Page 39 of 41
Congealing and warming, the Earth released the methane, ammonia, water and hydrogen gases that had been trapped within, forming the primitive atmosphere and the first oceans. Starlight from the Sun bathed and warmed the primeval Earth, drove storms, generated lightning and thunder. Volcanoes overflowed with lava. These processes disrupted molecules of the primitive atmosphere; the fragments fell back together again into more and more complex forms, which dissolved in the early oceans. After a time the seas achieved the consistency of a warm, dilute soup. Molecules were organized, and complex chemical reactions driven, on the surface of clays. And one day a molecule arose that quite by accident was able to make crude copies of itself out of the other molecules in the broth. As time passed, more elaborate and more accurate self-replicating molecules arose. Those combinations best suited to further replication were favored by the sieve of natural selection. Those that copied better produced more copies. And the primitive oceanic broth gradually grew thin as it was consumed by and transformed into complex condensations of self-replicating organic molecules. Gradually, imperceptibly, life had begun.
Single-celled plants evolved, and life began to generate its own food. Photosynthesis transformed the atmosphere. Sex was invented. Once free-living forms banded together to make a complex cell with specialized functions. Chemical receptors evolved, and the Cosmos could taste and smell. One-celled organisms evolved into multicellular colonies, elaborating their various parts into specialized organ systems. Eyes and ears evolved, and now the Cosmos could see and hear. Plants and animals discovered that the land could support life. Organisms buzzed, crawled, scuttled, lumbered, glided, flapped, shimmied, climbed and soared. Colossal beasts thundered through the steaming jungles. Small creatures emerged, born live instead of in hard-shelled containers, with a fluid like the early oceans coursing through their veins. They survived by swiftness and cunning. And then, only a moment ago, some small arboreal animals scampered down from the trees. They became upright and taught themselves the use of tools, domesticated other animals, plants and fire, and devised language. The ash of stellar alchemy was now emerging into consciousness. At an ever-accelerating pace, it invented writing, cities, art and science, and sent spaceships to the planets and the stars. These are some of the things that hydrogen atoms do, given fifteen billion years of cosmic evolution.
It has the sound of epic myth, and rightly. But it is simply a description of cosmic evolution as revealed by the science of our time. We are difficult to come by and a danger to ourselves. But any account of cosmic evolution makes it clear that all the creatures of the Earth, the latest manufactures of the galactic hydrogen industry, are beings to be cherished. Elsewhere there may be other equally astonishing transmutations of matter, so wistfully we listen for a humming in the sky.
We have held the peculiar notion that a person or society that is a little different from us, whoever we are, is somehow strange or bizarre, to be distrusted or loathed. Think of the negative connotations of words like alien or outlandish. And yet the monuments and cultures of each of our civilizations merely represent different ways of being human. An extraterrestrial visitor, looking at the differences among human beings and their societies, would find those differences trivial compared to the similarities. The Cosmos may be densely populated with intelligent beings. But the Darwinian lesson is clear: There will be no humans elsewhere. Only here. Only on this small planet. We are a rare as well as an endangered species. Every one of us is, in the cosmic perspective, precious. If a human disagrees with you, let him live. In a hundred billion galaxies, you will not find another.
Human history can be viewed as a slowly dawning awareness that we are members of a larger group. Initially our loyalties were to ourselves and our immediate family, next, to bands of wandering hunter-gatherers, then to tribes, small settlements, city-states, nations. We have broadened the circle of those we love. We have now organized what are modestly described as superpowers, which include groups of people from divergent ethnic and cultural backgrounds working in some sense together—surely a humanizing and character-building experience. If we are to survive, our loyalties must be broadened further, to include the whole human community, the entire planet Earth. Many of those who run the nations will find this idea unpleasant. They will fear the loss of power. We will hear much about treason and disloyalty. Rich nation-states will have to share their wealth with poor ones. But the choice, as H. G. Wells once said in a different context, is clearly the universe or nothing.
