An Incomplete Education
Page 61
The story told, bear in mind that Archimedes’ principle, as opposed to Archimedes’ bath, applies not only to bodies immersed in water but to bodies floating on it, and not only to solids but to liquids and gases. It explains, in addition to why ships float, why balloons rise, and it warns that in determining what will and what won’t sink, float, or fly away, both weight and volume must be considered, not to mention shape and position. If you can manage to remember anything here beyond “Eureka!” you might go for “specific gravity,” the ratio of a given density of a solid or a liquid to the density of water (and of a gas to air), and a term that, while unknown to Archimedes, pretty much sums up what his principle winds up being all about. FIBONACCI SERIES
He started with 1, as who among us wouldn’t? Then he repeated it: 1, 1. Then he added the two: 1, 1, 2. And he kept on adding, always the concluding pair of what was now, unmistakably, a series: 1, 1, 2, 3; 1, 1, 2, 3, 5; 1, 1, 2, 3, 5, 8. And so on, all the way to 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. And why stop there? So … 233, 377, 610, 987, 1,597, but you get the picture. Now, if you’re wondering what, apart from the lack of late-night television in thirteenth-century Pisa, accounts for Fibonacci’s perseverance, we can report that he was working out the solution to the problem of how many pairs of rabbits can be produced from a single pair of rabbits, if every month every pair begets a new pair, which from the second month on is itself productive, assuming that none of the rabbits die (or become bored) and each pair consists of a male and a female.
Wait, it gets interesting. Turns out that the Fibonacci series not only connects with controlled population growth, it also keeps popping up on nature walks. For instance, the ratio of scales distributed in opposing spirals around a pine cone is 5:8; of bumps around a pineapple, 8:13; of seeds in the center of a sunflower, 21:34. All of which are adjacent Fibonacci pairs. If nature doesn’t do it for you, try culture. Seems the ratio between any two adjacent Fibonacci numbers (after 3) is roughly 1:1.618—none other than the ratio behind the celebrated Golden Section, in mathematics the division of a line segment into two parts such that the whole segment is to the larger part as the larger part is to the smaller. The Golden Rectangle, whose length and width are the two parts of a line segment divided Golden Section–style, is a big deal in art and architecture because its proportions are so satisfying to the eye, incorporating, to hear the aestheticians tell it, both static unity and dynamic variety. The classic example here is the facade of the Parthenon, but Golden Rectangles have also been found in paintings by everybody from Leonardo to Mondrian. THE LINNAEAN SYSTEM OF TAXONOMIC CLASSIFICATION
What’s it to you whether or not we have an orderly, scientifically sound method for cataloguing plants and animals? Not much. But it comes in awfully handy for scientists who, up until the middle of the eighteenth century, had to say something like “that little yellow flower with the spots on its petals” every time they wanted to compare notes. It was the Swedish botanist Carolus Linnaeus who came up with the idea, shocking in its day, of dividing plants into twenty-four “classes” distinguished by their sex (for which he has been called “the Freud of the botanical world”), that is, according to the length and number of stamens and pistils in their flowers, then subdividing those classes into “orders” based on the number of pistils. The sexual approach made sense in light of the then-novel idea of dividing God’s creatures into broad categories, called “species,” according to their individual characteristics; Linnaeus grouped his species by their ability to reproduce the same characteristics, generation after generation.
Besides coming up with a game plan, Linnaeus hit on the idea of using binomial nomenclature—that is, two names, one for the genus (as in “Homo”), one for the species (as in “sapiens”)—as a shorthand labeling system to replace the cumbersome and confusing descriptive names botanists were using at the time. Then he got everyone he knew to run around frantically naming plants and animals as fast as they could, hoping to get absolutely everything in creation named before some unsuspecting botanist or zoologist somewhere used the same name for a different species, thereby messing up the system. As a result, many of these names were arbitrary, spur-of-the-moment affairs that may, today, strike us as unsystematic and not all that helpful.
