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.

Against Humanism

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.

The Analytical Language of John Wilkins

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.