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.

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."

The Most Important Image Ever Taken

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.

An interesting essay and a comic.

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

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.