A few million years ago there were no humans. Who will be here a few million years hence? In all the 4.6-billion-year history of our planet, nothing much ever left it. But now, tiny unmanned exploratory spacecraft from Earth are moving, glistening and elegant, through the solar system. We have made a preliminary reconnaissance of twenty worlds, among them all the planets visible to the naked eye, all those wandering nocturnal lights that stirred our ancestors toward understanding and ecstasy. If we survive, our time will be famous for two reasons: that at this dangerous moment of technological adolescence we managed to avoid self-destruction; and because this is the epoch in which we began our journey to the stars.
The choice is stark and ironic. The same rocket boosters used to launch probes to the planets are poised to send nuclear warheads to the nations. The radioactive power sources on Viking and Voyager derive from the same technology that makes nuclear weapons. The radio and radar techniques employed to track and guide ballistic missiles and defend against attack are also used to monitor and command the spacecraft on the planets and to listen for signals from civilizations near other stars. If we use these technologies to destroy ourselves, we surely will venture no more to the planets and the stars. But the converse is also true. If we continue to the planets and the stars, our chauvinisms will be shaken further. We will gain a cosmic perspective. We will recognize that our explorations can be carried out only on behalf of all the people of the planet Earth. We will invest our energies in an enterprise devoted not to death but to life: the expansion of our understanding of the Earth and its inhabitants and the search for life elsewhere. Space exploration—unmanned and manned—uses many of the same technological and organizational skills and demands the same commitment to valor and daring as does the enterprise of war. Should a time of real disarmament arrive before nuclear war, such exploration would enable the military-industrial establishments of the major powers to engage at long last in an untainted enterprise. Interests vested in preparations for war can relatively easily be reinvested in the exploration of the Cosmos.
A reasonable—even an ambitious—program of unmanned exploration of the planets is inexpensive. The budget for space sciences in the United States is shown in the table above. Comparable expenditures in the Soviet Union are a few times larger. Together these sums represent the equivalent of two or three nuclear submarines per decade, or the cost overruns on one of the many weapon systems in a single year. In the last quarter of 1979, the program cost of the U.S. F/A-18 aircraft increased by $5.1 billion, and the F-16 by $3.4 billion. Since their inceptions, significantly less has been spent on the unmanned planetary programs of both the United States and the Soviet Union than has been wasted shamefully—for example, between 1970 and 1975, in the U.S. bombing of Cambodia, an application of national policy that cost $7 billion. The total cost of a mission such as Viking to Mars, or Voyager to the outer solar system, is less than that of the 1979–80 Soviet invasion of Afghanistan. Through technical employment and the stimulation of high technology, money spent on space exploration has an economic multiplier effect. One study suggests that for every dollar spent on the planets, seven dollars are returned to the national economy. And yet there are many important and entirely feasible missions that have not been attempted because of lack of funds—including roving vehicles to wander across the surface of Mars, a comet rendezvous, Titan entry probes and a full-scale search for radio signals from other civilizations in space.
The cost of major ventures into space—permanen
t bases on the Moon or human exploration of Mars, say—is so large that they will not, I think, be mustered in the very near future unless we make dramatic progress in nuclear and “conventional” disarmament. Even then there are probably more pressing needs here on Earth. But I have no doubt that, if we avoid self-destruction, we will sooner or later perform such missions. It is almost impossible to maintain a static society. There is a kind of psychological compound interest: even a small tendency toward retrenchment, a turning away from the Cosmos, adds up over many generations to a significant decline. And conversely, even a slight commitment to ventures beyond the Earth—to what we might call, after Columbus, “the enterprise of the stars”—builds over many generations to a significant human presence on other worlds, a rejoicing in our participation in the Cosmos.
Some 3.6 million years ago, in what is now northern Tanzania, a volcano erupted, the resulting cloud of ash covering the surrounding savannahs. In 1979, the paleoanthropologist Mary Leakey found in that ash footprints—the footprints, she believes, of an early hominid, perhaps an ancestor of all the people on the Earth today. And 380,000 kilometers away, in a flat dry plain that humans have in a moment of optimism called the Sea of Tranquility, there is another footprint, left by the first human to walk another world. We have come far in 3.6 million years, and in 4.6 billion and in 15 billion.
