'Limited CPU Power…'
by Daniel M. Ryan The title phrase is a common one in mathematical circles; it’s a polite way of saying "math is the best academic subject to make the clever person feel stupid." I’m surprised that it isn’t known as the "great equalizer" in other areas of the academy. This moniker is apt, because an immersion in math is a real, if not proven, way for the intellectual, with a Mensan (or higher) level of aptitude, to empathize with the struggler back in the old high school. If you’re looking for a psychological reason why so many mathematicians are inclined towards socialism, this one will do it. Whether it be caused by acquired guilt or drained loneliness, the realization that you’ve bumped into a subject that is as hard, for you, as most of the courses were for the burly fellow who went off to trade school, makes identification with "the worker" a real temptation. Even if the commiserative overlap doesn’t reach very far, as many a mathie who has "seen the proletarian light" discovers when he takes his newfound insight to the sports bar. The temptation to socialism is one of two that exist in the somewhat lonely math world. The other is the myth of the "math god." Just like the investment-field myth about the (usually apocryphal) fellow who bought near the bottom and sold near the top, the "math god" is a character who managed to construct a proof, sometimes from known mathematical techniques, that is not only true, but unprecedented as well. "How in blazes did he (sometimes "she") come up with that?" If said proof unlocks a great mystery, and/or provides a new tool to crack open previously uncrackable problems, then the gates of Math Immortality are truly open for the prover. Like many myths, the "Myth of the Math God" tends to stabilize on a particular image, an archetype. The typical eponymous archetype, I suspect, looks a lot like a French aristocrat. There has been at least one in math annals, although I am sad to relate that the "mathematician" of highest rank in the ranks of the immortals was a mathematical physicist, the 7th Duc de Broglie. If it’s any consolation to mathies, though, Louis de Broglie’s title was only titular, due to France previously becoming a republic. Guillaume, the Marquis de l'Hôpital’s, was the real deal. Of course, for Anglophiles, there is the alternate eponym, but dreams of sitting at table, where one of the tablers is a noble Frenchman, is standard enough to make this gibe a useful deflater: "Wow! If you study hard enough, you may win the favor of the legitimate pretender to the throne of France!" It is easy to make fun of people who seem motivated by social climbing, but that’s because the incentives offered to the social climber work reliably for the lonely, a group of people who are easy to make fun of anyway. Admittedly, social climbing is an incentive like any other, and tends to produce both intended and unintended consequences resulting from the effort it calls forth. This incentive is gradually being replaced by professional pride in the present-day math world. Increasingly, mathematics is becoming a standalone trade, a self-sustaining profession. Its place in modern society and economy is actually a supplier of intellectual capital goods. That’s right; mathematicians supply intellectual capital goods – producers’ goods – that go into the making of products. The more abstract part of it, the most being number theory, exists for three reasons: as a standalone consumers’ good for people in, or interested in, the field for its own sake; as a retooler for more quotidian math fields; and, as a means to make teaching math easier. The mathematician whose life’s work is centered on creating a magnificent "Principles of All Math" actually aspires to be a supplier of teaching techniques for math pedagogues. The profession is slowly becoming self-aware of this status. The professionalization of the field isn’t confined to the selection of winners of the Fields Medal. The emergence of the "gotcha" is a more humble means of remaking the discipline of math into a free-standing profession. The use of "gotchas" is an informal means of professional development. The aim of a particular gotcha is to scotch out a particular kind of stupid-think: being suckered by a plausible non sequitur. Some of them are of old standing. The most known class of them slyly smuggle in division by zero, to "prove" that, for example, 1 = 2. Another one, of old standing, is: "what does (x – a)*(x – b)*…*(x – z) equal?" In this one, the devil is in the ellipsis; the 24th term is (x – x), equaling zero, which makes the entire expression equal to zero. I have yet to hear of someone who actually sat down and calculated this by hand – it’d be a monster – and prove, by hand calculation, that the answer is indeed zero. Most of the entrapped give up, as I did when I first encountered it. A more sophisticated "gotcha" was recently published in The American Scientist Online. In the middle of the article, in the section "Who’s On First?", a mock dialogue between Socrates and a young boy is included. Unlike the real thing, though, Socrates winds up looking like somewhat of a fool. The specific non sequitur, which the poor fellow drifts into, is his inference that implies that there are more than 128 possible outcomes to a seven-game World Series where no ties are allowed. If you’re ignorant of, or have forgotten (as I did,) this (provably solid) conclusion, then you’ll fall right into a combinatorics "gotcha." Working out the possibilities that Socrates includes in his attempt to use the Binomal Theorem, to analyze what turns into a "meta-World-Series," will reveal that Socrates did indeed double-count along the way to his first odds computation. The boy’s correct answer is provable, by going through a count of all 128 possible outcomes and deriving an equation from them. This more "ideal" (formal) method is different from the one the kid used, but both give the same (correct) result. As mathematics becomes fully professionalized, there’ll be more gotchas a’coming, like this one, an exchange between a bright kid and the teacher: B.K.: "I’ve found a repeating fraction!" Teacher: "Oh, a fraction that gives you a repeating decimal? Good! What is it?" B.K.: "Expressed in normal fractional numbers, 1/5." Teacher: "Uh-h-h…1/5 is 0.2." [Pause, as the youngster in questions seems unabashed.] "I suppose it is possible to say that it can be a repeating decimal, but you’d have to have all zeroes behind it. [He regains his confidence.] If you include that kind of decimal in the definition of repeating decimals, you drain the meaning from the term, as all decimal numbers are repeatable if you allow repeating zeroes. Do you understand what I mean?" [B.K nods.] Teacher: "What’s wrong?" B.K.: "What’s wrong is that it’s possible for what we call ‘1/5’ to repeat." [Now, the teacher is wondering is one of them is unhinged, but a small part of him isn’t sure which one.] Teacher: "1/5 is 0.2. By the generally-accepted definition of ‘repeating decimal’ it isn’t one. Would you care to explain how you got yourself into denying that?" [Now, B.K. is showing a somewhat winning smile.] B.K.: "1/5 in base 2 is 1/101. In base 2, this equals 0.00110011…" [As B.K. rattles on, the teacher is trying to recall the Board-accepted euphemism for "ARE YOU FINISHED BRAGGING YET!?"…] Another kind of possible "gotcha" is one that flushes out a too-hasty quest for symmetry, the source of a lot of thoughtlessness in mathematics. One that tripped me up twice is the proven result that eiπ = –1. This famous result is only one single element of a whole slew of them, though: –1 = eiπ+2iπ*n, where n is an integer. Not realizing this got me two "brownoff" points, on two different occasions. There are a quite a few "gotchas" in applied mathematics, too. Here’s one derived, courtesy of my own creaking brain, from Friedrich Hayek’s Pure Theory of Capital. It’s based upon the graph on p. 117 – figure 6. "Money has time value. In order to get the value of a sum of money, therefore, you have to specify a time, relative to an arbitrary but fixed ‘time zero’ where present value is computed. If you graph the value of a fixed sum of money over time, you’ll see an inverse exponential curve, if the inflation rate exceeds the interest rate. "Now, the paradoxical but neat thing about such a curve is that, because it’s an inverse exponential curve, integrating it will lead to the inverse exponential curve being inverted with respect to the x-axis. This means that the integrated value of money over time, as integrated by the t variable, is actually a negative amount! Now, an odd result such as this one, which is proven in today’s handout, has to, of course, be properly interpreted – Yes?" "Uh – sir? If money has to have a time value, then isn’t integrating it over time sort-of detaching it from the time component? How is it possible for money to have a sort of meta-value over a time continuum when we already assumed that its value has to be pinpointed with respect to a specific time?" No, the sir in question is not really glad that the fellow in the student’s desk asked the question. He’s just being polite.
December 28, 2006 Daniel M. Ryan [send him mail] is a Canadian whose reach has long exceeded his grasp. He's currently wearing out his thumb with pen and paper. Copyright © 2006 LewRockwell.com |