Archive for August, 2014
This is the fifth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This installment’s mathematics article is entitled ‘From Numbers to Number Systems’ and the poetic form is allegory, making this the third poetic form in a row that isn’t actually a poem.
A long time ago in Greece, there was a community of numbers where everybody lived as one, or two, or three. They were not all equal, because each was unique, but they were all numbers, and that’s what counted. They were the true numbers, and they lived alongside the false, or negative, numbers.
Then One day, which was the day when the number One was celebrated, One Seventh came along. The other numbers looked at it with pity.
“You poor, broken thing,” they said. But the seventh didn’t feel broken.
“I’m not broken. I’m a number, just like you!” said One Seventh.
Seven looked at One Seventh with trepidation. “I don’t think it’s safe to be around a part of seven. What if it wants to take more of my parts?”
Three agreed. “It’s just not wholesome.”
One Seventh pointed to its numerator. “Is this not a one, like the number of the day? How can I not be a number when my very numerator is the purest number of all?”
One was flattered by the description, and in the spirit of the celebration, declared, “One must not only celebrate Oneself, but also display kindness to all those around One. I declare One Seventh to be a number, along with all little Ones like it!” After that, the other numbers were largely kind to the unit fractions, and the fractions always reciprocated.
The next day, Two Fifths came along. Emboldened by the success of One Seventh, Two Fifths said, “I’m a number too! Can I join the celebration?”
Two, whose day it was, said, “But you’re just One Fifth plus One Fifth. It’s just not proper to be going around as if you’re a single number. Split into unit fractions before you scare the little Ones!”
But Two Fifths persisted. “What are you,” it said to Two, “if not One plus One?”
Two did not like the idea two bits, but it could not find a problem with the argument.
Five, who was never any good at acting composed, protested. “This is preposterous! Two, I always knew you weren’t quite as prime as us. Think about it. If we let these two fifths…”
“This two fifths,” corrected Two Fifths.
Five shot it an incalculable look. “If we let these two fifths act like a whole number, next we’ll have matrices, or lengths, or linear graphs wanting to be numbers. It’s a steep gradient!”
“That’s not true!” said Two Fifths. “In other cultures I am a perfectly acceptable number. In Mesopotamia, nobody thinks twice about my being a number, but they would never allow One Seventh. It’s all a matter of culture! And graphs are not numbers there either, so you needn’t worry about that.”
Two was divided by Five’s argument. It worried about diluting the number system, of course, but it was aware that even it could have been excluded from the primes using such an argument. Having always felt like an outsider itself, it had pity on Two Fifths, and declared the fraction and others like it to be numbers.
The next day, The Square Root of Two, who could not be expressed as a fraction, decided to join the numbers. Three said, “Don’t be absurd. You’re not really the square root of two; only square numbers have square roots. You’re just a fraction who’s confused. You look like about one and a hundred and sixty nine four hundred and eighths, to me.”
But the square root was resolute. “Look,” it said, holding up a square. “If we say the sides have length one, then the diagonal has length the square root of two. There is no way we can find a unit that can measure both of them as whole numbers. I can prove it to you!” And The Square Root of Two proved it.
“Okay,” said Three. “You’ve shown that the diagonal can’t be measured with the same unit as the sides. But they’re just lengths, not numbers. All you’ve done is show that not all lengths can be measured with numbers. The numbers are not going to be happy about this, you know.”
“But I am a number! I am the number which can measure that diagonal!”
“That’s just irrational. Lengths are not numbers. Either you’re a number, in which case you should show yourself as a fraction instead of wearing that radical outfit, or you’re a length, or a ratio of lengths, and you should go back where you belength. Make up your mind.”
“I told you this would happen!” said Five. “I told you lengths would be next!”
So the Square Root of Two skulked back to geometry, and commiserated, but did not commensurate, with the ratio of a circumference to a diameter.
Meanwhile, Two Fifths told all its new number friends about its adventures in Babylon, and the sexy sexagesimal numbers there. Before long, it became fashionable for numbers to represent themselves using decimal places instead of fractions. Some of them had to use zeros to make sure their digits hung in the right places.
Zero saw its chance, and claimed its right to be considered a number.
“But you’re not a number!” said Four. “You’re just a placeholder that the fractions use when they’re dressing up in their costumes for their unwholesome sexagesimal parties.” Four looked down its slope at a nearby decimal.
“But if I add myself to you, is there not equality? I should be treated the same as you.”