For we are the local embodiment of a Cosmos grown to self-awareness. We have begun to contemplate our origins: starstuff pondering the stars; organized assemblages of ten billion billion billion atoms considering the evolution of atoms; tracing the long journey by which, here at least, consciousness arose. Our loyalties are to the species and the planet. We speak for Earth. Our obligation to survive is owed not just to ourselves but also to that Cosmos, ancient and vast, from which we spring.
*The process is similar to, but much more dangerous than, the destruction of the ozone layer by the fluorocarbon propellants in aerosol spray cans, which have accordingly been banned by a number of nations; and to that invoked in the explanation of the extinction of the dinosaurs by a supernova explosion a few dozen light-years away.
*The word cosmopolitan was first invented by Diogenes, the rationalist philosopher and critic of Plato.
*With the single exception of Archimedes, who during his stay at the Alexandrian Library invented the water screw, which is used in Egypt to this day for the irrigation of cultivated fields. But even he considered such mechanical contrivances far beneath the dignity of science.
Besides those thanked in the introduction, I am very grateful to the many people who generously contributed their time and expertise to this book, including Carol Lane, Myrna Talman, and Jenny Arden; David Oyster, Richard Wells, Tom Weidlinger, Dennis Gutierrez, Rob McCain, Nancy Kinney, Janelle Balnicke, Judy Flannery, and Susan Racho of the Cosmos television staff; Nancy Inglis, Peter Mollman, Marylea O’Reilly, and Jennifer Peters of Random House; Paul west for generously lending me the title of Chapter 5; and George Abell, James Allen, Barbara Amago, Lawrence Anderson, Jonathon Arons, Halton Arp, Asma El Bakri, James Blinn, Bart Bok, Zeddie Bowen, John C. Brandt, Kenneth Brecher, Frank Bristow, John Callendar, Donald B. Campbell, Judith Campbell, Elof Axel Carlson, Michael Carra, John Cassani, Judith Castagno, Catherine Cesarsky, Martin Cohen, Judy-Lynn del Rey, Nicholas Devereux, Michael Devirian, Stephen Dole, Frank D. Drake, Frederick C. Durant III, Richard Epstein, Von R. Eshleman, Ahmed Fahmy, Herbert Friedman, Robert Frosch, Jon Fukuda, Richard Gammon, Ricardo Giacconi, Thomas Gold, Paul Goldenberg, Peter Goldreich, Paul Goldsmith, J. Richard Gott III, Stephen Jay Gould, Bruce Hayes, Raymond Heacock, Wulff Heintz, Arthur Hoag, Paul Hodge, Dorrit Hoffleit, William Hoyt, Icko Iben, Mikhail Jaroszynski, Paul Jepsen, Tom Karp, Bishun N. Khare, Charles Kohlhase, Edwin Krupp, Arthur Lane, Paul MacLean, Bruce Margon, Harold Masursky, Linda Morabito, Edmond Momjian, Edward Moreno, Bruce Murray, William Murnane, Thomas A. Mutch, Kenneth Norris, Tobias Owen, Linda Paul, Roger Payne, Vahe Petrosian, James B. Pollack, George Preston, Nancy Priest, Boris Ragent, Dianne Rennell, Michael Rowton, Allan Sandage, Fred Scarf, Maarten Schmidt, Arnold Scheibel, Eugene Shoemaker, Frank Shu, Nathan Sivin, Bradford Smith, Laurence A. Soderblom, Hyron Spinrad, Edward Stone, Jeremy Stone, Ed Taylor, Kip S. Thorne, Norman Thrower, O. Brian Toon, Barbara Tuchman, Roger Ulrich, Richard Underwood, Peter van de Kamp, Jurrie J. Van der Woude, Arthur Vaughn, