year again ... Zombies: The Movie.

GENERAL FRED: They behave ... exactly like we do ... except that they're not conscious.

(Silence grips the table.)

COLONEL TODD: Dear God.

[Update, 2008-05-01: This didn't actually appear until several days after it was written, because Easyspace's FTP server was b0rked.]

gods

Yet more NCS notes, for our next concert.

Someone posted a stupid list of alleged pagan parallels to the Jesus story (desired conclusion: the Christians made it all up, basing their stories on earlier myths) in uk.religion.christian, where I still hang out for sentimental reasons. I did a little light debunking. (It probably has mistakes; corrections welcome.)

Wikipedian time travel.

more light

In case the previous hint was too oblique:

Come to my choir's concert tomorrow (if you're reading this the same day as I'm writing it). Church of St Edward King and Martyr, 3pm, Saturday 2008-02-09, about an hour. £7, or £3 if you have the good fortune to be a student and able to prove it.

Which reminds me of a story from many years ago. I was at the railway station in Cambridge and wanted to get a Young Person's Railcard. The rules were something like "you can have one of these if you're N years old or less, or if you're over N years old but a student". So, I go to the window and explain that I'm a student and want a railcard. Do you have the paperwork to prove you're under N years old, sir? Er, um, no, it seems that I don't have the right documents on me. But look, I have this and this and this to prove I'm a student. So can you prove that you're over N years old?

Clearly Bradley (for some reason I have always remembered his name) was a devotee of intuitionistic logic, according to which "A or B" is provable only if either A is provable or B is provable.

light

Once again, some rambling notes on the text for my choir's next concert, at 3pm on Saturday 2008-02-09 at the Church of St Edward King and Martyr in Cambridge, entitled "Towards the light".

Primarily intended for the edification of other members of the choir, but it's possible that others might be interested.

Update, 2008-01-28: I've tweaked the translation of the Bach and fixed a typo in the Morgenlied; thanks to Natalie for spotting opportunities for improvement.

Update, 2008-02-04: one typo fixed, and the hideous paraphrase of the opening of "Hail, gladdening Light" made clearer (but no less hideous).

don

Today is Donald Ervin Knuth's 2·5·7^th birthday.

EXERCISES

1. [70] Have a life at least 1/70 as productive as Knuth's.

know

From Mark Abley's book Spoken Here about endangered languages:

The Inuktitut language goes further, much further. Inuit distinguish utsimavaa (he or she knows from experience), sanatuuq (he or she knows how to do something), qaujimavaa (he or she knows about something), nalujunnaipaa (he or she is not ignorant of something), nalunaiqpaa (he or she is no longer unaware of something), and two or three other verbs that mean, roughly speaking, "know".

Or, as Mark Abley inexplicably failed to put it, the Inuit have eight different words for know.

bags

On a recent trip abroad, I had to choose whether to pack everything into one bag or take two. One factor: how much longer will I wait in baggage reclaim if I take two bags? Making the assumption (implausible for a couple of reasons, but never mind) that bags arrive independently at random times during some fixed interval, it turns out that if you have n bags then the expected (i.e., average) arrival time of bag k is k/(n+1) of the way through that interval. In particular, on average your last bag arrives n/(n+1) of the way through. So if you take two bags instead of one then you wait, on average, for 2/3 of the worst-case time instead of 1/2; you wait 4/3 times as long.

This isn't difficult to prove, but the simplest proof I can find still involves doing some integrals. Isn't there a one-line argument that makes it obvious?

Update, 2007-12-09: A friend emailed me with a nice intuitive proof for the case n=2. I haven't been able to make it work rigorously for arbitrary n, but reflecting on why not leads to this, which isn't a one-liner but at least involves few ideas and no integrals:

Suppose your bags (numbered in advance, rather than in order of eventual arrival) arrive at times t₁...t_n. They're independent and uniformly distributed. Now, suppose you learn that t₁ lies in a certain interval; then the distribution of t₁ conditional on this new discovery is uniform in that interval, and the distribution of all the other t_j is what it was before.

Learning that t₁=t for some particular t *and that t₁ is the smallest of all the t_j* is just a matter of learning that t₁ is in [0,t] and that all the others are in [t,1]; therefore, all the other t_j are now uniformly distributed in [t₁,1], and this is true whatever the value of t was so it's always true.

