by Judy Jones
NATURE:
A special category of irrational numbers: They’re real, all right, but they’re not algebraic.
EXAMPLES:
π, e, plus all the trigonometric ratios and logarithms they give rise to; also any quantity (except 0 and 1) raised “radically,” e.g., .
HISTORY:
In 1844 the Frenchman Joseph Liouville proved the existence of transcendentals— irrationals that would not serve as solutions to any of the infinite number of polynomial equations in any of the infinite number of degrees possible—but couldn’t come up with an example. In 1873 somebody else showed that e, the base of the so called “natural” logarithms, was transcendental. In 1882 still another somebody showed the same of π. Since then, so many transcendentals have turned up that a lot of us have simply stopped answering the doorbell.
PRACTICAL USES:
π (or pi, from the first letter of perimetron, Greek for “measurement around”) is indispensable in carpentry and construction. e figures in statistics and nuclear physics, and, in the days before slide rules and pocket calculators put logarithms out of business, enjoyed a certain vogue. They tell us that trigonometric functions allow you to gauge the height of a telephone pole from the length of its shadow, but where we live all the phone lines are underground.
MATHEMATICAL RESONANCES:
The transcendental numbers constitute an infinity even greater than the algebraic ones. Also, since π is transcendental, nobody has to try to “square the circle” using only a straightedge and compass anymore. Flash: eπi—1. COMPLEX
NATURE:
The real numbers, plus all the imaginary ones, based on i, defined as . But especially, these numbers when seen to consist of both real and imaginary components.
EXAMPLES:
HISTORY:
What do you do with an equation like x2 + 1 = 0? In 1777, Leonhard Euler, the most prolific mathematician of all time, introduced the symbol i (for “imaginary”) to stand for the . Like the X in “X-ray,” it was both a shriek of victory and an admission of perplexity. But imaginary numbers, like negative numbers, caught on.
PRACTICAL USES:
The description and handling of vector quantities, which have not only magnitude but direction.
MATHEMATICAL RESONANCES:
The complex numbers constitute a plane and correspond in a perfect one-to-one way with the points in it (cf the line of real numbers). Also, with them on board, you’ll always have the solutions to polynomial equations of the Axn + Bxn-1 + Cxn-2 + …+ Z = 0 variety, no matter how high n gets, and you’ll always have exactly the same number of solutions as the degree of n. With the complexes, man can solve any such polynomial, providing it has rational coefficients. In general, mathematicians were so excited by all of this that they couldn’t not push their luck: The result was quaternions (see next page). ALGEBRAIC
NATURE:
So called because they turn up as the solutions of algebra-style equations. Which means all of the above numbers except the transcendental ones.
EXAMPLES:
To reprise:
HISTORY:
The nineteenth century was the big one here: Gauss, Kummer, and Dedekind. Ideals, unique factorization, and Abelian number fields.
PRACTICAL USES:
There were repercussions in mathematical physics, but we admit that’s not rewiring the living-room lamp. Frankly, lingering over this category is probably a mistake.
MATHEMATICAL RESONANCES:
First and foremost, a way of gathering all of the above together so that they make sense and can breathe a bit. However, what was originally a scheme for investigating the solution of problems had now become a whole thing, an end in itself. The beat goes on. TRANSFINITE
NATURE:
“Styles” of infinity. And the sky’s the limit.
EXAMPLES:
(The letter is aleph, first of the Hebrew alphabet. Say “aleph-null,” “aleph-one,” “aleph-two,” and so on. Or maintain a discreet silence.)
HISTORY:
In 1895, the German mathematician Georg Cantor worked out “the arithmetics of infinity,” a whole series of endlessnesses. Aleph-null is simple infinity: the familiar ∞, roughly equivalent to the endlessness of the natural numbers (or, in some systems, the integers or even the rationals). Aleph-one is the infinity of the Continuum, C, roughly all the real numbers or, depending on how you look at it, all the points in a line (or a square or a cube). Aleph-two may or may not be the endlessness of all curves. And nobody has been able to figure out what aleph-three, let alone aleph-thirty, might stand for.
