Find the Fallacy

Two mathematicians begin a discussion by assuming that A=B. Then they begin making changes. They add an equal term to both sides of the equation. A+A=A+B or 2A=A+B. Then they subtract an equal number from both sides. 2A -2B = A+B -2B. They look at this and agree that this all makes sense. Then they agree that this equation can be simplified by a simple operation. 2(A-B) = A+B-2B. This again simplifies to 2(A-B) = A-B. The result astounds them. Two times (A-B) equals (A-B)? Two equals one????

The answer is that the fallacy occurs only at the final step.

We get to 2(A-B) = A-B. But at this point we should recall that if you subtract something from itself you always get zero. So A-B = 0 and 2(A-B) also equals 0.

It is surely the case that 0 = 0, but thus stated the air of paradox has disappeared.

You obviously can’t get from there to the conclusion 2=1, only to the conclusion that

2(0) = 1(0).

The final step would be division by zero, which the basic rules prohibit. Precisely to avoid such paradoxes as this!

My point? Just that zero, and the rules governing its use are more exotic human conceptual inventions than one might think. The casualness with which we usually treat zero comes from familiarity, not from simplicity. This confirms the point I sought to make yesterday, that our most successful inventions are also discoveries, and vice versa.

This turns out to be, not especially Kantian, but certainly Jamesian, or well within the area of their overlap.

Interacting with the World, and Ciceronianus

This weekend, Ciceronianus, author of a wonderful neo-Stoic blog, posted on epistemology. He wrote: "I'm bemused from time to time by the view that the world (as in "reality" or the universe) is, in part at least, our creation, or perhaps is created by each of us for himself/herself."

He named first Kant and later Wilfred Sellars as examples of the sort of epistemologist he has in mind. Let us use the broad term "constructivism" for the broad PoV that Kant and Sellars share, and that bemuses our Stoic.

Ciceronianus, if I understand him, then proceeds to the assertion that such constructivism is either pointlessly obvious or wildly wrong. The obvious and uninteresting point is that "we are human beings, and as such interact with the world as human beings do." Yet those who are most serious about urging that "they shape the world" seem to want to go much further than this, and that furtherness is what bothers Ciceronianus.

I contributed a thought of my own to his comment section.
 

There is a tee shirt that bears upon some of the issues you raise. It shows somewhat anthropomorphized versions of the Greek letter pi (Π) and of the expression √-1.

Pi is saying to √-1, “Get real.” And√-1is replying, “Be rational!”

I’ll pause now while you slap your knees.

The joke, of course, is that pi is an example of an “irrational” but real number, while √-1 is the definition of i, the foundation of the imaginary numbers.

Fermat's Last Tango

I recently watched a DVD of the musical comedy Fermat's Last Tango,  produced by The Clay Mathematics Institute.

I may discuss the production in greater depth at some point next week.  Right now I'd just like to record the lyrics of one of the songs: Your Proof Contains a Hole.

Just a bit of background information first. This scene takes place in a quasi-heaven called "The AfterMath," where famous mathematicians live out their immortality. The bulk of the song is sung by Fermat, who believes his own immorality requires the insolubility of his theorem. He is singing to Daniel Keanes, a fictionalization of Andrew Wiles, who though still very much alive is visiting the AfterMath.

The chorus consists of AfterMath regulars: Euclid; Pythagoras; Newton; Gauss.

Without further ado, then....

Fermat: Your proof contains a flaw, Professor Keane,
               It destroys the whole foundation of your finely tuned machine:
               I hate to be a spoilsport, I know it was your goal
               But your proof contains a big fat hole.

Keane: A hole?
Chorus: A hole?
Keane: My proof contains a hole?

Fermat: I didn't want to be the one to saaaay
               I know this is upsetting, please show some self-control
               But your proof contains a big fat hole.

Caring only about your work

The philosopher Nel Noddings once (in 1984) wrote: "Most of us commonly take as pejorative, 'He cares only about money'; but we have mixed feelings when we hear, 'He cares only about mathematics' or 'She cares only about music.' In part, we react this way because we feel that a person who cares only about money is likely to hurt others in pursuit of it, while one who cares only about mathematics is a harmless and, perhaps, admirable person who is denying himself the pleasures of life in his devotion to an esoteric object."

