Marvellous Merchiston

Squeeze Theorem

2011-09-20T17:12:00.000-07:00

What did you do last Friday night? Went to the club? Dancing? I see.

It wasn't on the floor, was it? Table? I thought so.

And you were asked to leave? Naturally.

But you didn't, did you? Why not? Oh yes, a sustained period of bad judgment. Because....

Of course.

How big were the security people? Six-foot-seven or so, with weights approaching four hundred pounds?

And I suppose none of it was fat, either.

They got you by the arms? And headed for the door? I'm sure you didn't want to.

You had no choice about it though? I suppose not. They had you between them and they went to the door, so you did too.

What are you doing this Friday night?

Worse than Meaningless

2011-09-20T16:54:00.000-07:00

Worst Graph of the Year. If it was completely devoid of any information, it would be a whole lot better, simply because it wouldn't be shockingly historically inaccurate and blatantly propagandistic. I also wonder what the units are for "militancy." Apparently the person who didn't label the axis doesn't know either. Relationships have ended for less.

Truth, Fiction, and Strangeness

2011-09-18T05:46:00.000-07:00

So I was re-reading some Subnormality and encountered the one about Princess Washburn. That got me thinking. The debunkery on display here is nice, but what exactly is she debunking?

Well, she's trying to debunk the idea that "truth is stranger than fiction". That's an aphorism, and debunking one of those is a little like debunking poetry. And if you look at how she goes about it, you can tell how she thinks about the aphorism.

She's using a counterexample, which tells me she thinks of the aphorism as a statement of the type "P always has property Q," specifically, "Fiction is always less strange than truth." She then provides a counterexample consisting of a character who is more strange than truth. QED!

Still, I wonder...

First of all, how do you measure strangeness? If it has to do somehow with averages, then any discrete object you choose is likely to fall away from that average on some scale of measure. The "normal" is an abstraction, even an illusion. Strangeness is more of an intuitive sense of difference from lived or imagined experience. A red, fire-breathing dragon seems less strange than a plaid, gravel-breathing one even though both are equally nonexistent. The red kind is just more common in the domain of fantasy.

Princess Washburn is certainly strange as far as fantasy people go, relative both to other fantasy and reality. But there are real people just as singular relative to other real people, as she is to fantasy people. She just gets to be extra weird by having properties no real person could have. There is a sense in which fiction has the luxury of being stranger because it need not exist. Truth is constrained by reality; fiction only by imagination. Fiction has another degree of weirdness freedom.

But that leads to another question. Is Princess Washburn really fictional? I don't mean, of course, that I think she might be real. I'm not talking about the axis of real vs. fictional. I'm talking about what it means to be fictional, given that one is known to be imaginary. Anything made up is imaginary, but what is fictional?

It seems reasonable to claim that the fictional is anything pertaining to fiction. And except for the fact that she is being discussed by fictional characters in a comic, Princess Washburn has only a tenuous connection to any narrative structure or fictional construct. The details that make her so weird are chosen randomly for the effect of strangeness. What connects coins, spiders, the Rolling Stones, and ultraviolet light, except for being related somehow to Princess Washburn?

Not a whole lot. Her personal details have the broad, scattered effect that the details of a real person do, the effect of springing from a deep, complex, and unstructured development process that is largely opaque to outside observers. Fictional characters, on the other hand, are generally expected to have fairly clear motivations within the confines of the narrative arc.

I think that may be where the aphorism comes from. Saying that truth is stranger than fiction is acknowledging that fiction is a human artifact with fairly simple rules and an expected structure. Fiction, as people say, has to make sense. True stories also make sense, just in a different kind of context. All the motivations are richer, and the character development is more organic, because it is real. Princess Washburn is strange because everything that makes her who she is, is hidden from us. Any real person would also seem strange if all you know about them was a dozen or so random facts.

The personal details of fictional characters are meant to advance the story. But Princess Washburn is a character without a story, or at least not an accessible one. She is more like a real person, in a sense, than a fictional one, except for the part where she doesn't exist. She isn't fictional. She's imaginary.

Fiber arts and Mathematics

2011-08-28T04:16:00.000-07:00

"Have you ever wanted to explore the symmetries of the cube and octahedron through an old Japanese art form? Or to investigate fractals using tatting and string art? Or to study the helix by knitting bed socks?" Plus magazine has a review of a book called Crafting by Concepts.

When I saw that, it reminded me of this TED talk from 2009 involving coral, crochet, and hyperbolic geometry.

And writing this post reminded me of the old favorites, Klein Bottle Hat (and Mobius Scarf!)

Henry VIII

2011-08-18T18:07:00.000-07:00

A jolly comic!

Plush statistical distributions?

2011-06-05T18:30:00.000-07:00

Oh yeah, here you go.

Word Problem of the Week: Unlikely Maternal Scenario

2011-05-18T19:52:00.000-07:00

"The past is a foreign country," says the first line of The Go-Between. "They do things differently there." In the case of Gilbert Dyer's Most General School-Assistant, do they ever. On page 182, question 7, we have this amazing scenario:

Hodge, to inherit, must marry.
That is, he must marry a specific person.
His first cousin.
And gets more money the more kids she has.
She has 24 boys.
In 12 years.

Now, I think everyone expects algebra problems to be a bit contrived, possibly to abstract out some real life detail for the sake of the simplicity of the solution. Dyer certainly does this. But something about the little details he does include, makes me find this hilarious. Hodge is honest, Polly is pretty, and they start having kids "honestly, one Year after Marriage."

There are several questions in here that say a lot about the differences we have with those in the 18th century. The next question is about a four-day family beer party and the one after that concerns, as far as I can tell, a patent medicine seller who, for some reason, is operating at a loss.