Modern naturalists also disagree with certain of Linnaeus’ categories, and, to the old “kingdoms” of Plantae (plants) and Animalia (animals) they’ve added a third, Protista, to cover single-celled amoebas, bacteria, slime molds, and the like. Nevertheless, the Linnaean system, arbitrary and artificial as it may be, is still the best we’ve got. BROWNIAN MOVEMENT
The zigzag, irregular dance done by minute particles of matter when suspended in a liquid; named for Robert Brown, the botanist who established its existence in 1827, while watching microscopic pollen grains float around in water. Brown noticed that although the direction any particular pollen grain would take was unpredictable, all the grains moved faster when the water got hotter and slowed down as it cooled. Einstein later did a paper on Brownian movement, theorizing that the grains were always in motion because they were being batted around by water molecules; the hotter the water, the faster the water molecules moved, and the more direct hits to the pollen grains. Eventually, Brownian movement became an important substantiating factor of the kinetic molecular theory of matter, which states that matter is composed of tiny particles (a.k.a. molecules) that are constantly in motion. THE DOPPLER EFFECT
The change in the frequency of a wave (whether of sound or of light) that occurs whenever there is a change in the distance between the source and the receiver; named for the early nineteenth-century Austrian physicist Christian Doppler. If the source of the waves and the receiver are approaching each other (or one is approaching the other), Doppler observed, the frequency of the wavelengths increases and the waves get shorter, producing high-pitched sounds and bluish light. If the source and receiver are moving farther apart, sound waves are pitched lower and light appears reddish. (The most commonly cited example of the Doppler effect: the train whistle that screeches in the distance, dropping in pitch as it approaches the platform where you’re standing.) Used in radar to track the velocity of a moving object; in astronomy to measure distances between and rotations of stars, planets, and entire galaxies; also to track satellites. When, in 1929, the astronomer Edwin Hubble noticed that the light from distant stars was becoming redder, he took this “Doppler shift” or “red shift” to mean that the stars were rushing away from earth. His conclusion, known as Hubble’s Law and now generally accepted, was that the universe is expanding. BOOLEAN ALGEBRA
Ever since Aristotle—as George Boole, the nineteenth-century English mathematician, would have been the first to tell you—language has been getting in the way of truth. That is, the way we talk, with all its inaccuracy, ambiguity, and potential for hysteria, tends to mess up the way we think.
Boole’s solution: Get rid of the words altogether, instead letting symbols (a and b, x and y, P and Q, whatever; it doesn’t matter that they’re arbitrary as long as they’re precise) stand for the components of thought, which, not entirely co-incidentally, often turn out also to be the elements of formal logic. Then, manipulate the symbols mathematically, in a kind of mental algebra that’s based on such simple operations as negation (corresponding roughly to what you and I mean by “not”), conjunction (our “and”), and alternation (“or”); and that—here’s where you’re supposed to stand up and applaud—always reduces things, no matter how complicated or abstruse, to either a “1” (standing for “all” or “true”) or a “0” (standing for “nothing” or “false”). Thus was not only Boolean algebra born, but also the device known as the truth table (which lists all possible combinations of true and false values that can accrue in the interplay of two or more statements) and the intellectual specialty known as symbolic logic (which succeeded in wresting logic from the philosophers and delivering it over to the mathematicians).