I do think this is a fair general statement of the usual connotations of such sentences. In the spirit of expanding Noddings' comment, we might observe: Someone who cares only about mathematics may be a harmless drudge, or may end up discovering a mathematical anomaly in radio waves that in turn improves worldwide communications for the good of us all. Someone who cares only about money may obtain it by violence or fraud, doing active harm to others.

More generally, we often extend respect, even if it is a grudging respect, to anyone who is "married to" his occupation, especially if it is an occupation which may derive some of its appeal from the intellectually challenging nature of the work.  "He cares only about his work" said of a lawyer or an engineer, is no bad thing.

Still, I think Noddings goes a bit too far when she writes about how such a person is "denying himself" pleasures of life outside the job: suggesting that this is a praise-worthy sacrifice. If our mathematician prefers the blackboard and the computer lab to the pleasures of, say, sexual relations, child-rearing, neighborly comaraderie, adopting the former over all of the latter is no sacrifice.

OTOH, if a mathematician believes that he has a duty to stay at the blackboard hour after hour, because the human race needs better exploitation of radio frequencies, then I can imagine that his doing so would be a praise-worthy sacrifice. But in such a case, we would not in any case express the situation by saying that he "cares only for" the mathematics.  

On Eternity

This quotation has recently become popular in the blogosphere, or that portion of it intrerested in religious/areligious argumentation.

It has been posted recently by the blogger poor taste for example, and by the estimable Epicurean Dealmaker, and by Aaron Kinney of Kill the Afterlife!.

Why should I buck a trend? Especially a wholesome one?

The passage going the rounds comes from A PORTRAIT OF THE ARTIST AS A YOUNG MAN by James Joyce, in which he grapples with the notion of forever.

"For ever! For all eternity! Not for a year or for an age but for ever. Try to imagine the awful meaning of this. You have often seen the sand on the seashore. How fine are its tiny grains! And how many of those tiny little grains go to make up the small handful which a child grasps in its play. Now imagine a mountain of that sand, a million miles high, reaching from the earth to the farthest heavens, and a million miles broad, extending to remotest space, and a million miles in thickness; and imagine such an enormous mass of countless particles of sand multiplied as often as there are leaves in the forest, drops of water in the mighty ocean, feathers on birds, scales on fish, hairs on animals, atoms in the vast expanse of the air: and imagine that at the end of every million years a little bird came to that mountain and carried away in its beak a tiny grain of that sand. How many millions upon millions of centuries would pass before that bird had carried away even a square foot of that mountain, how many eons upon eons of ages before it had carried away all? Yet at the end of that immense stretch of time not even one instant of eternity could be said to have ended. At the end of all those billions and trillions of years eternity would have scarcely begun. And if that mountain rose again after it had been all carried away, and if the bird came again and carried it all away again grain by grain, and if it so rose and sank as many times as there are stars in the sky, atoms in the air, drops of water in the sea, leaves on the trees, feathers upon birds, scales upon fish, hairs upon animals, at the end of all those innumerable risings and sinkings of that immeasurably vast mountain not one single instant of eternity could be said to have ended; even then, at the end of such a period, after that eon of time the mere thought of which makes our very brain reel dizzily, eternity would scarcely have begun."

Mathematics and Reality

Exists there a world of Platonic form aside from (transcending) the world of matter and particulars?

Today I'll give a three-part answer: no, and no, and maybe. The "maybe" is the interesting part.

1. I don't think there's any broad case to be made for a Platonic treatment of universals as such.

2. Nor do I think its helpful to morality to think of virtues as disembodied essences -- Justice, Courage, Temperance, the Good.

3. On the third hand, there IS a strong case to be made that numbers and the equations made from them ought to be considered as an independent world, which mathematicians discover rather than invent.

This case is based largely on the experiences reported to us by the greatest mathematicians since the days of Pythagoras.

Heinrich Hertz put it well. "One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them."

Who was Hertz that we should pay heed to his views on math? A physicist who obtained his Ph.D. from the University of Berlin in 1880, and whose work helped refine the mathematical theory of electromagnetism developed before him by Faraday and Maxwell. He earned his bones in terms of the relationship between physical reality and abstract formulae employed to describe it. If he believed (as have many others of equal eminence) that the formulae have an independent sort of existence, this isn't testimony that those of us who have shown less facility in their use than he should take lightly!