Word Problem of the Week: Whiskey Rations Aboard Ship

2011-04-29T05:48:00.000-07:00

There are two things I find interesting about this problem. First, the detail offered gives it a level of realism that is not typically found in these sorts of texts. On the other hand, the quantities are all represented as variables, which adds a level of abstraction. Thus, the answer gives us a formula for any possible scenario of this type. The problem is number 23 on page 479:

Word Problem of the Week(s): Classical History and Mythology

2011-04-16T16:56:00.000-07:00

These problems come from Mathematicall Recreations, or

A Collection of many Problemes extracted out of the Ancient and Modern Philosophers, as Secrets and Experiments in Arithmatick, Geometry, Cosmographie, Horologiographie, Astronomie, Navigation, Musick, Opticks, Architecture, Staticks, Mechanicks, Chemistry, water-works, Fire-works, &c.

Basically, a seventeenth century Dangerous Book For Boys.

Several of the arithmetic questions have to do thematically with the stories of ancient Greece. These are:
Of the number of Souldiers that fought before old Troy.
Of Pythagoras his Schollers.
Of the number of Apples given amongst the Graces and the Muses.
Of the Cups of Croesus. (He gave the Temple 6 golden cups weighing a total of 600 drams, with each one a dram heavier than the next.)
Of Cupid's Apples. (The Muses steal them.)

Word Problem of the Week: Map the Ohio River

2011-03-09T03:06:00.000-08:00

It's nice to see a little connection between geometry and geography.

From Graphic Algebra, by Andrew Wheeler Phillips and William Beebe, 1904

p. 8:

Draw a map of the Ohio River from the following latitudes and longitudes, which are reckoned from the Equator and the meridian of Washington respectively:

This is the earliest book I have seen that emphasizes graphing equations as an algebraic tool. Then, 80 years later, the graphing calculator is invented.

Word Problem of the Week: Problem of the Lights

2011-03-03T15:35:00.000-08:00

From Elements of Algebra, by Charles Davies, 1854.
p. 162:

"Find upon the line which joins two lights, A and B, of different intensities, the point which is equally illuminated by the lights."

This is a great little simple sounding problem that lends itself, in this case, to three and a half pages of discussion of various cases. At the end of this, the author states that "the preceding discussion presents a striking example of the precision with which the algebraic analysis responds to all the relations which exist between the quantities that enter a problem." I should say so.

Word Problem of the Week: Say, Lovely Woman, the Number of Bees.

2011-02-15T17:37:00.000-08:00

From Lilavati, translated by Henry T. Colebrooke, 1817.
p. 211:

The square-root of half the number of a swarm of bees is gone to a shrub of jasmin ; and so are eight-ninths of the whole swarm : a female is buzzing to one remaining male, that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.

The whole book isn't like this though; the very next problem is about a guy shooting arrows at his enemy.

I highly recommend this review of the twelfth century Indian algebra text Lilavati, by Bhaskara. It references the Colebrooke Translation from 1817, which also mentions this gem from a commentary:

Whilst making love a necklace broke.
A row of pearls mislaid.
One third fell to the floor.
One fifth upon the bed.
The young woman saved one sixth of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?

Although Colebrooke refers to her as a "wench" which is a bit less romantic.

Word Problem of the Week: French Degrees

2011-02-03T14:30:00.000-08:00

I was telling the students the other day how dividing a circle into 360 pieces is a bit arbitrary. In fact, it comes from the Babylonians, and it could have been different. There is nothing special about 360.

The French, it turns out, actually did propose a different system of angular measure: one in which the fundamental unit was one percent of a right angle. Thus there are four hundred of them in one full revolution.

I remember years ago I saw a calculator with the familiar "degrees" and "radians" settings but also something called "gradians." This is the name for the "French Degrees." They are also called "grades" in a lot of old textbooks. I personally like what I called them in class, which is "People's Revolutionary Degrees."

Anyway, there seems to have been some confusion as to whether gradians were actually ever used. Some are of the opinion that the unit was "frequently used in France and ocasionally elsewhere" whereas others are convinced that they were not.

I suspect they must have been, because it is otherwise difficult to explain the enthusiasm shown for the French Degrees by a certain Mr. Isaac Todhunter. His trigonometry text is full of strange abstract problems involving equivalencies between the English and French units. Fairly typical is this one:

Divide two-thirds of a right angle into two parts, such that the number of degrees in one part may be to the number of grades in the other part as 3 to 10.

This combines the slightly obscure and uncommonly used unit with the proportionality question so common in 18th and 19th century texts to produce a masterpiece of bizarre irrelevance. Really, this is an interesting puzzle, but trig texts are usually much more practical than this.

Word Problem of the Week: From the Notebook of Benjamin Banneker

2011-01-22T17:17:00.000-08:00

This morning I had the pleasure of attending a talk by John Mahoney concerning his experience teaching at Benjamin Banneker Academic High School in the District of Columbia. Part of the talk concerned Banneker himself, who is a fascinating personality about whom I'm going to have to write more later.

Banneker kept a notebook in which he recorded interesting problems. This one is interesting because there is an elegant solution that is unintuitive to someone educated in the modern way. At least to me, it was unintuitive. Problems such as this seem to have been popular in the 18th century, though.

Question by Elliot Geographer General
Divide 60 into four Such parts, that the first being increased by 4, the Second decreased by 4, the third multiplyed by 4, the fourth part divided by 4, that the Sum, the difference, the product, and the Quotient shall be one and the Same number.

You start by making a guess as to the number mentioned at the end, then adjust things based on the error you get.

Word Problem of the Week: Wainscoting a Gentleman's House

2011-01-16T11:57:00.000-08:00

From A Compendium of Algebra by John Ward, 1724.

p. 51:

Three Joiners undertook to wainscot a Gentleman's House in 150 days. One of then (being esteem'd the best) was to have 5 s. a day for every day he work'd; another was to have 4 s. 6 d. a day, and the third was to have but 4 s. for every day we wrought.
When the Work was finish'd, every one of them had just the same Sum of Money to receive; Quere, how many Days each of them work's? &c.

