I’ve decided to publish here some of the many essays I’ve written on mathematical and philosophical subjects.  I realize that kind of stuff bores a lot of people (or, possibly, intimidates them–it often seems to me that so many people are afraid of math because they were taught to fear it–I think most people have a lot more ability at math than they give themselves credit for).  But, what the hey, this sort of thing occupies a large part of my thinking, and you might as well know about it.  You can always just skip them.

In fact, for a lot of them, that may be a good idea.  These essays are more or less the finished products.  The real fun has been on the scratch sheets, the rough and tumble of chasing down an idea and working it out, and the finished product is way more dense than the fun stuff.

And saying that gives me an excuse to copy in an old poem (written in a house backed by woods in Jasper, Texas).  I like to think the poem could apply equally well to mathematics or poetry itself.


When I used to work deferential equations

I had a neat sheet I kept track of it on

inking in ordered chains of tuneful logic

like dew collecting on a rusty screen

or sugar crumbling grain by grain to nectar

but when the next change stumped me as it did

more times than I’ll mention, out came Stubby

the friendly yellow # 2 chewed pencil

with round blunt lead and sweat-stained foreskin wood

long unwhittled in the sharpener’s whirling knives

and Eraser’s thin brass jacket bitten flat

to raise just one more day’s meniscus of correction

and out came Oscar, the mangy scratch-sheet

and all his haypile fat comedian friends

glad to see me as always and so we romp

we tussle in the briar-patch scratchy grass-fields

until a girl without an ounce fat on her

strides by in a pure white muscular gown

and I am dressed in wedding-white for church again

with an ink string tie the height of fashion

over the starchy ruffle of my beating chest

when I used to work D for any ol’ equations


It occurred to me that one ought to be able to fit a circle to any parabola, thus creating a smooth two-dimensional shape more or less like a plane section along the long axis of an egg. In fact, the resulting graph is not merely smooth, but continuous, which makes it interesting, since the continuity derives from the seamless fusion of two functions.

I decided to work with the basic parabolic equation, y = x2, since all of the others are transformations, and similar principles would apply. Besides, the curve I was most interested in was parabolic at its small end and circular at its large, and I wanted the bottom of that curve to pass through (0, 0). (It’s possible to construct a closed curve circular at each end and parabolic on the sides by calculating a smaller and a larger circle for each parabola and replacing the bottom of the parabola with the bottom of the smaller circle.)

In order to make the fusion seamless, it would be necessary to join the two curves precisely where they had the same slope, or tangent—in other words, where the derivatives of the two functions were equal.

Since the absolute value of Dx for the function y = x2 is always rising but never reaches infinity, and since the absolute value of Dx for the general equation of a circle can only reach infinity when x = the radius of the implied circle, the fusion of the two curves must always occur before that point. In other words, no matter how great the absolute value of x, you can always close the loop of the parabola by fusing it to a circle.

It develops that a circle fused to a parabola at x and –x will have its center at (0, x2 + ½ ) and a radius of (x2 + ¼ )1/2 (absolute value). From this it’s obvious that the radius of the circle can never be equal to or greater than the height of the center above (0,0), which verifies the assertion that one can close a parabola of any size with a circle. The single exception occurs when x = 0. When x = 0, the parabola disappears completely, leaving only a circle of radius ½ centered at (0, ½ )—the bottom of the curve still passing through (0,0).

The greater the absolute value of x, the larger the circle surmounting the parabola, obviously enough, and the more elongated the egg shape. I had been thinking all along of rotating the curve on the y-axis to generate a more or less ovoid three-dimensional surface. Now I wondered what the most pleasing proportions for such an object would be—in other words, what two-dimensional closed curve would produce the most pleasant proportions when rotated.

I appealed to the Golden Mean, and decided to find what value of x would produce a curve whose major axis (along the y-axis) was approximately 1.61803399 times the length of the diameter of its surmounting circle (twice the radius)—in short, a curve approximately 1/.61803399 times as tall as it is wide at the “shoulders.”

It turns out that to produce the desired curve, x must = approximately 2.05817103, the radius of the circle = approximately 1/.61803399 + ½ (or 2.11803399), and the major axis = approximately 6.85410197.

One could argue whether those are indeed the most pleasing proportions, but regardless, I got a good title out of it for this discourse.   

Whatever I Want It to Mean

Occasionally a creative writing student would comment, “A poem (or other piece of writing) means whatever you want it to mean.”  I hated this brainless attitude, and still do.  It is not true.  The author intends one or more, usually more, way more, specific effects.  It takes work and it takes intelligence.  I blame what I call the hidden meaning theory of poetry, a favorite of hack teachers everywhere.  You’ll recognize it.

The hidden meaning theory of poetry substitutes ingenuity, however preposterous, for understanding.  It arose with Eliot’s generation, the culture’s defense against the perceived “difficulty” of contemporary poetry.  Its basic tenets are that the job of the poet is to hide as many meanings as possible in the poem, and the job of the student is to dig them out.  The more “meanings” you can invent, the more likely you are to get an A.