Math notation

I'm not accustomed to thinking of mathematical notation as having a history. After all, the equals sign, "=," just is what it is. When did I first encounter that sign? First grade? I can't remember, and this "time out of mind" quality is what protects it from historical inquiry.

But of course such notation does have a history. The number zero has a history that supports a legion of scholars, as does the history of pi. How we write a number is connected with the way we think about numbers, so that writing "0" is something very different from writing "zero."

Still, let's not drift away into number theory. Let's stick with a sample of what one might call "pure notation," the equal sign. This = was invented in 1557 by Robert Recorde, of Wales, in what was actually the first book on algebra printed in the British isles.

Here's the URL for an image of the page in which the equal sign first appears.

http://members.aol.com/jeff94100/witte.jpg

You'll notice if you go to that page that Mr. Recorde used two rather longish parallel lines.

The old-fashioned fonts can make comprehension difficult. So I'll reproduce the final paragraph for you, just before his list of seven equations.

"Howbeit, for each alteration of equations, I will propound a few examples, because the extraction of their roots, may the more aptly be wrought. And to avoid the tedious repetition of these words, 'is equal to,' I will set as I do often in working, a pair of parallels, or remote lines of one length, thus: ===========, because no two things can be more equal. And now mark these numbers."

We should all have as solid a claim to immortality as Mr. Recorde and his scheme for the avoidance of tedious repetition.

Uncontrolled Growth

Our "thought for the day" comes from Michael Crichton, and from the novel that made him famous, The Andromeda Strain (1969).

"The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth."

The mathematics of geometrical progression is what famously freaked out the Reverend Malthus, too.

Anatoly Fomenko

I'd like to recommend the drawings of Russian mathematician Anatoly Fomenko.

Here's a URL. http://anatoly-fomenko.com/art/main.php

From what I understand, Fomenko is a brilliant mathematician, who has taken it into his head that he's also a historian -- although his theories in the latter sphere (his "New Chronology") have the whiff of the crackpot about them.

Still, the drawings have a wonderful quality to them, combining minimalism and surrealism if you will. What is the minimum you can do on a two-dimensional surface to produce a surreal result?

String Theory

Until a couple of years ago, string theory (or "superstring theory" as it is also known in grander moments) got very respectful, sometimes adoring, attention from those reporters whose job it is to convey some sense of cutting-edge science to the general public.

String theory involves ten or eleven space-time dimensions (the exact number varies with the theorist)-- and the "strings" or one-dimensional objects that twitter about in space-time so conceived show up as mere points, or point-like sub-atomic particles, within the usual old-fashioned four dimensions.

String theory was designed to serve as a "theory of everything," and in particular to integrate the other forces of nature with the one force that seems to resist any detailed integration into a unified theory: gravity.

One problem, though, is that string theorists haven't been able to suggest any experiments that might bear upon their theories at all. There is only so much blackboard scribbling that scientists will attend to before they start getting impatient for data. Is this all just pure mathematics, and not any part of the physical sciences at all?

Unfortunately, no. Mathematicians don't recognize string theorists are part of their domain either. Mathematics is about rigorous deductive proof, and string theorists don't have that. They have suggestions and hypotheses. What they're doing looks too much like science for the mathematicians, just as it looks too much like mathematics for the scientists.

That doesn't establish that it's wrong, only that if its right the disciplinary name for its sort of rightness has yet to be invented.

Poincare Conjecture

For those who, like me, have heard talk of the recently proven Poincare conjecture, but who have no grasp of higher mathematics, this is a neat story which conveys, at least, some sense of why people who DO have such a grasp are fascinated by the subject.

http://blog.360.yahoo.com/blog-5CcC9sQlcqhUDN1P5SbXoWa5x.0-?cq=1&p=4

Derbyshire & Clifford

I've recently been reading a book by John Derbyshire on the history of algebra, called "Unknown Quantity." It's a follow-up of sorts to his book on the still unproven (and unfalsified) Riemann Conjecture concerning prime numbers, which I believe I mentioned earlier in this blog.

Anyway, in the more recent Derbyshire book, I came across a brief passage about the 19th century English mathematician William Kingdon Clifford.

Clifford is an important name for those of us who admire William James. Not for his mathematical accomplishments (which are undoubted) but for his venture into philosophy (which was unfortunate). It was Clifford who wrote that it is our general duty to our species to guard our minds from unproven beliefs, even if they should happen to be true. Believing something on insufficient evidence is like accepting into one's body "a pestilence which may shortly master our own body and then spread to the rest of the town."

