Frederik Pohl on science fiction

Frederik Pohl is one of the important writers and editors from the “Golden Age” of science fiction. He began writing in 1937, and seventy years later is still going strong. I read his somewhat-obscure story Shaffery Among the Immortals when I was 12 (my own Golden Age), and I have long considered it the funniest story about a scientist ever written. (Disagreements most welcome.) Of course, at that time, I imagined all scientists to be heroic characters, every one a genius—so the story was indeed all fiction. It seems less funny when I reread it as a struggling graduate student myself. :-)

In his introduction to the collection Day Million, Pohl offers the following pleasant description of science fiction. Writing in December 1969, he notes:

I put “science fiction” in quotes because I’m not always sure what that is, either. “Science fiction” is a poor name for a field of writing. It would be an uninspired one even if it were exact, and of course it is far from exact: there is a great deal of “science fiction” that doesn’t contain any science at all. (You will find some specimens herein.) But it is not entirely a misnomer, because just as “science” is a state of mind and a systems approach to inquiry rather than test tubes and facts, so “science fiction” is a way of writing stories. Harlow Shapley, talking of something else, once described this perspective as “the view from a distant star”. It is a look at the human race and all its affairs from outside.

One of the most popular sports at science-fiction gatherings is defining science fiction: it is that kind of story which deals with events that may happen, but as far as we know haven’t happened; it is that kind of story which takes some real event or trend and extrapolates it to its logical conclusions; it is that kind of story which would not exist if it were not for some central supposition which is based on scientific theory. Et cetera. I play this game as little as possible, because I am an inclusivist and try to avoid setting up barriers which might make me refrain from buying a story for a magazine or anthology I may be editing, or refrain from writing a story of my own, because it could be excluded by one of those barriers. But I do have a suggestion toward a definition, which seems to me attractive if only because it employs that favorite writer’s trick of standing a question on its head. It goes like this:

A science fiction story is that story which might really occur anywhere in space and time, except that stories of our own real world are a special and less imaginative kind of science fiction.

In those terms, Hamlet and War and Peace and Little Women are examples only of a subclass within the general framework of sf. For reasons of vanity as a science fiction writer, it gives me some pleasure to think that this is so. But vanity is not the only reason. Our world is but one of a very large number of planets—no one on Earth knows exactly how many there are, but a reasonably good guess puts it at 60 million or so—in our own galaxy, which in turn is but one of some hundreds of billions of galaxies in the universe. The events that Shakespeare and Tolstoi and Louisa May Alcott wrote about pertain to the history and customs of a certain kind of vertebrate, mammalian, carbon-based, two-sexed, air-breathing creature. We know a great deal about this species, and we can see how complex its societal and ecological systems are; but since they are the systems we move in, it is hard for us to see that they are the product of chance. Science fiction gives us the perspective that makes the job a little easier. Not always perfectly, in fact not always even very well, it does give us a look at our churches, politicians, addictions, morals, family relationships, vices and pleasures from the point of view of a frame of reference that takes none of them for granted.

When you think of how many millions of human beings have shot, stabbed, gassed, clubbed and burned other human beings because they thought their way of life was the uniquely best and proper one, it appears that this point of view could have saved us all a lot of heartburn over the centuries. It could save us some right now.

As I write, we are in the last days of the year in which human beings first walked on the surface of another world. It’s only the Moon. It’s really just our own back yard. There’s not much there to want, and little enough of even that worth the cost of hauling back to Earth.

But it’s a doorstep to the universe, and out there are many very wonderful things indeed, and to reach and master them we shall probably need all the wisdom and objectivity and freedom from parochial prejudice we can come by. We need them badly enough here on Earth, heaven knows, and if science fiction can help us attain these goals, it will have done more than a good many Messiahs.

Who said that?

One of my favourite pastimes obsessions is trying to find the correct attribution for quotes, phrases, stories, etc. [For example, “Premature optimization is the root of all evil” was said by Knuth, not Hoare; I haven't been able to track one Asimov quote despite a bit of looking, etc.]