It’s a thoroughly boring activity, of interest only to hacks and suck-ups, and if that’s what most people think poetry is about, no wonder they hate the stuff.

But the malaise runs even deeper, is even more insidious.  When I was in college and grad school, it was dogma among intellectuals that we could not discover the actual nature of things, that language itself, with its insistence on conceptualization, was a barrier to “the truth.”  (I view language as a path into reality instead a screen that filters it out.)  Philosophers insisted that we could not know how we knew things, and therefore all language was suspect, and therefore no viewpoint could be impeached.  They insisted that one’s language shaped one’s perceptions, was the dominant force.  (Among linguists, this is known as the now widely discredited Whorfian hypothesis–usually phrased as “The Eskimos have thirty words for snow,” or some similar and equally misleading statement).

Again and again the academics of my day insisted that perception was primary, that one viewpoint was good as another.

I’m think of this now because my daughter just mentioned a television channel called Belief Net, which is apparently devoted entirely to “Christian” programming.  When exactly did “belief” come to mean “contrary to evidence and reason”?  My other daughter once described to me a member of her church who declared that “faith is believing in things that don’t make any sense.”

No it isn’t.  Faith is more nearly holding to the truth no matter what the fashions may be.  Sticking to the facts, no matter how widely disparaged you may be for doing so.

There IS such a thing as Christ-like forbearance and sacrifice.  It’s a true facet of human nature.  The story of Christ presents that capability in transforming myth.  (When I say “myth,” I do not mean “silly little story,” but something so powerful it can reshape society.  Neither do I necessarily mean “historically inaccurate.”

It’s also true that sort of forbearance and sacrifice delivers better results than any of the alternatives, especially the alternatives of violence and revenge.  (It’s extremely rare and difficult to practice, though.  One must first learn to factor oneself out of the calculations.)

I do ridicule the “believer” who thinks the world was created six thousand years ago.  No it wasn’t.  The Bible was never intended as a physics manual.  I do not hesitate to mock anyone who insists that his or her opinion is as valid–simply because he or she “believes” it–as painstakingly acquired evidence, the fruits of long and difficult reason, or fidelity to fact.

But I blame those midcentury philosophers and academics as much as anyone.  They were the ones who floated the notion in the first place that all points of view were equally valid.  They were largely humanists, and feeling the hurt because science was getting all the attention.  It was their take on relativity, their attempt to be as cutting-edge and theoretical as the physicists.

But it leaves them with no comeback to the fruitcake fundies.  Typically what is fashionable in philosophy gradually leaks into the cultural substrata and becomes a given several decades later.  Now, although the philosophers and academics have long since moved on, everybody “knows” that any point of view is as good as any other.

No it aint.

I’m Glad You’re Dead, You Rascal You

Okay, bin Laden is dead.  I, like most people, am glad of that.  He murdered 3000 people on 11 September 2001, and maybe as many as a thousand more in various other incidents.  He was a terrible human.  How awful must it be to be bitter and violent still when you get old?  I don’t envy Al-Zawahri, who looks at least as murderously dyspeptic, and who will surely soon die as well.

But what are we to make of the people who made billions of dollars during our tragedy and disarray?  What are we to make of the Wall Streeters who crashed the economy, the greedheads at insurance companies, people who did not hesistate to profit while millions of Americans not only went through shock and mourning, but lost their savings and their jobs and their hope?  Many of us not only mourned the dead, but mourned a lost country–my first reaction on 11 September was horror and shock and a deep sick feeling, and my second was fear for my country, how angry and vicious we were now, in the aftermath, almost certain to become.

What did most people think of arms profiteers on either side during World War II?  And yet arms profiteering is now our major industry, while taking care of our own citizens doesn’t even make the to-do list.  If I had contempt for Osama Bin Laden, how much more contempt must I have for those who never hesitated not only to strip the vulnerable average citizen of as much money as possible, but to portray anyone who opposed their thievery as unpatriotic?  How much contempt for those who lobbied to change the laws even further in their favor (and often succeeded)?

Greed is NOT good.  This is the firm opinion of the species over the thousands of years of its existence.  Greed serves only itself, looks after only itself, ignores facts and probabilities.  Greed will do anything, anything at all, to further itself.

Can we begin to talk about this now?  Now that the obvious lie has been exploded, the lie that opposition to crooks means you hate the U. S.  Can we begin to talk about these creeps for the sordid little spirits they are, now that they can no longer wrap themselves in the flag and declaim, while taking our money to offshore banks, that they are the only true patriots?

Never mind punishment.  As the KJV puts it, verily they have their reward.  I would not have been willing to be Osama bin Laden for all the money in the world.  Neither would I have been willing to be Anthony Mazilo.

What about the truth?  Is it okay to face the facts about these creeps now?