So, they all made the same amount, which makes me wonder: Is the highest-paid worker the best because he's the most efficient, or is he doing a bit of slacking off?

Word Problem of the Week: Arraying the Troops

2011-01-08T11:16:00.000-08:00

From The Elements of that Mathematical Art Commonly Called Algebra, Expounded in Two Books by John Kersey, 1741.

p. 70:

A General of an Army having set his Soldiers in a Square Battel, there happened to be 500 (or b) Soldiers to spare; but to increase the Square so as that its side might consist of 1 (or c) Soldier more than the Side of the former Square, there would be 29 (or d) Soldiers wanting. The Question is, to find how many Soldiers the General had in his Army.

Word Problem of the Week: A Brandy Smuggler

2011-01-01T07:45:00.000-08:00

From Ray’s Algebra, Part Second by Joseph Ray, 1852. p. 91:

A smuggler had a quantity of brandy, which he expected would sell for 198 shillings; after he had sold 10 gallons, a revenue officer seized one third of the remainder, in consequence of which he sells the whole for only 162 shillings. Required the number of gallons he had, and the price per gallon.

My favorite part of this problem is the story we aren't getting. Why did the revenuer only confiscate a third of the brandy? My two best guesses are that the "seized" brandy was actually given to the revenuer as a bribe to ignore the rest, or that only one stash of several got raided.

This is apparently the sort of thing that seemed relevant to algebra students living at a time when the Whiskey Rebellion is in living memory.

Word Problem of the Week: Age of Father and Son

2010-12-21T14:17:00.000-08:00

From Ray’s Algebra Part First by Joseph Ray, 1848.
p. 212:

If 4 be subtracted from a father's age, the remainder will be thrice the age of the son; and if 1 be taken from the son's age, half the remainder will be the square root of the father's age. Required the age of each.

Word Problem of the Week: Height of a May-Pole

2010-12-15T02:48:00.000-08:00

From Pleasure With Profit: Consisting of Recreations of Divers Kinds by William Leybourne, Richard Sault, 1694, p. 35:

There was a May-Pole which consisted of three pieces of Timber, of which the first (or lowermost) was 13 foot long, the third (or uppermost piece) was as long as the lowermost piece, and half the middle piece; and the middle piece was as long as the uppermost and lowermost together: How high was this May-Pole, and how long each piece?

This is not the only problem in the book involving a maypole. I like how something so "Old-Europe" was apparently common enough in the 1690s to warrant mention in word problems. The height, by the way, turns out to be 104 feet. I've seen maypoles, but never one that tall.

Calculus Made Easy

2010-12-11T09:52:00.000-08:00

The prologue of a Calculus book from 1914:

This is still true today.

Word Problem of the Week

2010-12-08T03:39:00.000-08:00

This is from An introduction to algebra; with notes and observations; designed for the use of schools and places of public education, by John Bonnycastle, 1806.

A labourer engaged to serve for 40 days upon these conditions, that for every day he worked he was to receive 20 d. but for every day he played, or was absent, he was to forfeit 8 d. now at the end of the time he had to receive 1 l. 11 s. 8 d. it is required to find how many days he worked, and how many he was idle.

One reason I love these old word problems is that they raise so many other questions than what "it is required to find." For example...

Was this labor arrangement something that actually happened, or is it an abstraction of a more subtle agreement? This is not the only time I have seen a problem like this one in an old book; it seems to have been a stock question in 19th century algebra books.

If this is the "First American Edition" then why does it still use British monetary units and spelling? Was the publisher just lazy?

Against Humanism

2010-11-08T02:37:00.000-08:00

This article argues that Comte's attempt to create a religious humanism failed because we revere individuals and not the species. This is because individuals are the ones who change the way we view the world - greatness is an attribute of individuals.

There is no fixed, unalterable background map of the "familiar facts" that must survive all such shifts, and certainly no fixed schedule dividing real entities from fishy, imaginary ones. Entities like Fate and Progress and the Logic of History and the Hidden Hand of the Market come and go.

Unfortunately for the science of consciousness, Comte's materialism has yet to go:

The search for a "scientific explanation of consciousness" which goes on at the yearly conference at the Center for Consciousness Studies in Tuscon, Arizona still centres not on trying to be scientific in the sense of using suitable methods, but on making consciousness respectable by somehow bringing it within the range of physics and chemistry, mainly at present through neurobiology.

A real robust humanism would have to take the religious tendencies of people into account. Science isn't up to the task of myth-making:

It just works through old-fashioned personification. If there is no purpose and everything is impersonal, how can DNA be actively ruling our destinies? How can it feel "pitiless indifference" and make us "dance to its music"? The Cartesian drama of inert matter and active spirit is suddenly reversed here to show humans (and animals) as helpless objects – passive "lumbering robots" – stage-managed by plotting genes (and memes) that are sometimes helped by other entities such as market forces. The myth-building capacities that surround every new world-view are surely as busy here as they are in established religions. These visions perhaps offer the worst of both worlds – an ontology that is as bankrupt morally as it is scientifically. The imagery of science is used, not, as Huxley hoped, to ground a deep reverence for the natural world but to justify human alienation from it.

Architecture fail.

2010-10-01T05:56:00.000-07:00

A giant parabola-shaped hotel covered with reflective glass. What could possibly go wrong?

Fractal poverty

2010-09-14T18:15:00.000-07:00

I got this link from Andrew Sullivan's blog. Maps of income levels apparently look the same at any scale.

The Unimaginable Mathematics of Borges' Library of Babel

2010-08-22T14:33:00.000-07:00

I read this story by Jose Luis Borges years ago - in fact, the link where I first read it still exists. Just the other day I found this book in an actual library, so I checked it out. I have yet to start it but it looks like it will be good.