At this point we've reduced the situation for n to the situation for n-1 and a simple application of induction finishes the job.

yellow

The conclusion of an excellent rant about the appearance of IntelliTXT advertising links in online news articles:

The in-text ad links have a slightly different appearance than the legitimate news-content link supplied by the columnist herself. The advertising links are underlined in green text. The news link is not underlined and it's in blue text.

This is what the distinction between news content and advertising has come down to: the difference between blue and green. I suppose this is what they mean by "yellow journalism."

mean

I recently read the excellent Judgment under uncertainty edited by Kahneman, Tversky and Slovic. I was particularly struck by a simple observation that's mentioned in a couple of the chapters.

Suppose you're a parent or manager or teacher. When one of your charges does something particularly good, you're likely to respond positively (praise, reward, promotion, ...). When they do something particularly bad, you're likely to respond negatively.

Now, suppose that your positive and negative responses are equally effective (or ineffective) in producing improved performance. Because of regression to the mean, they'll tend to do worse after a really good performance and better after a really bad one. Therefore, they'll tend to do worse after you respond positively and better after you respond negatively.

Therefore, we are all repeatedly exposed to misleading evidence that tends to make us think that negative responses are more effective than they are, and positive responses less effective.

(The Wikipedia article on regression to the mean quotes Kahneman making the same point. He calls the moment when he noticed it "the most satisfying Eureka experience of my career".)

ut queant laxis

That was a bit surreal. Radio 3 just played a very nice setting of Ut queant laxis and didn't so much as mention Guido d'Arezzo or solfeggio.

(Apparently some cathedral somewhere has, or had, a musical clock that plays the hymn; Iain Burnside is celebrating The Day The Clocks Go Back with a set of time-related pieces. I don't think the version of the hymn they played has the same melody as the plainchant Guido used.)

change on 2007-10-28 at 01:55:42

Page changed: books/oosc.html

Correct an embarrassing slip that's gone unnoticed for ages:
I wrote "covariant return types" when I meant "covariant parameter types".

change on 2007-10-25 at 23:04:41

New page: thesis.ps.gz

My PhD thesis, which personal/maths.html has long claimed is here
but for some reason never was before.

comments

Right, I'm fed up of Haloscan. I've put together my own commenting system. It's flaky and broken and ugly, and right now it probably doesn't even work at all, but it'll get better. And even if it doesn't it's better than Haloscan.

oak

My choir had a rehearsal today at the chapel at Churchill College. I was pleased to find that there were trees nearby, so that I was able to lean my bike up against some oak.

The story of the founding of the chapel is interesting. Some of the fellows of Churchill were passionately opposed to there being a chapel at the college, and so the chapel is not in fact the chapel of Churchill College; it is owned and operated by a separate trust, and merely happens to be located on Churchill's premises.

This wasn't enough for Francis Crick, who resigned his fellowship in protest at the building of the chapel. He had an exchange of letters with Winston Churchill along the following lines:

Crick: I am resigning my fellowship in protest at the institution of a chapel at Churchill College.

Churchill: I'm sorry to hear that. I don't really understand the problem. The chapel will be an amenity for the benefit of those students who want to use it, and no one else will ever have to set foot in it.

Crick: Very well. I enclose a cheque for ten guineas towards the founding of the Churchill Brothel. I am sure you will agree that there can be no reasonable objection to this; it will be an amenity for the benefit of those students who want to use it, and no one else will ever have to set foot in it.

(Actually Crick, being an erudite chap, called it the College Hetairae. The "cheque for ten guineas" was a reference to the fact that when the colleged had decided some time before that it would not spend any of its own money on a chapel, some eminent chap -- I forget who -- had immediately sent them a cheque for the same sum towards a chapel-building fund.)

notes on texts for notes

As I did for our last concert, I've put together some rambling notes on the texts for my choir's next concert, in the hope that they'll be useful to other choir members. I'm putting a link here in case anyone else is interested.

Update, 2007-10-11: actually, not everything there is in our next concert; in particular, we aren't doing the Bach until December.

