The Usefulness of Mathematical Learning - Part 2

Part 1

Continuing on the the second page, we find that

In all ages and countries where learning has prevailed, the mathematical sciences have been looked upon as the most considerable branch of it. The very name "Mathematician" implies no less, by which they were called either for their excellency or because of all the sciences they were first taught, or because they were judged to comprehend "all things mathematical". 

I'm no scholar of Greek, so I'm taking a bit of a liberty with what I think Arbuthnot was trying to say here. I do know that the original Greek word from which the term "mathematics" is derived has senses of meaning "what is learned." So it's not really surprising that he goes on to write

And amongst those that are commonly reckoned to be the seven Liberal Arts, four are mathematical, to wit, Arithmetic, Music, Astronomy, and Geometry.

He is referring here to the Quadrivium, which was the second course of study in the medieval university. The first was the Trivium, and consisted of Grammar, Logic and Rhetoric. Of course, contemporary people don't really think of music as a mathematical study, even though things like rhythm, pitch, tempo, and duration can all be quantified, and despite the explicitly mathematical origins of the art in the Western tradition starting with Pythagoras.

Not only that, logic actually is a branch of mathematics now, but wasn't in the 17th century. It started to be formalized as a mathematical study in a serious way by thinkers such as Boole, Pierce, and Russell.

In the next part, the author starts to describe why he thinks people don't learn math as much as they should.

But notwithstanding their excellency and reputation, they have not been taught nor studied so universally as some of the rest, which I take to have proceeded from the following causes:

The reasons he gives all sound very modern: people don't like to think so hard, they don't realize how useful math is, they think that only geniuses can learn it, it is not encouraged, and there aren't enough good teachers.

The Usefulness of Mathematical Learning - Part 1

I have, for quite some time, admired "An Essay on the Usefulness of Mathematical Learning in a Letter from a Gentleman in the City to his Friend in Oxford."

Though it was first printed in 1701, the arguments it makes are remarkably modern sounding - possibly because the time we live in actually has a lot in common with the late 17th and early 18th centuries.

So I think that this book is worth reading, but the format, typesetting, and 17th century usage make it tricky. Therefore, I will transcribe it into slightly more modern English while doing my best to preserve the tone and rhythm of the prose. Also, I mean to offer such commentary as may be useful to a student who endeavors to fully comprehend the import of the argument.

I think I'm in the spirit now, so here we go.

AN

ESSAY

ON

The Usefulness of

MATHEMATICAL LEARNING, &c.

Sir,

I am glad to hear from you that the study of the Mathematics is promoted and encouraged among the youth of your university. The great influence, which these sciences have on Philosophy and all useful learning, as well as the concerns of the public, may sufficiently recommend them to your choice and consideration: and the particular advantages, which you of that place enjoy, give us just reason to expect from you a suitable improvement in them. I have here sent you some short reflections upon the usefulness of mathematical learning which may serve as an argument to incite you to a closer and more vigorous pursuit of it.

Spring Break: Alchemical Texts and Darwin Art

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.

The Innumeracy of Intellectuals

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."

Three Amazing Women

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.

Historical problem from Medieval China

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.