Math and Science Shirts

2010-03-14T13:57:00.000-07:00

Happy π Day!

I hope Professor Hawking knows about this, and thinks it's awesome.

Try and hear it in Bogart's voice.

This place is basically a cornucopia of science-apparel hilarity.

For some reason, several of the shirts in the store of the hipster-soap-opera webcomic Questionable content are science related.

Fractal broccoli and other links

2010-03-06T08:06:00.000-08:00

Fractal broccoli

The type of calculator I used in high school can still be found. Mine is still going strong after almost 16 years.

Before you click on the link, ask yourself: "Do actual socialists like Avatar?"

Going to the Bookstore

2010-02-10T16:13:00.000-08:00

An Anthropological Curiosity

2010-01-30T15:23:00.000-08:00

The reports predicted snow that night. So Friday in my classroom, the possibility was on my students' minds. Having finished her work, one girl asked to draw on the board and so I allowed her to do so.

She proceeded to draw a snow scene, complete with snowman. I thought I had seen something like this before, some event related somehow - and I realized what it was. Cave paintings.

One theory about the purpose of those paintings has it that they were magical in nature, as they often showed hunting scenes. If some of them were done as a way to attract success in hunting, what happened in my classroom was another example of the same thing - sympathetic magic.

In some ways, how little do we change, as a species, in thirty thousand years. The girl would most likely not be happy to think of her actions as magical, and it is possible she would doubt that humanity was so old. Nevertheless, without knowing it, she affirmed the connection we all still have to our ancestors in the ancient past.

James Joseph Sylvester: Mathematitian and "poet".

2009-11-11T07:03:00.000-08:00

There is a rather interesting transcribed speech here which discusses the poetical efforts of J. J. Sylvester, who was the first Jew to hold a professorship at Oxford. In particular, as mentioned by Bell, he wrote a poem called, "To a missing member of a family of terms in an algebraical formula" which starts as such:

Lone and discarded one! divorced by fate,
From thy wished-for fellows--whither art flown?
Where lingerest thou in thy bereaved estate,
Like some lost star or buried meteor stone?

So yeah, his math was better than his poetry.

What I'm Reading: Men of Mathematics

2009-11-07T17:26:00.000-08:00

Men of Mathematics by Eric Temple Bell is a collection of personal and professional biographies of thirty great mathematicians. The first three are from ancient Greece: Zeno, Eudoxus, and Archimedes. The others are from the seventeenth to nineteenth centuries, from Descartes to Cantor.

It was written in 1937, which shows. Not just when the author refers to Bertrand Russell in the present tense, but also, for example, when he mentions in the introduction that certain "writers and artists (some from Hollywood)" have been interested in "how many of the great mathematicians have been perverts." ("None.")

The great thing about this book is the richness of detail Bell gives in the details of the subjects' personal lives. One gets the sense that Bell cares for the mathematicians personally, and also that there is a distinct lack of objectivity here. Gauss in particular is the recipient of torrents of praise. Anyone who opposed or insulted him, or who he disliked, is savaged, usually hilariously.

For example, Napoleon apparently once told Laplace that he would read his book "the first free month he could find." Bell, after relating this, proceeds to write that "Newton and Gauss might have been equal to the task; Napoleon no doubt could have turned the pages in his month without greatly tiring himself."

Later in the chapter, Bell mentions that Gauss didn't like Lord Byron, then goes on to describe the poet in terms of "posturing", "reiterated world-weariness", "affected misanthropy", and "histrionics". He then points out that "no man who guzzled good brandy and pretty women as assiduously as Byron did could be so very weary of the world as the naughty young poet with the flashing eye and the shaking hand pretended to be."

Right now, I'm on the chapter about William Hamilton. It starts with relating that when Hamilton was thirteen he knew a language for every year of his life. Bell does not approve. ("Good God! What was the sense of it all?") At fourteen he wrote a letter of welcome to the Persian Ambassador, in Persian, which Bell imagines may have been responsible for the Ambassador giving an excuse to avoid meeting him. According to an article of Nature (from 1883!) I found on Google Books, it read like this:

As the heart of the worshiper is turned towards the altar of his sacred vision, and as the sunflower to the rays of the sun, so to thy polished radiance turns expanding itself the yet unblossomed rosebud of my mind, desiring warmer climates whose fragrancy and glorious splendor appear to warm and embalm the orbit about thee, the Star of the State, of brilliant lustre.

The article says it was received warmly.

So all in all, this book is amusing and a good read for those interested in the personalities behind the major mathematical discoveries of the seventeenth to nineteenth centuries. But while Bell was an accomplished mathematician in his own right, Men of Mathematics is in large part a work of opinion, and does not in all cases possess the rigor that history is capable of having.

An issue of terminology.

2009-05-21T13:34:00.000-07:00

Wheel in the Sky

2009-05-09T11:58:00.001-07:00

The Analytical Language of John Wilkins

2009-04-29T18:08:00.000-07:00

The article can be found here. The main idea can be summed up as

...it is clear that there is no classification of the Universe not being arbitrary and full of conjectures. The reason for this is very simple: we do not know what the universe is.

I can think of four systems of elements that supposedly categorize all things. Two are from history, the Western classical elements and the Chinese version. One is from a game, and one is something else entirely. There are surely more, as many as imagination allows, and that is the key.

At the end of Maybe Logic, there are exercises. One of them is to gather a bunch of arbitrary objects and divide them into two categories in as many different ways as possible. This is a concrete demonstration of the principle that categories are a mental phenomenon and not a property of the universe-as-such. Thus philosophical attempts to discover the "true nature" of the "real categories" of the objects in the universe are misguided.