Other random remarks:

Robin Hanson has enough faith in markets' ability to make accurate predictions that he thinks they could form the basis for an effective form of government; he also believes that by paying a few hundred dollars per year to Alcor (to freeze his brain when he dies) he's "buying a >5% change of living for thousands of (subjective) years". Unless he really thinks that a few hundred dollars per year is comparable in value to a 5% chance of living for thousands of years (which seems to me like it requires a very steep discount rate indeed), or that Alcor is run by extreme altruists, something's wrong with this picture.

I thought Jim Macdonald's detailed analysis of Betty and Barney Hill's story of alien abduction (from back in 1961), over on Making Light, was rather excellent.

causelation

This is nicely done.

To determine if the inclusion of a meta-analysis in itself is the usual scientific practice, we constructed an exhaustive list of all meta-analyses that don't list themselves (Appendix B).

change on 2007-09-02 at 22:08:14

Page changed: software.html

Bump versions of disarm.c (trivial bugfix) and qsort.c (major bugfix).

fed

(Disclaimer: I am not an economist, nor a political analyst, nor a statistician.)

J K Galbraith (son of the J K Galbraith and a notable economist in his own right) and two co-authors have published an analysis of the US Federal Reserve's monetary policy, purporting to give strong evidence that, since 1983,

• the Fed's goal is to keep unemployment from getting too low, not to keep inflation from getting too high; when both unemployment rates and inflation rates are included in their model, the dependence on the former is much stronger than on the latter (and this is true robustly across multiple models);
• the Fed does not ease its monetary policy when unemployment is very high, as you might expect if it aims to fight recessions;
• the Fed systematically, consistently, manipulates interest rates as presidential elections approach, lowering them when the incumbent is a Republican and (to a lesser extent) raising them when the incumbent is a Democrat;
• this last political consideration plays as big a role in determining the Fed's monetary policy as inflation and unemployment together do.

It seems to me that the first point could have a not-so-odious interpretation: if low unemployment is (or is believed to be) a good leading indicator of high inflation, reacting to low unemployment might be a better way of keeping inflation down than waiting for high inflation rates. Or reacting to some other leading indicator of inflation might look like reacting to low unemployment. However, the paper also purports to show that low unemployment is not in fact a good predictor of high inflation in the future. (Maybe it isn't one because the Fed reacts so promptly to low unemployment. But that seems like a stretch.)

Galbraith talked about this to the House Committee on Financial Services, and his comments offer some useful further insights, especially on inflation.

I wonder what a similar study of the Bank of England would find.

polyhedra

Here's a beautiful little bit of mathematics that isn't as well known as it should be. First, by way of introduction, there's a rather elementary theorem that says: if you have two polygons of equal area, then you can cut one of them up into finitely many polygonal bits and rearrange them to make the other one. Obviously you can't do this if the areas aren't equal, so we have a necessary and sufficient condition. Yay for necessary and sufficient conditions. Now, what about polyhedra?

It turns out that equal volume isn't enough. We might hope to prove this by assigning numbers to polyhedra according to some rule, so that (1) when you put some polyhedra together to make a bigger one the numbers add up and (2) polyhedra of a given volume don't all get the same number, because #1 would mean that two "equidecomposable" polyhedra have to have the same number and then #2 would mean that you can have two polyhedra of equal volume that aren't equidecomposable.

That's more or less what we do, except that what we associate to each polyhedron is a bit more complicated than a single number. The idea, crudely, is to measure how much edge the polyhedron has.

Think about what happens when you chop up a polyhedron into smaller ones. The small ones have, between them, all the same edges as the big one, plus some others (created by the chopping-up) that fit together and thereby disappear in the bigger polyhedron. What we're going to do is to find a way of counting "edginess" so that the new edges all cancel out.

There are two different ways to measure the "size" of an edge. The obvious one is its length. The less obvious one, which will be crucial here, is what's called its dihedral angle, which is the fraction of space taken up by the polyhedron near to the edge. To be more precise: imagine a very thin cylinder with the edge for its axis, and ask what fraction of that cylinder is inside the polyhedron. There may be some funny business at the ends, but as the radius of the cylinder goes to 0 the funny business does too, and the limiting fraction is the dihedral angle -- in slightly unorthodox units, because what I'm calling 1 is usually called 2π.

Let's write L@f for an edge of length L and dihedral angle ("fraction") f. I'll call this the "edginess" of the edge.