Quite by coincidence, I found in the last hour three great webpages doing an impressive and thorough-looking (but alas, we’ll never know when something is thorough) job of it:

• Alan P. Scott trying to track down “Talking about music is like dancing about architecture”
• Martin Porter trying to track down “All that is necessary for the triumph of evil is that good men do nothing”. (It has so many variations so frequently quoted that A Bit of Fry and Laurie in Series 4 had “For evil to flourish, all that is required is for good men to spout clichés”.)
• Duen Hsi Yen on The Blind Men and the Elephant

There’s also a book by Ralph Keyes called The quote verifier: who said what, where, and when — looks great, must read.

If you have more examples, please let me know.

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More examples:

• Jeffrey E.F. Friedl on ‘Some people, when confronted with a problem, think “I know, I’ll use regular expressions.” Now they have two problems.’ (thanks to Karthik below)
• William J. Rapaport on Buffalo buffalo buffalo buffalo buffalo (it’s his own sentence, so perhaps it shouldn’t count)

Extreme image compression: the Twitter challenge

If a picture is worth a 1000 words, how much of a picture can you fit in 140 characters?

Mario Klingemann (Quasimondo on Flickr) had a fascinating — call it crazy if you like — idea: can you encode an image such that it can be sent as a single Twitter message (”tweet”)? Twitter allows 140 characters, which seems like nothing. It’s pretty much guaranteed that you’ll be able to get nothing meaningful out of so few bits, right?

Well, he came up with this, using a bunch of clever tricks: using the full Unicode range for “characters” (Chinese, etc.) to squeeze a few more bits’ worth, representing colours as blends of just 8 colours (3 bits!), and arriving at a Voronoi triangulation through a genetic algorithm:

“Mona Tweeta” © Quasimondo:Flickr/CC-BY-NC (210 bytes?)

The one on the right is the real Mona Lisa, and the left one is what fits in 140 characters, specifically the message: “圑嘌婂搒孵怤實恄幖戰怴搝愩娻屗奊唀唭嚟帧啜徠山峔巰喜圂嗊埯廇嗕患嚵幇墥彫壛嶂壋悟声喿墰廚埽崙嫖嘵奰恛嬂啷婕媸姴嚥娐嗪嫤圣峈嬻尤囮愰啴屽嶍屽嶰寂喿嶐唥帑尸庠啞彐啯廂喪帄嗆怠嗙开唅恰唦慼啥憛幮悐喆悠喚忐嗳惐唔戠啹媊婼捐啸抃岖嗅怲幀嗈拀唹坭嵄彠喺悠單囏庰抂唋岰媮岬夣宐彋媀恦啼彐壔姩宔嬀”

This is pretty impressive, you’d think, for 140 characters. But it gets better. Brian Campbell started a contest on Stack Overflow, and some brilliant approaches turned up.

Boojum wrote a nanocrunch.cpp, based on fractal compression, which can do this (on the left is the original, for comparsion):

by Boojum (490 bytes)

Sam Hocevar wrote img2twit, which segments the image into square cells and tries to randomly assign points and colours to them until something is close. It can do this:

img2twit by Sam Hocevar (250 bytes?)
You can watch a movie of the image evolving; it’s pretty cool!

There were also attempts at converting the image to a vector format and encoding that instead. Needless to say, it works well for vector-like images:
(almost perfect!)

but it’s hard to even convert some images to vector form:
by autotrace (before compression!)

Finally, this is how Dennis Lee’s record-holding “optimizing general-purpose losy image codec” DLI does:

Comparison: JPEG at 536 bytes, img2twit at 534 bytes, DLI at 534 bytes

Or if you want to be fair and compare at 250 bytes, here’s img2twit and DLI:

img2twit at ~250 bytes, DLI at 243 bytes

Amazing!

For silly amusement, you can read a liberal translation of the original message, or the Reddit thread with ASCII porn.

Disclaimer: I did not participate in any of this, and I know nothing about image compression, so no doubt there are errors in the above. Please point them out. All images are copyright the respective owners, and the quote in the first line is by Brian Campbell on Stack Overflow.