Spring Break: Alchemical Texts and Darwin Art

2009-04-22T13:37:00.000-07:00

I went to new Haven for spring break.

The Beinecke Rare book and Manuscript Library has several interesting past, current, and future exhibitions dealing with the history of math and science. For example, "Trees in Fact and Fable" examines its subject from several disciplines, including botany. There is one about mathematics in early modern England, an International Year of Astronomy exhibition called "Starry Messenger", and some works on alchemy in the European imagination. (I got to see that one.)

Also at Yale, the Yale Center for British Art is hosting an exhibition called "Endless Forms": Charles Darwin, Natural Science, and the Visual Arts (site here). It deals with the effect the publication of Origin of Species had on the arts world. It includes scientific diagrams, paintings depicting prehistoric humans and other creatures, fanciful illustrations of past humanity, Romantic depictions of "the struggle for existence", and photographs of non-European people meant for the scientific study of race. Or I might say pseudo-scientific.

Science and Math Tattoos

2009-04-18T11:45:00.000-07:00

Check out the Science Tattoo Emporium. One interesting thing is that many people who get tattoos involving the DNA molecule use it for its heavy symbolic content. For example, here, here, here, and my favorite here.

2009-03-16T17:24:00.000-07:00

A long, detailed, and nuanced discussion of the affair between Galileo and the Church may be found here. Part of the interest is in the exposure of some of the "mythology" that surround this issue. Such as this:

According to one of these readings, Galileo knew the Earth to go round the Sun, as Copernicus had written, rather than the converse as implied in several Biblical passages. The Church would not allow science to disprove the revealed truth of Scripture, however, and hence threw Galileo to the Inquisition where he was forced under threat of torture to disclaim this opinion and never speak of it again. He was then imprisoned under house arrest for the remainder of his life, a clear example of the conflict between scientific investigation of the world around us and the presumed infallible authority of the Bible.

Which is a story often cited as the truth of history.

2009-03-16T16:46:00.000-07:00

A comic.

Double comic after a long break!

2009-02-04T17:36:00.000-08:00

Double comic. There are two.

10 Remarkable Female Mathematicians

2008-11-17T13:46:00.000-08:00

I wrote a post much like this one, a while back.
The post is here.

Mathy comics

2008-11-07T15:48:00.000-08:00

In Questionable Content, Hannelore reveals one of her hobbies: theoretical knitting. If you've ever wondered why bubble-form tests ask you to use a #2 pencil, XKCD reveals the awful truth. And Saturday Morning Breakfast Cereal pits A 17th Century Explorer vs. Calculus.

Science, meet Politics. Charmed, I'm sure

2008-10-25T10:11:00.000-07:00

This was never meant to be a political blog, but what happens now will become history. And in this day and age, scientific research is intertwined with politics. So yeah. I can't think of anything witty to say.

A 29th twofer.

2008-09-29T16:23:00.000-07:00

First, an interesting little article on a very strange research project that, strangely, seems to have produced some results. Strange ones. Basically, they are trying to map personality traits according to geography in the US.

Second, an article about the mistakes made by Einstein. This, in my opinion, is an important thing to write and read about, because it defuses the idea that science is ideology. If a physicist whose name and countenance are symbols of intelligence itself can be shown to have been wrong - well, that's only to be expected really.

All the other great scientists of the past were wrong about things, because advancing the state of the art in knowledge is like that - better theories, better data, better models - but never The Truth. Just a useful approximation. This is the power and glory of science: to recognize and correct its errors.

This is why I dislike terms such as "Darwinism" and "evolutionism" so intensely. They attempt to crystallize the ever changing ideas of how species originate into an ideology that, if somehow shown to be false, can then be defeated. It surely must be comforting for opponents of science and knowledge to think that this could possibly work. However, they are just as wrong about this as they are about so many other things.

Beauty's Truth

2008-09-28T12:52:00.000-07:00

Find here an old but very interesting essay about what makes a scientific theory beautiful. Essentially, the author argues that the beauty of a theory comes from its truth, and not the other way around. That is, having preconceived notions of what a beautiful theory is can prevent scientists from accepting one that may describe reality better.

Especially interesting given the recent news about the Large Hadron Supercollider. Let's see what happens - maybe we will live to see new theories emerge. This is a historic time.

The Innumeracy of Intellectuals

2008-08-12T15:51:00.001-07:00

Magnolia Sitter has linked to this very interesting article. The double standard of engineering students who don't like humanities versus humanities students who don't like math or science is especially interesting.

Also interesting to me is that in medieval education, the trivium and quadrivium consisted of grammar, logic, and rhetoric; and arithmetic, geometry, music, and astronomy, respectively. Of the seven courses in the original liberal arts curriculum, two were explicitly mathematical and three were closely related to mathematics.

To paraphrase Robert Anton Wilson, "Some people at the university specialized in manipulating mathematical symbols, others in verbal ones. Because the people who manipulated verbal symbols were better with language, they got to define themselves as the intellectuals."

The Most Important Image Ever Taken

2008-08-12T14:54:00.000-07:00

According to this video, it is the Hubble Deep Field, and I'm inclined to agree. Whenever I look at it, I feel chills. This image, and the even higher resolution Ultra Deep Field, contain hundreds and hundreds of points, blobs, and smears of light. Each one is not a star, but an entire galaxy.

On the less inspired side of things, if you want to see something that manifests such intense stupidity and willful, possibly malevolent ignorance that it is capable of causing physical pain in those who watch it, go here.

The basic "idea", for those unwilling or unable to subject themselves to such gibbering imbecility, is staggeringly nonsensical, and incandescently moronic; a train wreck of epic cognitive ineptitude: Rainbows are a government conspiracy.

Or as the video says: "Everywhere we look, the visible spectrum, is rainbows. This cannot be natural." This is why we need math and science education with tough, rigorous, and reality-based curricula.