PRACTICAL USES:
Hmmm?
MATHEMATICAL RESONANCES:
The mathematical imagination rents the video of Yellow Submarine, then goes out for Indian food. While everybody continues to talk about this one, it was really Cantor’s baby, and he’s with his alephs now; a kind of dead end. QUATERNION
NATURE:
The complex number concept extended from two dimensions to four.
EXAMPLES:
HISTORY:
First suggested by William Hamilton in 1843 in response to his personal successes with complex numbers. Subsequently taken into even more dimensions—just as you suspected they would be.
PRACTICAL USES:
Engineers get into them. But you’re up past your bedtime.
MATHEMATICAL RESONANCES:
The feeling that, once having learned to walk, one can run. Also fly. The lesson here is that when you extend numbers beyond the complex stage, you do so at the expense of something called permanence; one by one, properties you took for granted fall away. For instance, with quaternions, you have to give up either the role 0 plays or multiplicative commutativity (i.e., x times y no longer equals y times x). Say good night, Gracie.
Double Whammy
As it happens, two of the biggest deals in modern science start with the letter e: entropy and evolution. Each spans two centuries (plus this one), and each has the kind of reverberations that can’t really be done justice to on rest-room walls. These are their stories. ENTROPY, THE LAWS OF THERMODYNAMICS, AND WHY YOU MAY HAVE BEEN FEELING TIRED AND LISTLESS LATELY
Entropy is what the Second Law of Thermodynamics is all about. Unfortunately, most of us seem to have gotten bogged down shortly after the First Law. Nothing surprising about that—understanding the First Law doesn’t demand much in the way of concentration or stamina. The basic principle of thermodynamics, the branch of physics dealing with the transformation of heat into work and other forms of energy, it simply states that “energy is conserved”; that is, it’s indestructible—there is always the same total amount of it in the universe. The First Law of Thermodynamics was a big deal back when people were trying to build better steam engines; today, it serves chiefly as the solid platform from which scientists and philosophers like to hurl themselves into the abyss of the Second Law.
The latter states, with deceptive simplicity, that “the entropy of the universe tends to a maximum.” Already, whether you know it or not, you’re in trouble. Entropy is a measure of the total disorder, randomness, or chaos in a system. In thermodynamics, it crops up every time any work gets done, since the only way work ever gets done is through heat transfers—hot water coming into contact with cool air, for instance, to produce the steam that drives a steam engine. At the outset, the system is said to be at a low level of entropy: The fast-moving water molecules are distinct from the slow-moving air molecules, and the whole thing has a kind of order to it. But as the heat flows into the cooler medium—as heat naturally does—the fast-moving molecules begin to spend themselves, mixing with the slow ones until eventually all the molecules are moving at approximately the same speed. At this point, we’re at maximum entropy; everything is at the same temperature, all the molecules are milling about without any particular order, and nothing more can be accomplished—the system has, literally and figuratively, run out of steam. The energy within the system is still there, but unless you separate the mo
lecules again, returning them to a state of tension by heating some and chilling others, it can’t be used to make things happen anymore.
Not your problem, you think, since your car runs on gas? Guess again, mush-for-brains. Outside thermodynamics, increased entropy—things going from a state of relative order to one of disorder—is the upshot of all natural actions. It’s what happens when you let your vodka on the rocks sit around for a while and before you know it the ice cubes have melted into the vodka and the only thing worth dealing with is the lemon twist. It’s what happens when you move into a nice neighborhood and within a few years it turns into the South Bronx. It’s what happens to your body from about age thirty on. And, as scientists predicted back in the nineteenth century, it’s what’s going to happen to the universe—or at least, to our galaxy—in the scenario known as the “heat death” of the universe: Solar energy, the product of a hot sun turning in cold space, will inevitably run out, molecular chaos will prevail, and we’ll all be left sitting in a lukewarm cosmic bath watching our toes decay.