Reading aloud

There’s this poem, which you can read comfortably but trying to read which aloud is torture:

I take it you already know,
Of tough and bough and cough and dough.
Others may stumble, but not you,
On hiccough, thorough, laugh and through.
Well done! And now you wish, perhaps,
To learn of less familiar traps.

Beware of heard, a dreadful word,
That looks like beard and sounds like bird.
And dead: it’s said like bed, not bead —
For goodness’ sake, don’t call it ‘deed’!
Watch out for meat and great and threat,
(They rhyme with suite and straight and debt).

A moth is not a moth in mother,
Nor both in bother, broth in brother.
And here is not a match for there,
Nor dear and fear for bear and pear.
And then there’s dose and rose and lose –
Just look them up – and goose and choose.
And cork and work and card and ward,
And font and front and word and sword.

And do and go and thwart and cart –
Come, come, I’ve hardly made a start!

A dreadful language? Why man alive!
I’d mastered it when I was five.

Alternative last verse:

A dreadful language? Why, man alive!
I’d learned to talk it when I was five.
And yet to write it, the more I tried,
I hadn’t learned it at fifty-five.

So this poem is about spelling not corresponding to pronunciation and vice-versa. Wikipedia cites it as “From a letter published in the London Sunday Times in 1965 [...] The author was only listed by T.S.W.”, but it’s at least as old as 1961, possibly much older.

The other poem is called The Chaos, and it’s by the Dutch teacher Gerard Nolst Trenité, illustrating how impossible it is to deduce pronunciation from spelling. It makes you doubt the pronunciation of many words you think you know. :-) He first published it in 1920, with 164 lines, and revised it many times until his death in 1946 (274 lines). It’s quite painful to read, so skim when it gets unbearable and save the rest for another sitting. Prof. David Madore has a version here, with the first few verses in IPA for AmE and BrE. What follows is a random excerpt only!

Dearest creature in creation
Studying English pronunciation,
I will teach you in my verse
Sounds like corpse, corps, horse and worse.

I will keep you, Susy, busy,
Make your head with heat grow dizzy;
Tear in eye, your dress you’ll tear;
Queer, fair seer, hear my prayer.
[...]
Sword and sward, retain and Britain
(Mind the latter how it’s written).
Made has not the sound of bade,
Say—said, pay—paid, laid but plaid.

Now I surely will not plague you
With such words as vague and ague,
But be careful how you speak,
Say: gush, bush, steak, streak, break, bleak,
[...]
Say, expecting fraud and trickery:
Daughter, laughter and Terpsichore,
Branch, ranch, measles, topsails, aisles,
Missiles, similes, reviles.
[...]
Billet does not end like ballet;
Bouquet, wallet, mallet, chalet.
Blood and flood are not like food,
Nor is mould like should and would.
[...]
Liberty, library, heave and heaven,
Rachel, loch, moustache, eleven.
We say hallowed, but allowed,
People, leopard, towed but vowed.
[...]
Stranger does not rhyme with anger,
Neither does devour with clangour.
Pilot, pivot, gaunt, but aunt,
Font, front, wont, want, grand and grant.
[...]
Say inveigh, neigh, but inveigle,
Make the latter rhyme with eagle.
Mind! Meandering but mean,
Valentine and magazine.
[...]
Don’t be down, my own, but rough it,
And distinguish buffet, buffet;
Brood, stood, roof, rook, school, wool, boon,
Worcester, Boleyn, to impugn.
[...]
Nor are proper names included,
Though I often heard, as you did,
Funny rhymes to unicorn,
Yes, you know them, Vaughan and Strachan.
[...]
Please don’t monkey with the geyser,
Don’t peel ‘taters with my razor,
Rather say in accents pure:
Nature, stature and mature.
[...]
Though the difference seems little,
We say actual, but victual,
Seat, sweat, chaste, caste, Leigh, eight, height,
Put, nut, granite, and unite.
[...]
Say aver, but ever, fever,
Neither, leisure, skein, receiver.
Never guess—it is not safe,
We say calves, valves, half, but Ralf.

Starry, granary, canary,
Crevice, but device, and eyrie,
Face, but preface, then grimace,
Phlegm, phlegmatic, ass, glass, bass.
[...]
Mind the O of off and often
Which may be pronounced as orphan,
With the sound of saw and sauce;
Also soft, lost, cloth and cross.

Pudding, puddle, putting. Putting?
Yes: at golf it rhymes with shutting.
Respite, spite, consent, resent.
Liable, but Parliament.

Seven is right, but so is even,
Hyphen, roughen, nephew, Stephen,
Monkey, donkey, clerk and jerk,
Asp, grasp, wasp, demesne, cork, work.
[...]
Pronunciation—think of Psyche!—
Is a paling, stout and spiky.
Won’t it make you lose your wits
Writing groats and saying ‘grits’?

It’s a dark abyss or tunnel
Strewn with stones like rowlock, gunwale,
Islington, and Isle of Wight,
Housewife, verdict and indict.

Don’t you think so, reader, rather,
Saying lather, bather, father?
Finally, which rhymes with enough,
Though, through, bough, cough, hough, sough, tough?

Hiccough has the sound of sup.
My advice is: GIVE IT UP!

There’s another version here.

All of which reminds me of the following (true) story about reading aloud. In the 4th century, where apparently it was common practice for everyone to read aloud, St. Augustine encountered a man (Bishop Ambrose) who read silently! He didn’t even move his lips! You couldn’t hear his voice while reading even if you stood very close to him! As Augustine reports (and it’s a matter of debate whether in amazement or in distaste):

When he read, his eyes scanned the page and his heart sought out the meaning, but his voice was silent and his tongue was still.

(Might want to take a look at this chapter from Alberto Manguel’s A History of Reading.)

Finally, if I may rant again about spelling pronunciation: the character ~ is written “tilde”, but I wish people would stop calling it “till-day” or “tilled”! It is pronounced “til-duh”, as in the Australian ballad: “Waltzing Ma~, Waltzing Ma~…” or the name of the actress ~ Swinton (literally?)

TODO: Read about “the very notion of silent, individualized reading is scarcely known prior to the advent of the printing press (Goody and Watt: 42)” That’s Goody, Jack, and Watt, Ian, 1968, “The Consequences of Literacy.” In Literacy in Traditional Societies, edited by Jack Goody, pp. 27-68. Cambridge University Press. This is from Thomas Coburn, “Scripture” in India: Towards a Typology of the Word in Hindu Life, p.437. He goes on:

there has never been a happy marriage between the holy words of India, composed and transmitted orally, and the writing process. Particularly in contrast with, say, China, scribes in India have been of low social standing (Lancaster: 224-25), and the very act of writing was held to be ritually polluting: a late “Vedic text, the Aitareya Aranyaka (5.5.3) states that a pupil should not recite the Veda after he has eaten meat, seen blood or a dead body, had intercourse or engaged in writing” (Staal, 1979:122-23). The profoundly spoken character of India’s holy words is a matter on which we will reflect below, but for the moment it will suffice to note that we should not be misled by the fact that most of these words have eventually found their way onto the written or printed page. This is not their primary home, and Staal is not simply being mischievous in discerning a symbolic significance to the fact that Indian books “still tend to fall apart” (1979:123).

mplayer: changing speed without changing pitch (avoiding the chipmunk effect)

You can change the playback speed with [ or ], but that probably changes the pitch as well (naturally). Can be amusing the first time, but not after you realise that it is actually possible to do something sophisticated to avoid this. (Wikipedia calls it Audio timescale-pitch modification) I’ve heard Windows Media Player(?) can do this automatically (is this true?), here’s how to do it in mplayer.

Short answer:

Start mplayer as mplayer -af scaletempo
That’s it. The catch is that you need to get an mplayer which has the scaletempo filter, and we know how much the mplayer project loves making releases. (It’s not in Ubuntu at the time of writing.)