The Quest to Solve the Hardest Math Problem

2008-08-08T06:22:00.000-07:00

A recent article (which appears not to be available online) in Mental Floss talked some about the quest to prove the Poincaré conjecture. It's the sort of thing that, at some level, seems to be true intuitively but is a nightmare to prove. One of the people who attempted to do so eventually quit math and took up poetry, another swore he would not marry until he proved it - and died a bachelor.

Between 2002 and 2003, a reclusive mathematician named Perelman posted online an outline of a proof, which was then filled out and verified by others. He refused the Fields Medal which the ICM offered him for this feat. Given that the conjecture was proposed in 1904, and this only after Poincaré himself found a flaw in his own proof, the problem went unsolved for over a hundred years.

As for Mental Floss, it is awesome. And they have a couple of articles in the "More" section that are math related: 5 Rap Songs That don't Make the Grade and a proof that Vampires are Mathematically Impossible. Which is flawed because the quoted author assumes that vampires turn everyone they feed on instead of killing most of them. To use Anne Rice's terminology, not everyone deserves the Dark Gift.

“I don’t imitate nature. I try and understand her operating principles.”

2008-08-08T05:53:00.000-07:00

As John Todd quotes Bucky Fuller in this interview. This is a vision of ecology, industry, and design that uses natural processes as inspiration. It will take me a while to absorb so much exciting thought. There is an example of this sort of design at the NC Estuarium, which is next to a wetlands that was built to clean the runoff from the town of Washington before releasing it into the Pamlico Sound. It is home to muskrats, dragonflies, turtles, birds, fish, lily pads, and algae.

univers revolved

2008-07-25T11:23:00.000-07:00

Check out this font based on taking letter forms and revolving them though the third dimension. The infinitely symmetrical forms allow the letters to be placed into three dimensional and nonlinear arrangements. Very cool, mind bending stuff.

Steampunk Star Wars

2008-07-14T18:40:00.000-07:00

While not technically the sort of thing this blog is supposed to be for, this is far too cool to leave alone. My favorite part is the steampunk name for a light saber.

Three Amazing Women

2008-07-09T12:59:00.000-07:00

In chronological order.

1. Hypatia of Alexandria is the first known female mathematician. She lived during the fourth and fifth centuries in Roman Egypt, which was long enough ago that her life has had time to become legendary, as well as fodder for propaganda. Her father was Theon, the last mathematician to work at the Library of Alexandria before it was shut down by the Patriarch (which is what we now call the Pope, more or less) in 391. Having a distinguished father, she also had considerable mathematical and philosophical gifts. She edited her father's commentaries on the Almagest and the Elements, among other things.

Clearly a woman who went far beyond the sort of roles that women were expected to fill in the ancient world, she was a follower of Plotinus and taught the works of Plato and Aristotle. To men, in public. (shocking!) Despite this, and her paganism, there were Christians who respected her and attended her lectures. Being a Neoplatonist, she was not so much into "the delights of the flesh", "carnal desires", or that sort of thing, and this earned her a reputation for virtue among Christians and pretty much everybody else. Socrates Scholasticus describes her thus:

"There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner, which she had acquired in consequence of the cultivation of her mind, she not unfrequently appeared in public in presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more."

In one story she gets rid of a man who is interested in her by showing him one of her used pads (so to speak, women used cloth rags at the time) and telling him that such things were not beautiful. It's really the ultimate rebuttal, in a way. I can just imagine:

Suitor: Pleased would I be, Hypatia, if you would allow me to recieve your favors.
Hypatia: And what would those be?
Suitor: The favors of marriage, Hypatia.
Hypatia: And why do you want such favors from me? Please tell me, because I do not know why it is that you would desire such fleshy congress.
Suitor: For you are beautiful, O Hypatia, and it is in the nature of Man to desire and pursue beauty.
Hypatia, getting out rags and waving them around: Even now do you think carnal desires are a source of beauty? Is this a beautiful sight to you?
Suitor, runnng away: GAAAAHHHHHH!
Hypatia: Right then, time for some Ptolemy.

(Props to Plato, these things are hard to write.)

Eventually, a rumor got around that Hypatia, by talking to a man named Orestes, was preventing him from coming to terms with the Archbishop. So a mob of Christians led by someone calling himself "Peter" tracked her down and murdered her.

Later writers would claim that she was somehow doing magic on people using music or astronomy or something. Even later than that, it would be suggested that the story of St. Catherine (whose symbol is the wheel) was a Christianized version of Hypatia's life. Since then, her story has been used to illustrate the brutality of mob violence, to show the irrationality of religious fervor, and as an archetype of a powerfully intellectual woman, which she surely was.

2. As far as I know, Ada Lovelace is the only woman to ever have a programming language named after her. She, along with Charles Babbage, did work that helped inspire a whole subgenre of speculative fiction - steampunk. Babbage designed a mechanical computer called the Analytical Engine. Lovelace translated an Italian article about the machine and then described how to use it to do mathematical calculations like finding the Bernoulli numbers. For a number of reasons, Babbage was not able to build the machine, which is the might-have-been that inspired the literary genre. So Lovelace wrote a program for the computer before it existed, which was also before any computers existed. Not only that, she had foresight about the not-really-scientific roles computers might someday play:

"the Engine might compose elaborate and scientific pieces of music of any degree of complexity or extent."

So there she is anticipating experimental computer music in a time before the Rite of Spring caused riots and a sudden demand for smelling salts at the apothecary. As QC points out, steampunk music would be awesome.

I don't want to neglect some other things that only make the story more amazing. Her maiden name was Byron, because she was the daughter of Lord Byron. The name Lovelace is from having married the Earl of Lovelace. So basically the noble daughter of a famous poet was an accomplished mathematician and, using the overwhelming force of her intellect, wrote programs for doing advanced mathematical calculations on a nonexistent computer. She then predicted correctly that computers would be useful in the arts.