Now, we would like the total edginess not to change, or at least to change in a well controlled and well understood way, when we chop a polyhedron up. So we'll need some rules for adding edginesses. For instance, you can cut an edge into two pieces "across" its length, so we want (L+M)@f = L@f + M@f. Or you can cut it "along" its length, producing two sharper edges, so we want L@(f+g) = L@f + L@g. More subtly, you can combine these operations: split it into k pieces lengthwise and stack them "around" the edge, or split into k thinner pieces and glom them together to make an edge k times longer. So we end up wanting kL@f = L@kf for any rational number k.

The set of sums of the form L₁@f₁ + ... + L_n@f_n, with these rules and no others, has the concise mathematical name "R ⊗_Q R". The total edginess of a polyhedron, or a set of polyhedra, is such a thing. This isn't yet quite the right quantity, but we're nearly there.

So, what happens when we chop up a polyhedron into smaller polyhedra? The rules described above account for what happens to the original polyhedron's edges, but there are new edges as well. Each such edge is either internal to the original polyhedron, in which case all the space surrounding it is neatly occupied by the smaller polyhedra around it for a total edginess of the form L@1; or it's internal to a face of the original polyhedron, in which case only half of the space around it is occupied by the smaller polyhedra and we get a total edginess of the form L@1/2, which equals L/2@1 by the third of our rules. If only these were zero we'd have found a quantity that doesn't change when you chop up a polyhedron.

So, make them zero. Instead of working with the fraction f, work with "f mod Q", which just means treating two fractions as equal if they differ by a rational number. In other words, two bits of edge are equivalent if they're the same length and their angles differ by a rational amount. We're now working in "R ⊗_Q (R/Q)". I'll call the quantity we have now the "reduced edginess".

So what we've found is that when you chop up a polyhedron, the total reduced edginess doesn't change. We'll be done if we can find two polyhedra of equal volume but different reduced edginess. It turns out that a cube of volume 1 has total edginess 12@1/4 = 3@1 = 0, whereas a regular tetrahedron of volume 1 has total edginess L@arctan(1/3)/2π where L is the length of one side of that regular tetrahedron (whatever that turns out to be), which isn't 0 because arctan(1/3)/2π is irrational. Therefore, those two polyhedra can't be converted into one another by chopping them into smaller polyhedra and rearranging, even though their volumes are equal.

Even better: volume and reduced edginess between them tell the whole story, as area alone does in two dimensions: if two polyhedra have the same volume and the same reduced edginess, you can convert one into the other by chopping them up.

If you allow arbitrary pieces (not just polyhedra) and believe in the Axiom of Choice, then you don't need either equal volume or equal edginess; any two polyhedra are equidecomposable. (In fact, any two bounded sets with nonempty interior are.) This is one form of the famous Banach-Tarski paradox.

Real mathematicians say "Dehn invariant" instead of "reduced edginess"; the proof is due to Max Dehn, a student of David Hilbert.

i say i say i say

The final sentence of this BBC News article

The explorers also carried with them a church organ from Dorset as a gift to local Bolivians in order to secure their help with finding the meteorite.

... already sounds like the sort of thing contestants on I'm sorry, I haven't a clue or My Word might be required to work into a story ("... and when one of the contestants manages to say his line, I'll do this * H O N K * ... or else I might blow my horn").

But, as if that weren't enough, the actual subject of the article is the alleged spotting by Colonel John Blashford-Snell of a Crumple-Horned Snorkack Double-Nosed Andean tiger hound, a dog with two noses.

It should come as no surprise that Blashford-Snell is the Honorary Life President of the Centre for Fortean Zoology.

draw

Gosh.

(I'm a couple of months behind the times with this.)

signs and wonders

On a recent bus journey, I passed two churches with signs outside them. One said:

Whatever your place
Tend it with care
God put you there

and the other said:

Reason is the worst enemy faith has

I am not, in general, an atheist of the hostile or crusading sort. But sometimes I do get rather cross.

newcomb

I discovered a little while ago that at least one intelligent and well-informed person who sometimes reads this stuff had never come across Newcomb's paradox before. It's worth knowing about, if only because it will make your head spin.