For fifty years nobody but Lewis Carroll seems to
have gotten all that turned on by Boole’s “laws of thought,” but those laws did, shortly before World War I, knock the socks off Alfred North Whitehead and Bertrand Russell, who relied on them to establish more or less persuasively that not only was logic the proper domain of mathematics, it was indeed where the latter’s roots lay Since then there’s been no stemming the Boolean tide. First lawyers took to constructing truth tables in the courtroom. Then telephone-company engineers began thinking of parallel “on/off” switches and “open/closed” circuitry in 0-vs.-1 terms. To the point that today it’s the rare computer that doesn’t proceed along the relentlessly binary lines first laid down by Boole—as stunning an example as the atom bomb of how theory, and the wackier the better, fuels technology. MÖBIUS STRIP
Take an ordinary flat strip of paper. Give it a half-twist. Now Scotch-tape the two ends together to form a loop. OK, ready? Take a red Magic Marker and color in one side of the loop, and a green Magic Marker and color in the other. Whoops! That’s right: The strip has only one “side”—or do we mean “one” side? Easy to construct and hard to imagine, the so called Möbius strip (named after the nineteenth-century German mathematician and astronomer who first described it) also reacts strangely to scissors: Cut it along a line drawn lengthwise down its middle and you’ll get not two Möbius strips but a normal two-sided strip twice as long as the one you started with. (The mathematicians’ explanation for this: A Möbius strip has only one edge; the cut adds a second edge and with it a second side.) Now try cutting a new strip along a line one-third of the way in from its edge; we won’t tell you what you’ll wind up with, but it’s pretty weird, too.
As to what’s really going on here: The subject is geometric transformations undertaken in space, collectively known as topology. Superficially a series of exercises in paradox reminiscent of all those old M. C. Escher drawings, topology nevertheless crystallizes a bunch of the century’s top issues. For instance, in the case of the Möbius strip, what does it purport that something that appears at a given point to be two-sided, when traced round its continuum, is in fact onesided? One theory: that life is holistic, not reductionistic; that persisting in trying to consider everything in terms of its component parts (the brain as a collection of neurons, say) risks missing the whole (the brain as mind). GÖDEL’S INCOMPLETENESS THEOREM
The problem here—if you, unlike most twentieth-century theoreticians, still want to look at it as a problem—is self-reference, a historic stumbling block for logicians. In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms (statements like 1 = 1 that are accepted without proof) of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so, you’ll only create a larger system with its own unprovable statements. The implication is that all logical systems of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.
Gödel’s theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths. (Maybe, but we’ve met pocket calculators that are smarter than some of the people we went to high school with.) It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express new ideas. And it has been taken to imply that you’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows.
Fun—or at Least a Few Minutes—with Numbers
Either mathematics is poetry, all density and precision, or it’s the most flagrant type of entrepreneurship, in which nobody’s satisfied until his headquarters is taller than anybody else’s headquarters. Exhibit A: Numbers—the symbols we use to count up all the things around us that aren’t unique or continuous—an area where mathematicians have gotten either more lyrical or more rapacious over the years, depending on how you look at it. Below, some of the number “types” they’ve managed to corner and/or construct for themselves.
NATURAL
NATURE:
The counting numbers.
EXAMPLES:
1, 2, 3, 4, …
HISTORY:
In use since man first felt a need to know how many of something—goats, pots, issues of Penthouse—he owned. Some primitive societies still make do with “one,” “two,” and “many.”
PRACTICAL USES:
Conducting censuses, scoring basketball games, seating people at dinner parties.
MATHEMATICAL RESONANCES:
One of the first creative highs. The sense among early “numbers” types that they were on to something big, something real and abstract. Also, the inevitability of distinctions between “odd” and “even,” and “cardinal” (1, 2, 3; the counting numbers) and “ordinal” (1st, 2nd, 3rd; the ranking numbers). Ditto, of considerations of “infinity,” the quality of endlessness, symbolized as ∞, toward which the natural numbers were seen to progress. PRIME
NATURE:
Those natural numbers higher than 1 that can’t be evenly divided by any number other than themselves and 1. (Opposed to the composite numbers.)
EXAMPLES:
2, 3, 5, 7, 11, 13, 17, …
HISTORY:
Arduous. Also, odd. Euclid proved there was no end to the primes. Somebody else constructed a “sieve” to isolate them. Any number of Renaissance-and-later men tried to devise a formula that would generate all possible ones: no dice. Then there’s the Goldbach conjecture of 1,742. The largest known prime (as of this writing) has 7,816,230 digits; needless to say, they now have the computers on the case.