James thought this simply ridiculous, with "too much of robustious pathos in the voice" as he said, and it inspired him to write the famous/notorious essay "The Will to Believe."

But Derbyshire's passage doesn't mention any of that. He's interested in Clifford as a mathematician, one responsible for generalizing "Hamilton's quaternions into a whole family of n-dimensional algebras."

I won't explain here what that means, what Hamilton's quarternions are. It is fully discussed in the context of the book. I was simply impressed by this example of what seems to have been a general 19th century truth. One could still get a reputatiuon as a considerable philosopher without any academic credentials in the field, but one did have to have academic credentials in some field.

Clifford earned his bones in math, as James did in psychology.

Gravity without the Apple

I heard the story this way. Isaac Newton, as a young man, was very interested in optics. He got into an argument with Robert Hooke, an older man who had a reputation in the field to defend and wasn't happy with this bright whipper-snapper.

One day, at a coffee shop, (Ye Olde Starbucks), Hooke encountered a friend of his named Halley. An astronomer, as you've probably guessed.

Said Halley, "I've just discovered a comet. Also, I see in some old records from 86 years ago that there was another one like it then. It might be the same comet.

"If I could prove they were the same, I could name this returning rock after myself and attain immortality. There's a problem, though.

"To prove the point, I'd have to know of a general mathematical principle or law that explains why objects moving around the sun follow the path they do. Do you know of such a law?"

Hooke didn't. But he knew a bright young man who should be given some problem to work on that would take his mind off optics for a few years.

So Hooke introduced his rival to his friend, Newton did the work that helped Halley establish the two comets were the same one, and creating calculus and the laws of motion in the process, and the world lived happily ever after.

Okay, that story is a tad over-simplified. But It's fun, and truer than the one about the apple.

Riemann hypothesis

My recent reading has included a book by John Derbyshire, who bravely attempts to explain the Riemann hypothesis, and efforts to prove or falsify it, to those of us without much formal education in math.

This problem seems to have been the inspiration for the drama Proof, which won the 2001 Tony Award for best play and became the basis for a Hollywood movie. That movie starred Gwyneth Paltrow as Catherine, a young woman who learned mathematics from her father, and who may have devised an extraordinary solution to a long-standing puzzle about prime numbers. [On Broadway, Mary-Louise Parker was Catherine.]

In the meantime, and in the real world, the Riemann hypothesis remains unproven.

What is it? Just to test whether I've been wasting my time with the book, I'll try to explain it now. It involves the distribution of prime numbers, and the fact that primes are bunched up in the lower numbers but become more scarce further up the line. The early primes are: 1, 2, 3, 5, 7, 11, and so forth. They're already beginning to thin out, even in that brief series.

The thinning continues. There are 26 primes of less than 100. Between 401 (inclusive!) and 500, there are just 17. Between 901 and 1,000, there are only 14 primes.

Is there a rule that governs this thinning out? Yes, the rule is (unsurprisingly) called the Prime Number Theorem (PNT). That isn't the Riemann hypothesis (RH) though. The RH proper is something Riemann suggested as a rather incidental spin-off of his work on establishing the PNT.

In order to state the PNT, one has to invoke complex numbers (numbers with both a real and an imaginary component), and the complex value of something Riemann called the zeta function, because he was too humble to call it the "Riemann zeta function" as everyone else now does, http://en.wikipedia.org/wiki/Riemann_zeta_function

The Riemann hypothesis is that the real part of any non-trivial zero of the zeta function is 1/2.

We're almost to the point where that bare statement makes some sense. The "zero" of any function is that value at which the function produces a result of zero. Simple example: x squared – 6x + 9 = y. [I write "x squared" because I haven't figured out superscripts here.] The "zero of f(x)" in the relevant sense is that value of x for which y = 0. In this case, 9 - 18 + 9 = 0, so the zero of f(x) is 3.

The zeta function generated by the search for the PNT though, turned out to have an infinite number of zeros, each complex numbers. Some of these zeros (the "trivial" ones) are generated when the variable s (don't ask) is an even negative integer: -2, -4, -6 and so on. But that leaves us with the non-trivial zeros and the question: do they all (as Riemann believed) have a real component of exactly 1/2?

No wonder the Anthony Hopkins character lost his mind, and Catherine seemed on the way to losing hers.