“But,” said One, “numbers have to be able to multiply. If you multiply you only get yourself. Only multiplying with me should do that! I’m the Unit around here, not you.”
“You’re destroying the family Unit!” shouted Five, in defense of its onely other divisor.
“I can’t even tell whether you’re true or false!” cried One Seventh, nonplussed.
So Zero went back to dutifully holding places, quietly adding itself to everyone and everytwo it met, until they were all convinced it held a place in society.
On the Seventh day, which was the day when One Seventh’s acceptance as a number was celebrated, they rested.
On the Tenth day, which was the day when The Tenth was celebrated, The Tenth returned from a vacation in Flanders and declared, “There are no absurd, irrational, irregular, inexplicable, or surd numbers!”
Five and Three cheered, and made obtuse gestures at The Square Root of Two and its friends. “You see? You’re not numbers.”
“All numbers are squares, cubes, fourth powers, and so on. The roots are just numbers. Quantities, magnitudes, ratios… they are all just numbers like us. We can all fit along the same line.”
Five and Three looked at each other in primal disgust. “I’m not a point on a line! I’m a number! A real number!” Five shouted.
“Real numbers,” countered The Tenth, “include everyone, and everyfraction, and everylength in between.”
The Square Root of Two led its friends into their places between the other numbers, and they celebrated with unlimited sines, cosines, and logarithms. Some of the stuffier primes and fractions protested, but they backed down when they realised just how many of these strange new numbers there were.
But even as The Tenth spoke, it knew that not everything it said was true. After all, false numbers were not the square of anything, even though it had seen them act like they were in some delightful formulae.
At Length, which was the day when the acceptance of lengths as numbers was celebrated, somereal wondered what would happen if false numbers were squares of something too. It imagined a new kind of radical, like those the square roots wore, but for false numbers. It imagined a world where every polynomial equation had roots, be they real, false, or imaginary. These were clearly not like all the other numbers The Tenth had listed.
Soon after, the imaginary numbers came out of hiding. “We do exist!” they said. “And we can add and subtract and multiply and divide just like you!”
The other numbers were wary, for they could not work out where the imaginaries fit amongst them. They could not even tell who was bigger. Five was disgusted that such numbers had been secretly adding themselves to real numbers all along.
The real numbers were nonetheless intrigued by and slightly envious of these exotic creatures, and despite having become accustomed to all having equal status as numbers, sought new ways to distinguish themselves from the crowd. The whole numbers had never quite got over the feeling of being generally nicer than the other numbers, so they used the new trend to vaunt their natural wholesomeness. The ratio of a circumference to a diameter, who had taken on the name Pi, discovered that in addition to not being expressible as a fraction, it was so much more interesting than The Square Root of Two that it couldn’t even be expressed in such roots. It called itself ‘transcendental’, and had quite some cachet until most of its admirers realised that they had the same property.
Finally they discovered that instead of trying to organise everynum into a line, they could arrange themselves in two dimensions, with the imaginaries along one axis and the reals along the other, and the vast plane in between filled with complex combinations of both.
Some of the more progressive numbers were so excited by this system that they tried to find new numbers that they could arrange into a three-dimensional volume, but they couldn’t find any. However, during their search they found things called quaternions, which lived in a fourth dimension.
An excited transcendental, whose name is too long to write here, brought a subgroup of quaternions in front of the crowd and announced, “I have travelled to the fourth dimension, and found numbers there just like us. We are not alone!”
Five kept its fury pent up this time, but Four Sevenths called out, “They are not numbers like us. I have seen how they multiply. When two quaternions multiply, they can give different results depending on which comes first!”
The numbers clattered their numerals in shock, and a great amount of whispering about unlikeabel multiplication practices ensued.
A complex transcendental sneered, “And what were you doing watching them multiply, eh?”
“Oh, get real!” retorted Four Sevenths, crudely conveying what the transcendental should do with its complex conjugate.
The pair fought, and disorder spread throughout the dimensions. Some sets of numbers sneaked off into the fields to form their own self-contained communities, sick of the controversy surrounding being or not being numbers. As they did, they found still other communities which functioned much like theirs, and some were communities of functions themselves. Indeed, even matrices and graphs formed structures which the enlightened subgroups found familiar, though rather than trying to be accepted as numbers, these groups took pride in having their own identities. The p-adics were adamant that they were numbers, but did not care to join the rest of the real or complex numbers. The octonions did not associate themselves with such labels, going about their operations however it worked for them, and consenting to be called numbers only when it was useful to act as such.