PRACTICAL USES:
None whatsoever, unless you find yourself at a mathematicians’ convention.
MATHEMATICAL RESONANCES:
A thorn in the side. Mathematicians can’t bear that they still don’t know why primes fall where they do, or, for that matter, how many there are. Then, too, primes are just one example of sets of numbers having special properties that the Greeks and others managed to isolate. Another time we’ll tell you about the perfects and the amicables. INTEGER
NATURE:
The natural numbers, plus 0 and all the negative numbers. Also called the whole numbers.
EXAMPLES:
…, —4, —3, —2, —1, 0, 1, 2, 3, 4, …
HISTORY:
Zero, an ancient Hindu innovation, arrived in Europe in Roman times, along with the Arabic numerals; it was an instant hit. Negative numbers (from the Latin negare, “to deny”) had a harder time: If the root of an equation was less than 0, the Greeks, so flexible in other respects, called it “fictitious” and threw it out. Only in the sixteenth century did an Italian, Cardano, systematically use negative numbers, pointing out that there can be less than nothing—a debt, for instance.
PRACTICAL USES:
Negative numbers make debit-heavy bookkeeping elegant. Zero makes 37 different from 307 or 3,700.
MATHEMATICAL
Now there was a genuine number line, stretching infinitely to both left and right.
RESONANCES:
Also, all equations of the A + x = 0 variety could be solved. And, there was 0, with all its inherent weirdness, to think about. RATIONAL
NATURE:
The integers, plus all fractions (or their decimal representations), positive and negative, a fraction being defined as the ratio of two integers, a/b, where b ≠ 0.
EXAMPLES:
HISTORY:
The Egyptians and the Babylonians could handle fractions, though (a) they didn’t know what they were doing, and (b) conservatives of the day sneered at the idea of a number like 7½, which was ne
ither 7 nor 8. As usual, it took the Greeks to dignify, codify, and promote the idea. Since then, second nature to everybody.
PRACTICAL USES:
Measuring (as opposed to counting), dealing with continuous quantities like age, Brie, and drapery material, which don’t always break into convenient pieces. Also, sharing.
MATHEMATICAL RESONANCES:
A flush of pleasure: Rationals felt good. Besides, all linear (or first-degree) equations, of the Ax + B = 0 variety, now had one—and only one—solution. Geometrically speaking, all you needed was a straightedge. REAL
NATURE:
The rational numbers, plus the zillions of irrational ones, which can’t be expressed in fraction form: imperfect square, cube, and higher roots; decimals that neither terminate nor repeat; the transcendentals.
EXAMPLES:
HISTORY:
, the first irrational, was discovered by Pythagoras, c. 500 b.c., when he constructed a right angle, each of whose sides was one unit long, and measured its hypotenuse. His proof that it couldn’t be put into fractional form was very upsetting, and resulted in the immediate sacrifice of a hundred oxen. π, the ratio of a circle’s circumference to its diameter, wasn’t proved to be irrational until 1761. In between, it was discovered that there are a whole lot more irrational numbers than rational ones.
PRACTICAL USES:
With the exception of π and e, few if any, despite all the brouhaha. On the other hand, square roots are as much a part of the collective consciousness as asparagus or the march from Carmen.
MATHEMATICAL
Like arriving at your surprise party with a 102° fever: elation tempered by shakiness.
RESONANCES:
But, the real numbers could be shown to correspond in a perfect, one-to-one way with all the points on a line (regardless of its length); this endless series of points was termed the Continuum (abbreviated C), and proved to be much more endless than the mere °° formed by the natural numbers. Also, you could now solve any quadratic (or second-degree) equation of the Ax2 + Bx + C = 0 variety. TRANSCENDENTAL