Imagine that there is some being (call it Bill) with a well-established ability to predict people's behaviour. In particular, Bill has demonstrated to your satisfaction that he can predict, hours in advance and very reliably (let's say well over 99%), what you will do in a wide variety of situations, and he has demonstrated a similar ability to predict other people's behaviour in the exact situation you're about to be placed in, which is this:

In front of you are two boxes. One is made of glass; you can see into it. It contains a cheque for £1000. The other is made of steel, and welded shut; you can't tell what's in it. Bill has put in it either a blank piece of paper or a cheque for £1000000. You may take (and keep) both boxes, or just the opaque one. But here's the rub: Bill has used his pred1cti0n sk1llz and has put the big cheque in the steel box if he thinks you'll take only that box, and the blank piece of paper if he thinks you'll take both. What do you do?

Let me recap, in two different ways suggestive of two different answers.

• There is almost certainly £1000000 in the steel box if you choose to take only that box, and almost certainly nothing of value in it if you choose to take both boxes. (Therefore: Take only the steel box. Obviously.)
• There may or may not be £1000000 in the steel box, but there is certainly £1000 in the glass box, and whatever is in the steel box there is therefore £1000 more in the two boxes together than in the steel box alone. (Therefore: Take both boxes. Obviously.)

Almost everyone finds it entirely obvious what it's rational to do in this peculiar situation. Unfortunately, there is no agreement on which of the two courses of action is the obviously rational one.

When we ask questions like "What should I do in this situation?", I think we are implicitly operating with a possibly-naïve notion of how the world works, and if Bill's predictive skills are possible then it's definitely too naïve. That notion is as follows: There are various ways the world (past and present) could be, all of which look just like the actual world in the past, and which differ in the future according to your choice; all the future differences are consequences of your choice and could in principle be traced back to that choice itself, via causes flowing forward in time.

Generally, this view of things works well. But in a world containing Bill, it breaks down: your future choices are strongly correlated somehow with things in the past, and by Bill's clairvoyant or simulatory prowess those things in the past are in turn strongly correlated with something else contemporary with (even preceding) your choice, with macroscopic consequences. Thus, our key assumption – that all the differences between the hypothetical futures of the world in which you chose differently flow from that choice – fails.

(Very likely it fails in the real world too, but so far as we can tell it generally fails benignly, or at least in ways we don't see because our powers of perception and prediction are unlike Bill's.)

But that assumption is a fundamental part of what we mean by asking questions like "What should you do?". It's hard to answer this question when it concerns Newcomb's situation because one of the question's presuppositions is false. It's like asking what colour electrons are.

You may notice that the question I actually asked was "What do you do?". Not being Bill, of course, I don't really know, even if "you" means me. But I'm pretty sure that I take only the steel box. Is that a rational decision? I think the question has little meaning. But it's rational at least in this sense: making that decision is strongly predictive of getting £1000000 instead of £1000.

random

Yesterday there was a little festival sort of thing in Cambridge, where we saw (inter alia) a Rastafarian sheep and a business called "Nutty Tarts" which disappointingly sells nutty tarts.

Elsevier have finally decided to get out of the arms trade.

I wrote up some rambling notes on the texts for my choir's forthcoming concert, in the hope that they might be useful to other choir members; but who knows, they might be of interest to someone else, hence the link here.

(For geeks only.) Draw a triangle (any triangle will do, but it'll be prettiest with an equilateral one). Divide each side in the ratio 1:2. Join each division point to the opposite vertex of the triangle. You'll get a smaller triangle in the middle. What's the ratio of its area to that of the original triangle? What if you divide the sides in some other ratio? This is an old question, and it's easy to solve ploddingly with coordinate geometry. Someone pointed me at figure 5 in George Hart's explanation of his lovely "artificial radiolarian reticulum" (actually, the someone was George Hart himself), which provides a near-instant proof once you check the side-length of the outer triangle. Well, it turns out that this generalizes nicely. (I expect George knew this when he drew that diagram.) So: draw an equiangular hexagon (all interior angles 120 degrees) with sides p, q, p, q, p, q. Label the vertices ABCDEF. Join AC, CE, EA, giving an equilateral triangle of side sqrt(p²+pq+q²). Join AD, CF, EB, producing (1) an equilateral triangle of sides |p-q| in the middle and (2) a subdivision of each edge of ACE in the ratio p:q. Conclusion: The ratio of areas is

synchronicity

It's curious the way things happen in parallel.