Babbage wrote about her:

Forget this world and all its troubles and if
possible its multitudinous Charlatans - every thing
in short but the Enchantress of Numbers.

So we can add "inspired an engineer to write poetry" to the list.
All this before dying from cancer at the age of 36.

3. Hedy Lamarr was a Hollywood actress born in Austria. She starred in films from the thirties through the fifties and during the war, helped invent a technology to control missiles. It was too advanced to be used effectively.

Lamarr became notorious first in 1933 for starring in a movie made in Czechoslovakia with overt nudity and sexuality. Later she married a fascist who attempted to buy all the copies of the film. He didn't let her act, instead bringing her to military and business meetings. Being talented mathematically, she became knowledgeable about technical matters. When not using her as eye candy for other fascists, he kept her in his castle. But her husband didn't exactly have supervillain effectiveness, and Lamarr escaped in 1937.

During the war, she and her neighbor George Antheil worked on and invented the first spread spectrum communications technique. It used the roll from a player piano to "hop" the frequency of a radio signal between eighty-eight different values (the number of keys on a piano.) Unfortunately, the technology of the time wasn't up to the task of actually using it to control a torpedo as it was intended to do. Lamarr and Antheil were granted a patent in 1942, but the technology was not actually used until 1962, when the patent had already expired. They made no money from it.

Lamarr made films during the forties, and the government considered her celebrity endorsement of War Bonds to be more important to the war effort than her inventing. Her output declined in the fifties, and in the sixties she was involved in a shoplifting scandal and left public life.

In 1997 the EFF gave Lamarr special recognition for her invention:

"Actress Hedy Lamarr and composer George Antheil are being honored by the EFF this year with a special award for their trail-blazing development of a technology that has become a key component of wireless data systems. In 1942 Lamarr, once named the "most beautiful woman in the world" and Antheil, dubbed "the bad boy of music" patented the concept of "frequency-hopping" that is now the basis for the spread spectrum radio systems used in the products of over 40 companies manufacturing items ranging from cell phones to wireless networking systems."

After that, she was recognized in various ways. Corel used her image to market CorelDRAW 8 (she sued them and reached a settlement), Elyse Singer wrote a play about the invention, and the Inventor's Day in German speaking countries is celebrated on her birthday, November 9.

Lamarr died in 2000, at the age of 86.

Getting some eats.

2008-07-07T19:24:00.000-07:00

At long last a comic!

Some thoughts about psychology.

2008-05-28T14:00:00.000-07:00

Cracked.com recently had an article about "Psychological experiments that prove humanity is doomed." It's one of those cases where the humor comes mainly from the sensation of truth attempting to flank the rational mind. In this case, uncomfortable truth.
Psychology doesn't always get a lot of respect, which is a shame, because we tend to discover interesting things about whatever we study with the scientific method. Human personality and behavior is no different. Science can shatter the myths we hold about ourselves and our own mental independence just as well as it can demolish literal interpretations of mythology.
I have even heard people go so far as to say that psychology isn't a science. This attitude is usually based on a definition of science that focuses on the subject matter instead of the method. There is no reason for such a distinction when it is easier to say that science is when people study something scientifically.

It's like a graphic novel.

2008-05-23T16:02:00.000-07:00

It's a comic on 5/23, appropriately irregular. I think this would make a great movie.

A Nightmare

2008-04-25T13:34:00.001-07:00

After a long pause, a comic. Friday this time.

Math Wizard in Action!

2008-04-19T11:31:00.000-07:00

Standing astride the World of Math, Quadratic Formula staff in hand and readying the Theta Bomb, he prepares to deliver a beatdown to a smack-talking triangle.

This is exactly what it feels like to solve difficult problems.

My current favorite math history book is Robert Kaplan's The Nothing That Is, which discusses the history of the development of the number and the philosophical issues surrounding it as well as the very concepts of nothing, nonexistence, and absence.

Towards the end he examines how these ideas can make what seem like sensible questions into utter hash. Treating "nothing" as any other object (though linguistically a noun) leads to questions like "Why is there something rather than nothing?" Speaking as though "nonexistence" is just an alternative activity, people are said to experience their own absence in proclamations regarding who would have been better off if they had never lived. The final chapter is called "The Unthinkable" for good reason.

The title comes from a Wallace Stevens poem.

The Yanghui Triangle

2008-03-31T13:44:00.000-07:00

What is that? Well, it's used to find the coefficients of a binomial expansion. Just like Pascal's triangle. Actually, it is the same thing. Yanghui was a Chinese mathematician from the thirteenth century who did work with binomial expansions and created a triangular array to contain them. One interesting thing about the document we have of the ancient Chinese version of the triangle is that it appears to contain an error. Here is a high resolution image where you can try to find it:

An ancient scribe must have been having an off day. Notice also that for some reason, the symbol for "ten" looks a lot like the Arabic numerals version, just turned on its side. There is more history of the triangle, and the answer to where the mistake is, here.

Mathematical Personalities: Gerolamo Cardano

2008-03-20T15:48:00.000-07:00

Sorry about the lack of updates, but the life has been busy.

Down to business.

The Italian Renaissance was one of those interesting times in world history in which the great increase in knowledge and trade led to an explosion of wealth, which led to more knowledge and trade, and so on...but we have to remember that these are not modern people. This was a time that was altogether more violent and chaotic than anything we see today (in the Western world, anyway) because for some reason the people of that time and place really knew how to work the chaos into something rather creative. I mentioned Cardano before, but in the context of the solution to the cubic equation, which is a great story. Now I want to mention some other things about the man. The events of his life illustrate the differences between that era and this.