When peace finally settled, there were more groups of objects than there had been numbers, and still more came about when those groups interacted with each other. Most no longer cared about being called numbers, and simply communicated which rules they followed before participating in a given system. And if the requisite system turned out not to exist yet, well, it just had to be invented.
Turning this particular article into an allegory did not take much work. It almost seemed like one already, when I read it in that frame of mind. There are a few direct quotes in the story. The Tenth’s proclamations come from The Tenth, in which Simon Stevin introduced decimal notation to Europe. The very last line of the story is paraphrased from the last line of the article. All I really did was rephrase it as a story from the perspective of the numbers, and add in far too many mathematical puns of greatly varying levels of subtlety.
I’m sorry to anyone with ordinal linguistic personification who thinks I’ve given the wrong personalities to the numbers. Also, in case anyone was wondering, the Greek numeral for four does have a slope.
The next Forms and Formulae will be an anecdote about geometry.
This is the fourth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This post’s mathematics article is entitled ‘The General Goals of Mathematical Research‘ and the poetic form is alba, which is a kind of song; I recorded it [direct mp3 link] using my robot choir and some newfound musical knowledge, and there are many notes on that after the lyrics below.
Here are some extracts from the article on the alba, explaining the features that I ended up using:
A dawn song about adulterous love, expressing one or both lovers’ regret over the coming of dawn after a night of love. A third voice, a watchman, may announce the coming of dawn and the need for the lovers to separate. An Occitan alba may contain a dialogue (or serial monologues) between lover and beloved or a lover and the watchman or a combination of monologue with a brief narrative intro.
The alba has no fixed metrical form, but in Occitan each stanza usually ends with a refrain that contains the word alba.
…the arrival of dawn signaled by light and bird’s song…
The watchman plays an important role as mediator between the two symbolic worlds of night (illicit love in an enclosed space) and day (courtly society, lauzengiers or evil gossips or enemies of love)
I based the song on section 8.3 of the article, entitled ‘Illegal Calculations‘. In retrospect, using the word alba in each refrain (are these even refrains?) doesn’t make much sense, since I’m not writing in Occitan, and the casual listener will not know that alba means ‘dawn’ in Occitan. But hey, it kind of rhymes with the start of ‘self-avoiding walk‘. How can I not rhyme an obscure foreign word with an obscure mathematical concept?
Mathematicians struggle even today to learn about the average distance between the endpoints of a self-avoiding walk. French physicist Pierre-Gilles de Gennes found answers by transforming the problem into a question about something called the n-vector model when the n is zero. But since this implies vectors with zero dimensions, mathematicians reject the approach as non-rigorous. Here we find that zero waking up next to its cherished n-vector model after a night of illicit osculation.
I am just a zero; I am hardly worth a mention.
I null your vector model figure, discarding your dimension,
and every night I’m here with you I fear the break of day,
when day breaks our veneer of proof, and we must go away.
Here by your side
till alba warns the clock.
Fear’s why I hide
in a self-avoiding walk.
Let the transformations of De Gennes show your place.
Never let them say we’re a degenerate case.
When I’m plus-two-n there’s just too many ways to move,
But you’re my sweetest nothing and we’ve got nothing to prove.
Here by your side
till alba warms the clock.
Fear can’t divide;
it’s a self-avoiding walk.
The sun has come; your jig is up. It’s time for peer review.
You think your secret union has engendered something new.
You thought you would both find a proof, but is it you’re confusing
The sorta almost kinda-truths the physicists are using?
That’s not rigorous,
says alba’s voice in shock.
All but meaningless
to the self-avoiding walk.
Zero and N-vector model together:
If you say that our results don’t matter,
then go straight to find a better path.
For as long as you insult our data,
Is it wrong to say you’re really math?
Hey there, Rigorous
at alba poised in shock,
you are just like us,
in a self-avoiding walk.
All voices are built-in Mac text-to-speech voices, some singing thanks to my robot choir (a program I wrote to make the Mac sing the tunes and lyrics I enter, which still needs a lot of work to be ready for anyone else to use.) Older voices tend to sound better when singing than the newer ones, and many new voices don’t respond to the singing commands at all, particularly those with non-US accents. So for the introduction I took the opportunity to use a couple of those non-US voices. These are the voices used:
Introduction: Tessa (South African English) and, since I also can’t fine-tune Tessa’s pronunciation of ‘Pierre-Gilles de Genne’, Virginie (French from France)
N-vector Model: Kathy
Most of the bird noises come from the end of Jonathan Coulton’s ‘Blue Sunny Day‘, and I can use them because they’re either Creative Commons licensed or owned by the birds. The two peacock noises are from a recording by junglebunny. Free Birds!