Case 1: Conway's "angel problem", open for for 25 years, suddenly seems to have been independently solved by four different people: András Máthé and Oddvar Kloster have proofs for an Angel of power 2, Brian Bowditch for an Angel of power 4, and Péter Gács for an angel of unspecified large power.

(Brief sketch of the problem: consider a game played by two players on an infinite chessboard. Play alternates. The first player has a single piece, called an Angel. On each move it can hop to a new square provided it doesn't move more than k units in either direction. The second player has an unlimited supply of pieces; on each move s/he places one of these pieces on any unoccupied square. The Angel isn't allowed to move to a square with a piece on. The first player wins if the Angel can keep moving for ever; the second player wins if s/he can box the Angel in. Who wins? Answer, it turns out: the Angel, even when k is as small as 2.)

The two strategies recently published for the Angel of power 2 are even very similar to one another: declare a connected region of the board (initially one half of it) unavailable, walk along its edge, and modify the region as squares get eaten. Perhaps the idea of this approach has been in the air for a while?

Case 2: within a couple of weeks of one another, two friends of mine independently make blog posts describing (or, in the latter case, referencing a Wikipedia description of) substantially the same programming technique and remark that it's a bit obscure and should be better known. The first happens to have a slightly more sophisticated version of the technique.

Edited, 2007-05-25, to fix a typo and to note:

Case 3: Cases 1 and 2 occurred at pretty much the same time. (I completely failed to notice this until a friend pointed it out.)

Case n+1: Cases 1, 2, ..., n occurred at pretty much the same time.

advertising at its stupidest

In Cambridge, they're building this new shopping-centre thing called the "Grand Arcade". The hoardings bear (several times) the following slogan:

I suppose that if they'd changed it slightly...

... then it would arguably have been rather clever, albeit in a stupid sort of way (absolute value, geddit?), but never mind.

some books

Some books are to be tasted, others to be swallowed, and some few to be chewed and digested;

My daughter agrees. Fortunately she hasn't got much further than tasting so far.

inconceivable

What odd things some philosophers think. Victor Reppert quotes a book by Edward Feser on the philosophy of mind, which attributes the following argument to W D Hart. (So apparently at least three philosophers take it seriously.)

[...] you can imagine that what you see in the mirror is not even a headless body, but nothing more than the wall behind you and no body at all [...] But seeing is a mental process, as is the frenzied thinking you'd now be engaging in; which means that what you've conceived of is your mind existing apart from a body or brain. So again, it's conceivable that the mind exists apart from the brain -- in which case they are not identical.

Lest there be any doubt about what's being said here, Reppert expands on it in his comments:

If the mind is identical to the brain, then the mind is necessarily identical to the brain. If the conceivability of the mind's existence apart from the brain entails the metaphysical possibility that the mind and brain are not identical, then the mind and brain are non-identical, since identity claims are necessarily true, and their denials necessarily false.

It's a neat trick, isn't it? Let's see what else we can prove this way. I can imagine electric current flowing without any charged particles being involved; therefore electric current is not identical to a flow of charged particles. I can imagine my computer continuing to do its processing without its circuitry and the things that happen therein; therefore what accomplishes my computer's processing is not identical with its circuitry and the things that happen therein.

One might hope that this is only meant to establish that there could be minds that aren't brains; I haven't read Hart or Feser, but Reppert calls it "an argument for dualism". Oh dear oh dear oh dear.

A few other comments: (1) I thought this argument went all the way back to Descartes, but I think Reppert is a Descartes expert and he didn't mention Descartes so it probably doesn't. (2) Reppert's expanded version of the argument is a nice illustration of what a mess the notion of de re necessity can get you into. (3) I am not claiming that the mind is identical to the brain, just pointing out what a silly argument this is. I think it's nearer the mark to say that the mind is an activity of the brain, or a pattern in the brain, or a pattern in the activities of the brain, or something of the sort; if the Hart/Feser/Reppert argument were valid, it would rule those possibilities out too.

halo

So, as of approximately now I have comments, thanks to Haloscan. I'm afraid you don't get the how-many-comments notification; HS's code for doing this uses document.write in a way that completely breaks in my web browser. Haloscan is a bit icky, but it'll do.