You can see why the concept of entropy made the Victorians, already obsessed with keeping their corset stays in place, rather sad. Today, however, we prefer to brood about our weight and take a more objective approach to the Second Law, which has all sorts of interesting ramifications. It has, for instance, given us the “arrow of time,” a metaphor that expresses the purely physical distinction between past and present; the fact that time is an observable, one-way progression from order to disorder, and not just a figment of our imaginations. If you look at two still photos of eggs, one showing them in their unbroken shells, the other, scrambled in a frying pan, you immediately know which came first; reverse the order and you know something’s screwy. Time flows in one direction only. Another thing about progressive disorder: It’s synonymous with increasing complexity. The picture of the eggs in their shells is neater, simpler, and easier to make sense of than the one showing a gloppy mess in the pan. These two concepts—the arrow of time and the increased complexity of high-entropy states— helped convince scientists earlier in this century that the universe did, in fact, have a beginning, and encouraged them to come up with the idea of the Big Bang.
Entropy is also a hot topic among information theorists, who point out that increased complexity seems to add up to more, not less, order; and anyway, if it leads to richness of thought and communication, more power to it. Besides, according to the Second Law, the world should be falling apart by now, and it’s not. But maybe that’s because the universe is expanding; who’s to say that the apparent order we see in our corner of it isn’t balanced by total chaos at some other extremity? Then there’s the whole business of probability, with which entropy is embroiled in a way that’s enough, in itself, to make your brains hurt: There is no law that states absolutely that entropy can’t be reversed, that the scrambled eggs can’t re-form into their shells or your body revert to a state of adolescent glory—only that the probability of any of that happening is so small as to be virtually nil. At least in the universe as we know it. But what do we know? Maybe what looks like disorder to us is really order. And then there’s the fact (don’t slow down now or you’re lost) that the Second Law doesn’t seem to operate the same way on the subatomic level, where time flows in at least two directions, if it flows at all. Oh, there’s a lot more to think about, including, yes, the Third Law of Thermodynamics, but by now, you’re probably mulling over the First Law again. As for us, we’re smart enough to know maximum entropy once we’re up to our eyeballs in it. EVOLUTION, THE LAW OF NATURAL SELECTION, AND WHY YOU MAY HAVE BEEN FEELING STRESSED OUT AND PARANOID LATELY
Evolution—literally, “unrolling”—wasn’t Charles Darwin’s invention, much less his personal property. In fact, the Enlightenment, a whole century before, had celebrated the notion of progress, a basically upbeat take on the inevitability of gradual human change. Hegel had introduced evolution into philosophy, and Marx brought it into politics. Even scientists—geologists and biologists—had been speculating, since 1800 or so, on the evolutionary development of the earth and of the things that lived on it. What Darwin did was to make evolution come off as science—first, in his Origin of Species, published in 1859; later, in his Descent of Man, published in 1871.
By evolution Darwin meant that all plant and animal species are by their very nature mutable, able (and, more than that, under some pressure) to undergo small changes in their makeup; and that all existing plants and animals have developed in such a fashion from others that went before them. Also, that all life is interrelated and subject to the same laws; and that the history of living things is a unified one, unfolding continuously over millions and millions of years.
Here is how Darwin said evolution worked: Given that nature is competitive, that more daisies and starfish and foxes are “produced than can possibly survive, there must in every case be a struggle for existence, either one individual with another of the same species, or with the individuals of distinct species, or with the physical conditions of life.” Constantly embroiled in fights to the finish, nature is thus, in Tennyson’s famous formulation, “red in tooth and claw.” Nor is it about to provide any “artificial increase in food” or “prudential restraint from marriage.” Darwin’s conclusion: “Any being, if it vary however slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have a better chance of surviving, and thus be naturally selected.” There’s more: “From the … principle of inheritance, any selected variety will tend to propagate its new and modified form.” Let such a process go on long enough, even randomly, and not just new species, but new genera, new families, and even whole new orders will be evolved.