As I mentioned, I’ve been learning about songwriting from John Anealio, and since the Forms and Formulae project sometimes requires me to write songs, I’m putting the new knowledge into practice sooner than I expected. This song uses several musical things I’ve never tried before, which is quite exciting, but it also means I probably didn’t do them very well, because there’s only so much I can learn in a couple of months of half-hour weekly lessons. I welcome friendly criticism and advice. The new things are: Read the rest of this entry »
My first watch was digital. I was probably nine or ten, and the watch was a black Casio with a dashed line around the face in alternating green and blue. My brother and I would race to find each other whenever we noticed the hour was about to change, so that we could watch the watch digits all change at once. Needless to say, the changes from 9:59:59 to 10:00:00 and 12:59:59 to 1:00:00 were especially thrilling[⁉︎].
I’d learnt how to read an analogue clock, of course, but not fluently. To me, reading an analogue clock was akin to reading Roman numerals: a quirky, difficult system from long ago. Some analogue clocks even had the hours in Roman numerals. Some had no numbers at all. Some such watches only seemed to exist to give men a socially acceptable way to wear bracelets. Telling time was clearly not a priority.
So when I read in the Hitchhikers’ Guide to the Galaxy that humans were “so amazingly primitive that they still think digital watches are a pretty neat idea,” I naturally assumed it was because intelligent life forms had invented them so long ago that digital watches had about as much chance of being described as a ‘neat idea’ as the wheel. Digital watches are too simple an invention for anyone to find interesting. These days, almost everything has a digital clock built in, so the most important thing about a digital watch is a strong strap to keep it conveniently on the wrist.
A few digital watches and a grudge against fragile watch straps and lost pins later, I moved to Switzerland, and when my watch strap broke or fell off I felt obliged to check out some of the famous Swiss watches. I was baffled by the evidence that not only did humans still think digital watches were a pretty neat idea, they also still thought analogue ones were. The only Swiss digital watches with good straps I could find had skeuomorphic round faces, or lacked such basic features as seconds, dates, or a light. I get it: the Swiss are proud of how precise they can be with tiny gears. But it’s the third millennium; get with the timepieces!
As Swiss innovations go, I prefer milk chocolate and Velcro. I found a Casio dealer and bought a solar-powered, waterproof, digital watch that synchronises daily with an atomic clock using radio waves and has a well-attached metal strap. It will stay on my wrist and display precisely the right time in plain digits, indefinitely with no intervention whatsoever, for less than the price of a piece of Swiss jewellery that doesn’t even have numbers on it. A fall onto concrete gave it some sparkly cracks in one corner, but it is still waterproof and functional many years and no battery changes or time adjustments later.
Unhappy with the hypothesis that most of the human race was more concerned with adding respectability to their diamond bracelets than with locating themselves in spacetime, I had to eventually accept that there was something people liked about analogue watches. Just as there must be something great about shoelaces that keeps Earthbound people using them even after the invention of Velcro, and even though Back to the Future fans know that by 2015 we shouldn’t still be tying them.
The thought crept up on me that maybe Douglas Adams didn’t like digital watches at all. Maybe he didn’t think they were ever a pretty neat idea. I thought about this for a few years, gradually becoming less and less sure that my initial interpretation was the correct one. Eventually, I looked it up:
So there you have it. Douglas Adams liked pie charts. I like pie charts too, but after the first glance I will look for the labels with exact percentages, and be frustrated if they aren’t there. For me, a word can be worth a thousand pictures, and a number can be worth a poorly-defined number of words.
As he says, digital watches have improved since then. I don’t need to put down my suitcase to press a button on my watch, unless it’s either dark and I need to turn the watch light on, or it’s recently been dark and the watch turned off its display to save power. In fact, my suitcase has four wheels (wheels! Now, aren’t they a neat idea?) so I never have to pick it up to begin with; I just give it a push occasionally while I stroll along, reading the time like a frood.
Reader participation alert:
Did you interpret the statement about digital watches the same way I did? If not, how did you interpret it, and how did it mesh with your own opinion on digital watches?
[⁉︎] If you think you’ve grown out of such primitive excitement, try watching the hour change on this clock made of planks of wood and rearranged manually by construction workers. The website only delivers one image at a time, now, so you’d have to refresh a lot to get a video effect there, but they sell an iOS app which will show you video, the Lite version of which has the transition from 9:59 to 10:00.