So, either

Get such an mplayer
e.g the deb from Sourceforge (here),

or

Start mplayer as mplayer -speed 1.5 -af ladspa=tap_pitch:tap_pitch:0:-33:-90:0 foo.avi

Seems even the latter might require installing the ladspa plugins.

For more on all this, see:

1. blog comments (with patches) at Pitch-Correct Play Speed with MPlayer
2. Change MPlayer Playback Speed
3. Mplayer FAQ: “How do i change mplayer speed but keep pitch the same?”

First thoughts on Google Wave

Just saw the demo for Google Wave. It’s impressive and ambitious. It’s hard to describe, but it’s a collaborative real-time thing (think Google Docs for everything) that can work like email, IM, blogs, forums, whatever you want — and can be embedded into, or integrates with, apparently everything: Orkut, Blogger, Google Maps, Google Code (the bug tracker), Twitter, etc. (They’ve already fulfilled the annoying word requirement, by creating “twave”.)

They say it’s a “product, platform and protocol”.

I can see myself using this. (And thinking of the privacy implications (or the having-your-data-out-there-in-the-cloud-somewhere implications), it’s bloody scary.)

They’ve got pretty amazing sync. Search results and messages get updated in real time character-by-character, and the latter seems to make people cheer as if they’ve never seen good old talk.

Finally someone had the “playback” idea I have been trying to propose for years. (I was calling it the “undo bar” or “edit history bar”, or more recently “Time Machine for Emacs”, but whatever.) You can “play back” the edit history of a document (”wave”), seeing what changes each person made and in what order, and when the “wave” is a chess game, you can play back the chess game. Perfect.

They variously say it will be open-sourced, or that “a lion’s share of the code” will be open-sourced, but let’s hold off believing that until we see it. It’s extensible, so you can add your plugins to it. It’s a protocol, so you can write your own implementations of it. It’s a platform, so you can run it on your own servers. Now someone add a LaTeX compiler to it, and collaborative work with LaTeX will finally be possible.

If you have 80 minutes to spare, here’s the video, or an article at TechCrunch.

Giving credit

V. I. Arnold, On teaching mathematics:

What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? [...]

What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion and that the “modern” definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.

For what sins must students try and find their way through all these twists and turns? Actually, in Poincaré’s Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the “abstract” one.

(Interesting talk, do read.)

Meanwhile…

Bill Poser at the Language Log:

Sir William Jones is incorrectly viewed as the discoverer of the Indo-European language family and founder of modern historical linguistics [...]

The second and more important point is that Jones cannot be considered the founder of modern historical linguistics because he did not use the comparative method, the crucial innovation that distinguishes modern historical linguistics from its predecessors.

Sigh. Let’s not forget people who actually caused us to perceive the world differently, and leave it to pedantic types to define who invented what.

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Music and lyrics

I attended a talk today by Adriano Garsia, which was part of the MIT combinatorics seminar. It was called “A New Recursion in the Theory of Macdonald Polynomials”, and while I didn’t know what Macdonald polynomials were, I went to the talk anyway, because I like polynomials and I like recursion and I like combinatorics (but primarily because it was a way of procrastinating). :-)

Even though I understood almost nothing of the deep mathematics in the talk (and still don’t exactly know what Macdonald polynomials are), it was a very pleasant and refreshing talk, and I felt very good after hearing it. The reason is that it had, of all the talks I’ve attended in recent memory, probably the best “music”. What does that mean? As Prof. Doron Zeilberger invented the term:

Human beings have bodies and souls. Computers have hardware and software, and math talks have lyrics and music. Most math talks have very hard-to-follow lyrics, [...]

But like a good song, and a good opera, you can still enjoy it if the music is good. The “music” in a math talk is the speaker’s enthusiasm, body-language, and off-the-cuff heuristic explanations.

Sometimes you can recognize a familiar word, and relate it to something of your own experience, whether or not the meaning that you attribute to it is what the speaker meant, and this can also enhance your enjoyment.

