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.

Word Problem of the Week: The Most Sexist Word Problem Ever

D. H. Hill, who wrote the book containing the "Bad Pork" problem, loved to put his opinions into his math problems. Later in life he was a Confederate officer, which is not a fact I had realized when I wrote that entry.

His disapproval did not end at the moral standing of the northern American character. He was also, not surprisingly, unimpressed by the nascent Women's Liberation movement. Even so, the way he characterizes the delegates of the Third National Woman's Rights Convention is so sexist it just about took my breath away (because I was laughing so hard.)

It's number 70.

Word Problem(s) of the Week(s) - Meat!

Meat related Algebra Problems!

When I showed this to my students, one comment was, "There is more than one kind of bacon?"

Word Problem of the Week: Bad Pork

One of my fundamental premises is that the themes of algebra problems bear some relationship to the life of the times when they were written. That's why this one from 1857 really caught my eye. Fraudulent practices of this type must have been fairly common in the days before the government had the strength or inclination to regulate them.

As it happens, we know what Cincinnati looked like around that time, at least near the waterfront. There are newly restored daguerreotype images of the city taken in 1848 which have exceedingly high resolution. I mean, high-end digital camera level resolution.

Some of my favorite old problems are the ones that, like this one, combine very specific detail with bizarre contrivance. In this case, the contrivance comes toward the end:

The price at which he sold the pork per pound multiplied by the cost per pound of the chemical process was 3 cents.

 So, what sort of quantity is this, anyway? The unit is "dollars-squared per pound-squared" which is more than a little weird. Especially since he could have just divided the two numbers to get a dimensionless ratio.

But of course, I know why he did it. He needed the equations for solving it to be of a certain type, and having the problem make a little more sense would have upset that situation.

So imagine it said that the selling price p of the pork was 12 times as much as the price c of the chemical process, each in cents per pound. Then we have that the profit, 45000 = the gain - the expenditure. The gain is simply 10000p, and the expenditure is 10000 + 10000c.

So, 45000 = 10000p - 10000c - 10000. Clear the thousands to make it 45 = 10p - 10c - 10. But since p = 12c, This becomes 45 = 120c - 10c - 10 or 55 = 110c. This makes the cost of the chemical process half a cent and the price he charges for the pork 6 cents per pound.

This is a nice little linear system problem. But setting it up the way D. H. Hill did it, you end up having to clear a denominator, making the equation quadratic and introducing an extraneous solution.

In other words, this problem sacrifices some of its plausibility to make a different, harder solution necessary.

Squeeze Theorem

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

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

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.