I have moderation turned on for comments here, in an effort to mitigate spam. I'll see how that goes.

when i survey

Percentage of the US population who, according to a recent poll, ...

was it a saw i saw

(No, it isn't quite a palindrome. Too bad.)

A friend of mine found yesterday that some malefactor had attached an extra lock to his bicycle. So, in the middle of town, in public view, he set to work with a hacksaw. It took him 15 minutes to saw through the lock. In that time, only one person made any attempt to challenge him. Still, it could have been worse.

resurrection

A parable

Let us suppose that you have an interest in the history of science. While browsing in a bookshop, you come across three biographies of a chap called Joe Bloggs. They all claim that he was the greatest scientist of his day (a few hundred years ago), and that he developed an astonishing new theory of physics that works much better than the theories that prevail today, which enabled him to transport objects faster than light. Apparently he demonstrated this to a few groups of his students, but the demonstration has never been repeated.

It turns out that none of the other reference works you have access to even mentions Bloggs's work. (A couple mention that his partisans speak very highly of him.) Still, there's no good reason to think they're complete, and you'd like to give these biographies a fair hearing. How can you assess their claims?

Are their authors credible?Well, on further investigation you find that all the biographies are published by the Joseph Bloggs Society, an organization whose declared aim is "to further the reputation of Joe Bloggs, the greatest scientist of his generation". Two of them are anonymous, so you can't check up on the authors' credentials; basically nothing is known about the third author other than that he wrote a number of books for the Joseph Bloggs Society. Furthermore, you notice a lot of near-identical passages in two of the books; it seems that one has cribbed greatly from the other, or both are copying some earlier work. And it's not clear that any of the authors ever met Bloggs, saw his demonstrations, or read his scientific publications. Hmm.

Do they agree with other information you already have? They don't really have much to say about anything other than Bloggs, so it's hard to tell for sure. But, basically, sometimes they're right and sometimes they're wrong. (Sometimes quite badly wrong.)

Do they agree with one another? No, it turns out that they frequently disagree. On the particularly important question of Bloggs's allegedly world-class scientific research, there's hardly any point on which they are in clear agreement. Two of the books allege that his faster-than-light demonstration was shown to be genuine by a clear instance of time travel, but the third doesn't mention that. They do broadly agree that his scientific work was very hard to understand and that even his own students often completely misunderstood his papers until he explained them.

Is what they say plausible in itself? Much of what they say about Bloggs's life is plausible enough, though it's not too encouraging that one of them says he was married twice and had three children, another says he renounced all human relationships to devote his life to his research, and the third doesn't even mention his family. But there's that thing about faster-than-light travel, which you feel could do with a little more evidence. Oddly, some senior members of the Joe Bloggs Society now say that actually he never exactly developed a practical means of faster-than-light travel, but he did develop new ways of thinking about the possibility and so in a very real sense he did achieve it.

Also, one of the biographies says that Bloggs gave a public demonstration of levitation and another that his work was the cause of large-scale rioting; odd, given that no other record of such things remains.

Gentle reader: in this situation, is there anything that those biographies could contain that you'd regard as sufficient evidence that Bloggs had a correct and revolutionary theory of physics that enabled him to make things travel faster than light?

Update: I find that I've been misinterpreted by at least one intelligent and sensible person, so let me clarify: this is not meant to be a refutation of Christianity, or anything of the kind; it's pointing out the weakness of any argument for the resurrection that has the form "we have these accounts of what happened, and the only decent way to explain them is to say that Jesus was really raised from the dead". This isn't a straw man; for instance, N T Wright's recent book on the resurrection makes an argument of just that form. And of course this sort of argument is a staple of less-intellectual apologetics, as with McDowell or Morison or Strobel. (Disclaimer: It's some time since I read McDowell, and with Strobel I'm going on the basis of the publisher's summary; and of course both have arguments for Christianity other than ones based on the Resurrection.)

gjm as gtm

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Bela Bollobas's Modern Graph Theory. I am an in-depth account of graph theory, written with the student in mind; I reflect the current state of the subject and emphasize connections with other branches of pure mathematics. Recognizing that graph theory is one of several courses competing for the attention of a student, I contain extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. Which Springer GTM would you be? The Springer GTM Test

(See also everything2 for some nice commentary.)