(read more at Zeilberger’s Opinion 78)

And so it was with this talk. Prof. Garsia clearly loved the subject, and even someone like me who had no idea what’s going on felt compelled to listen, fascinated. He told us how the problem came about (”long relationship with Jim Haglund: he makes brilliant conjectures and I prove them”), of false proofs they had had, of how their current proof was driven by heuristics and unproven conjectures, he even posed a problem and offered a $100 reward for an elementary/combinatorial proof. :-)
Far better than the talks with bad music and bad lyrics. (It also helped that although I couldn’t understand the lyrics, they sounded nice: permutations, Young tableaux, polynomials defined in terms of them…)

Edit: See also the recent research ’showing’ that gestures help students learn mathematics.

Convex and concave

A mnemonic for ‘convex’ and ‘concave’:

A convex function

A conCAVE function

Two mnemonics are better than one.

Good Will Hunting: a mathematician’s review

Good Will Hunting, reviewed by Mark E. Saul in the Notices of the American Mathematical Society, April 1998. (The most interesting part is Daniel Kleitman’s box on “My Career in the Movies”; also, please read the book review of The Curious Incident of the Dog in the Night-time if you’ve read the book.)

It feels very good to read the Notices. All its issues since 1995 are available online.

“Every good theorem must have a good counterexample”

Abhyankar[1] attributes the quote to Severi.

[1]: Historical Ramblings in Algebraic Geometry and Related Algebra, Shreeram S. Abhyankar, The American Mathematical Monthly, Vol. 83, No. 6 (Jun. – Jul., 1976), pp. 409-448. Also available here, because it won a Lester R. Ford Award (”articles of expository excellence”) and also a Chauvenet Prize (”the highest award for mathematical expository writing”).

Abhyankar, after distinguishing between the flavours of “high-school algebra” (polynomials, power series), “college algebra” (rings, fields, ideals) and “university algebra” (functors, homological algebra) goes on to present his fundamental thesis (”obviously a partisan claim”):

The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester.

Perhaps for this reason, Dr. Z calls Abhyankar (”one of my great heroes”) “the modern prophet of high-school-algebra”.

Anyway, enough rambling. Back to “Every good theorem must have a good counterexample”. Discuss.

[Edited to add: The statement in its original context was referring to a phenomenon where a pleasing conjecture is found to have counterexamples, until it is resolved by realising that we must, say, count multiplicities the "right" way—the right way turning out to be whatever makes the conjecture true. Thus Bezout's theorem, etc., and the quote just means, as he paraphrases, "don't be deterred if your formula is presently invalid in some cases; it only means that you have not yet completely deciphered god's mind". On the other hand, what I (mis?)remembered was that one must know "counterexamples" to a theorem in the sense that one must know why the conclusion is not true if the hypotheses are weakened: thus one doesn't really understand a theorem till one knows at least one “counterexample” (and at least two proofs).]

Dan Brown parody

Dan Brown is a hilariously bad writer. The Da Vinci Code was an outrageously successful book.
So it was only inevitable that in addition to all the delicious criticism of Dan Brown’s writing,¹ there would also be a number of parodies of his books published, and indeed there have been several.² While looking for something in the library, I found The Da Vinci Cod: A Fishy Parody by “Don Brine” (real name Adam Roberts) and quickly proceeded to borrow it and read it. It was a good two hours spent, which is more than can be said for Dan Brown’s books themselves. Although the author is a professor of literature at London University, the book manages to remain true to the awful writing and plot of the original. I heartily recommend reading the book if you come across it; for a taste of what it’s like, some excerpts follow. You can also see parts of the book at Google Books.

Mathematics and notation: the Hindu-Arabic numeral system

Quick: What is CCXXXVII × CCCXXIX?

From page 15 of The Life of Pi by Jonathan Borwein:

The Indo-Arabic system came to Europe around 1000 CE. Resistance ranged from accountants who didn’t want their livelihood upset to clerics who saw the system as ‘diabolical,’ since they incorrectly assumed its origin was Islamic. European commerce resisted until the 18th century, and even in scientific circles usage was limited into the 17th century.