About this thing called “natural selection.” First off, it has nothing to do with conscious, intelligent, or purposeful behavior on the part of an organism; in that sense, nobody’s “selecting” anything. A species changes, not through choice or will, but through chance, through the “play” of heredity. Inherit a useful characteristic—speed if you’re a cheetah, strength if you’re an oak tree, turn-on-a-dime savvy if you’re a virus or a human being—and you’ll be better set up to eat, compete, and mate; in short, to pass that characteristic along. Thus useful traits tend, more than useless ones, to be successfully passed on to the next generation, which profits from them in turn, until, in time, a whole species takes a slight turn to the right or left, adapting itself better to its environment, not through careful planning but through the cumulative luck of the draw. This is survival-of-the-fittest stuff; the “natural” simply points up that it’s without self-consciousness on the part of the species, or intervention on the part of God.
About Darwinism. It’s true that Darwin couldn’t prove his theory since, as he put it, the great span of evolutionary time was simply unrecoverable. But that was OK, the circumstantial evidence—in the form of fossils, species distributions, plant and animal structure, embryology—was pretty good. And, except to the religious fundamentalists of the day, the basic setup felt right, just as the theory that the earth was round had felt right. What was upsetting were evolution’s vibes: that nature had gone from being a sun-kissed harmony to being a tag-team wrestling match; that everything was always in flux, always on its way to becoming something else; that there was no such thing as virtue, just more and more adaptation; and that there were greater rewards for being “fit” than for being good or even for being right.
Darwinian evolution steamed intact into the twentieth century, alongside psychoanalysis and socialism, an intellectual tall ship that had made it through the straits of Victorian England and would make it over the shoals of Scopes-trial America. In fact, in the Twenties and Thirties, it was to be both bolstered and refurbished (but not structurally altered) by the new science of genetics. You remember genetics. Derived initially from the nineteenth-century work of Gregor Mendel, the pea-planting monk who first came up with the laws of genetic inheritance, it crystallized around the discovery of the
first gene (or unit of heredity), which, within fifty years, would be shown to make up each chromosome and itself to be made up of DNA.
Fitted out with genetics, Darwinism passed into Neo-Darwinism. It still couldn’t “prove” the theory of evolution, but now it could at least demonstrate exactly how evolution worked. Neo-Darwinism is considered a scientific up for two reasons: First, it gracefully ushered evolution into the twentieth century; second, it showed that the Master’s basic hypothesis had stood the test not only of time but technology. After genetics, as before it, natural selection was still the name of the evolution game.
Ten Burning Questions in the History of Science POSED AND FIELDED BY CONTRIBUTOR MARK ZUSSMAN WERE THE ANCIENTS REALLY SCIENTISTS OR DID THEY JUST MAKE SOME LUCKY GUESSES?
Unlike the Scholastic Aptitude Test and the Graduate Record Exam, science gives credit not just for the right answers but for the method, as in “scientific method,” by which the right answers are arrived at. Science, in other words, wants to see your worksheet. And by worksheet standards, the lowest grades go to the same ancient scientists who appear to have been the most on the ball: Democritus, who articulated an atomic theory, and Empedocles, who believed that something more or less like evolution occurred by a process more or less comparable to the survival of the fittest. Face-to-face with poetic insight, real science throws up its hands.
Now, Aristotle rarely made a good guess. Aristotle believed in spontaneous generation (e.g., that a fly might arise out of a dung heap, no help from Mom and Dad). He believed that heavenly bodies were attached to rotating mechanical spheres. (Think clear plastic). He believed that terrestrial motion was regulated by a principle of inertia such that all bodies desired to be at rest at the center of the earth. He believed that the chemistry of things could be explained in terms of the four elements: earth, air, fire, and water. By worksheet standards, however, Aristotle gets reasonably high grades for empirical observation in zoology (he classified some 540 animal species, at least 50 of which he’d dissected) and particularly for his chicken embryology.