Remember that the modern world comes largely from the Enlightenment with heavy doses of Modernism and Postmodernism thrown into the mix. This all happened after the Renaissance, and the 16th century was more like the medieval world than like ours. His mother fled the plague. He had trouble starting a career in medicine because his parents were not married. He was a gambler and wrote a book on the subject that was the first to make probability into a science. While he contributed to the sciences and mathematics, he was an astrologer as well and was arrested for heresy after casting the horoscope of Jesus.

People who are brilliant enough to make lasting contributions to mathematics have a tendency to be eccentric. Add to this the fact that many discoveries were made before what we would call the modern era, and we find that the people of the history of mathematics are incredible to study.

mathematics, astronomy, religion

2008-03-15T06:37:00.000-07:00

I found something on the Library of Congress website: a collection of images from the Vatican Library of mathematical manuscripts from between the ninth and fifteenth centuries.

Saturday Comic:

The background is an ultra deep field photo taken by the Hubble Space Telescope and credit is due to ESA and NASA.

Pascal's quote is from the Pensées, numbers 205-206.

Reputation

2008-03-12T15:24:00.000-07:00

Rooster image from here.
All kinds of quotations about mathematics and by mathematicians can be found here.

According to St. Augustine:

The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

This makes much more sense when you realize that in this context, a "mathematician" is what we would call an "astrologer".

An interesting essay and a comic.

2008-03-08T10:57:00.000-08:00

I found something interesting on Reddit today, which is somewhat connected to the content of yesterday's posting. Here is an essay by a teacher and mathematician arguing that math education is too focused on notation and formulas instead of teaching kids about the art of discovering elegant numerical patterns and relationships. He suggests that mathematics should be art and play.

Also, here is the math subreddit.

Solving the Cubic Equation

2008-03-07T14:47:00.000-08:00

Stories like this are why I like math history. Students, often rightly, think that the subject they study is too abstract and removed from the real human experience. Most high school math is not especially related to the everyday life of teenagers. Furthermore, it does rather seem like whenever a teacher presents examples of how the abstractions can be applied to practical problems regarding quantities such as mass or distance, the students quail at the prospect of the dreaded Word Problems. To be fair these do tend to be more complicated than the abstract ones, because life is complicated and the abstractions of math tend to simplify matters for the purpose of examining certain underlying numerical relationships. The question of whether education should inspire students to look past their daily life and expand their scope of vision is for later.

At any rate, the story of how Gerolamo Cardano came to publish the solution to certain cubic equations is full of rich human drama: secrecy, competition, betrayal, and an obscure poem, (which apparently sacrificed a certain amount of accuracy to maintain the proper Italian rhyme scheme.) The people who developed the mathematics we use today were people much like us, and the facts and techniques we try to teach to our teenagers were slowly developed over thousands of years. Mathematics is relevant to our lives not because we will need to solve equations in the course of daily life, but because using and developing the concept of number is a fundamental human activity. Much like poetry.

Theorem Loafing

2008-03-05T19:08:00.000-08:00

Time for the Wednesday comic.

Next comic early.

2008-02-29T16:40:00.000-08:00

More adventures with Calvin.

The most beautiful machine in the world.

2008-02-27T13:47:00.000-08:00

Calendars and astronomy have always been one of the main purposes to which people have put mathematical research. Being aware of time, we seem to feel the need to locate ourselves within it. Whether we conceptualize time as linear or cyclical, much of the ritual we concoct has to do with establishing a relationship between it and ourselves and culture.

The Long Now Foundation was established to help humanity think better in the long term. From their website:

The Long Now Foundation hopes to provide counterpoint to today's "faster/cheaper" mind set and promote "slower/better" thinking. We hope to creatively foster responsibility in the framework of the next 10,000 years.

One of the projects they are working on to that goal is the machine to which the title of the post refers - a clock meant to keep accurate time for the next ten thousand years. The prototype that has already been built is not only visually stunning, but the very idea of building an artifact meant to still be around - and functioning - in a hundred centuries...I find it practically too audacious to contemplate. The designer chose the time frame he did because that is about the age of the oldest known human technology: pottery shards.

Here is the reason I think this may be the most beautiful machine there is: this sort of long term thinking is the sort we must do if we hope to survive as a species. It is beautiful mechanically, but the idea that led to the design choices that led to the design itself are the source of the beauty.

Every technology is the consequence of a series of ideas and choices. We desperately need good ones.

Comic time.

2008-02-27T13:42:00.000-08:00

It's Wednesday and that means time for a comic.

Historical problem from Medieval China

2008-02-24T13:26:00.000-08:00

The book The Nine Chapters on the Mathematical Art is an ancient, anonymous Chinese work that gives techniques for solving various practical problems. In 1261, a mathematician named Yang Hui wrote a commentary called "Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications." This work supposedly contains the "Problem of the Broken Bamboo" which deals with properties of right triangles. If the height of the stalk and of the break are known, the distance from the base of the stalk to the place where the tip touches the ground can be found with the Pythagorean theorem, which was known to them. The angles involved can be found using trigonometry.

Now for a comic.

2008-02-23T19:28:00.000-08:00

The reason I like John Napier is that he was one of those people who was like a superhero. Between inventing logarithms, writing theology, and cultivating a terrifying reputation, he had plenty of time left over for general awesomeness.

A new blog.

2008-02-23T18:46:00.000-08:00

I created this to blog about the history of math and science, which is a subject I really enjoy. I teach math in a high school, and it is not always possible to talk much about this sort of thing, because it is not actually in the curriculum. Science has historical developments which are covered in science class, but the development of math is not.

One of my favorite parts of the subject is the personalities of the people who made historical developments. I like to tell my classes that anyone smart enough to make a lasting contribution to mathematics is likely to also be a bit colorful. One of my favorites is John Napier, hence the name of the blog.

If you have noticed that the description mentions comics, let's just say that public domain images from old artwork are very handy.