The prior difficulty of doing arithmetic is indicated by college placement advice given a wealthy German merchant in the 16th century: “If you only want him to be able to cope with addition and subtraction, then any French or German university will do. But if you are intent on your son going on to multiplication and division — assuming that he has sufficient gifts — then you will have to send him to Italy.” (George Ifrah, p. 577)

[The rest of the pages of the slides are also great and worth reading!]

Just to give some context of the time: The Hindu-Arabic system was introduced into Europe by Leonardo of Pisa (Fibonacci) — an Italian — in his Liber Abaci, written in 1202. Gutenberg (in Germany) invented the printing press around 1450. In Italy, Tartaglia lived 1500-1557, Cardano 1501-1576, Sturm 1507-1589, Giodano Bruno (1548-1600), and Ludovico Ferrari (1522-1565). (And outside Italy, Robert Recorde (as we’re talking about notation) (1510-1558) in Wales, François Viète (1540-1603) in France, etc. See this image.) Of course Galileo Galilei (1564-1642) was Italian too, but came later, as did Newton, Fermat, the Bernoullis, and all the others.

While on the topic of mathematics and notation, see also this post: Visual Clarity in the Naming of Variables.
[And while not exactly notation, Donald Knuth's Calculus via O notation]

A “Möbius” inversion formula

[Not the Möbius inversion formula, but something similar.]
As usual, define the Möbius function μ on the natural numbers as

Let be any function defined on the natural numbers, and let be the function defined as .
Then it is true that .

Note that f need not be multiplicative; it can be any arbitrarily defined function. I have no idea why it is true. Help?

Asimov on ‘The Last Question’

I like tracking down quotes, but find it terribly hard to track down quotes by Asimov about his writing: there are so many anthologies, and so many comments he has made about a single story in different places. Here, for example, are two comments I remember having read about what is definitely one of his two most famous stories, The Last Question:

• The first was easier to track down; it’s on Wikipedia with a date. In The Best of Isaac Asimov, published in 1973, he says:
  

  ‘The Last Question’ is my personal favorite, the one story I made sure would not be omitted from this collection.
  Why is it my favorite? For one thing I got the idea all at once and didn’t have to fiddle with it; and I wrote it in white-heat and scarcely had to change a word. This sort of thing endears any story to any writer.
  Then, too, it has had the strangest effect on my readers. Frequently someone writes to ask me if I can write them the name of a story, which they think I may have written, and tell them where to find it. They don’t remember the title but when they describe the story it is invariably ‘The Last Question’. This has reached the point where I recently received a long-distance phone call from a desperate man who began, ‘Dr. Asimov, there’s a story I think you wrote, whose title I can’t remember–’ at which point I interrupted to tell him it was ‘The Last Question’ and when I described the plot it proved to be indeed the story he was after. I left him convinced I could read minds at a distance of a thousand miles.
  No other story I have written has anything like this effect on my readers–producing at once an unshakeable memory of the plot and an unshakeable forgettery of the title and even author. I think it may be that the story fills them so frighteningly full, that they can retain none of the side-issues.

Fermat’s last theorem

Fermat’s last theorem has a long and exciting history. Which everyone knows, so I’ll not mention it here.¹ What I find to be a remarkable event that gets the least attention though, is the very first one. The fact that Fermat scribbled it in a margin of Diophantus’s Arithmetica. That Pierre de Fermat, in France in 1637, was reading an ancient book written by a Greek in the 3rd century. That he was reading it in such a manner that the book’s asking how to split a square into two squares should impel him to not only investigate the question of how to split a nth power into two nth powers, for all n, but to also do it until he believed he had a truly marvelous proof.

When was the last time you made margin notes in a book?

Off topic: The book only answers the question for 16(=4²). Wikipedia has pictures of the relevant page for a 1621 edition, and the 1670 edition that contains Fermat’s notes. (Fermat died in 1665.) I’m not sure I’ve deciphered the Latin correctly (the Greek is right out), but what it says is the following.
[BTW, in case you have been thinking so far and have the objection that 16 cannot be written as the sum of two squares, I should point that for Diophantus, "number" apparently meant "positive rational number", there were no other kinds of numbers. Negative and irrational numbers were "useless", "meaningless", and "absurd".]
Suppose one of the two squares that add up to 16 is Q=N². ["Q" because it is a square, "quadratum".] The other square is 16-Q. If the other square is (2N-4)²=4Q+16-16N, [um, why should it be?] then we get 16-Q=4Q+16-16N so 5Q=16N, or N=16/5 and Q=N²=256/25 (which is misprinted as 256/52 in the 1670 edition), and the other square is 144/25, which add up to 400/25=16. So the (an) answer is that 16 = (16/5)² + (12/5)².
You might notice this is not really an answer; all that Diophantus has done is take 3²+4²=5² and multiplied it appropriately to make two “squares” add up to 16. We could do the same for any square, e.g. for 49=7², we could write (7×3)²+(7×4)²=(7×5)², then divide out by 5² to say (21/5)²+(28/5)²=49. For any x, we could take any a and b such that a²+b²=1 (e.g. 3/5 and 4/5) and write x²=(ax)²+(bx)².

↑1. I found today (2008-11-27) there is an interesting event which is not well known either. The setting is this: it was April 1994. Andrew Wiles had first announced his proof in June the previous year, and sent it off to a journal, but a hole had been found. It seemed at first it would take only a few hours, then weeks, to fix it, but months had dragged on without success. And on April 3 1994, Gian-Carlo Rota sent out an email announcing that Noam Elkies had found a counterxample to Fermat’s last theorem! So it seemed that the hole was unfixable after all. There was some disappointment all around before it was realised that the email was an April Fool’s joke, that had somehow got incorrectly dated :-) I found it on Wikibooks, but see Lance Fortnow’s blog post for the email.

Thank you for the music

Just watched Mamma Mia! The Movie this week (twice!). It sucks, is completely ridiculous, Pierce Brosnan cannot sing to save his life, and there are far too many annoying characters, but because it’s ABBA, all is forgiven. Meryl Streep, perfect as always, appears to be having the time of her life, but maybe it’s just her acting. Some of it would have been better if they simply used “playback singing” the way only we in India seem used to, but it was mostly okay. Apparently there are even thoughts of a “sequel” (wouldn’t be the first time a movie with no plot has had a sequel, anyway), because “there are still plenty of ABBA songs left”.

Lattice points visible from the origin

[A test of LaTeX-to-WordPress conversion. Bugs remain, point them out. Original PDF]

Here is a problem I love. It is simple to state, and it has a solution that is not trivial, but is easy to understand. The solution also goes through some beautiful parts, so I can promise it’s worth reading :-)

[The solution is not mine. Also, the question is just an excuse for the ideas in the solution. :P]

Question. Suppose you are standing on an infinite grid in the plane. You can see infinitely in all directions, but you cannot see through grid points: a point is hidden from view if some other grid point lies in your line of sight. What fraction of the grid points can you see?

Let us first imagine that we are standing at the origin, and that the grid is that of the lattice (integer) points.

The blue points are visible; the grey points are not

A timeline ends

Sad news that was too easy to miss in the ongoing deluge: Michael Crichton passed away on Tuesday, at age 66. See my friend’s post.

No more Knuth checks

Have you seen the news? Knuth has announced that he will no longer be writing personal cheques, for security reasons. As an alternative, he suggests

Instead of writing personal checks, I’ll write personal certificates of deposit to each awardee’s account at the Bank of San Serriffe, which is an offshore institution that has branches in Blefuscu and Elbonia on the planet Pincus.

As most Knuth cheques are never cashed anyway (they “have apparently been cached”), this is perfectly as good and perfunctory as the old system, with the additional advantage that there is now a Hall of Fame :-)

Although it is brownie points that are being awarded now (”kudos, not escudos”), he also offers to find a way to send actual money to anyone who wants it :P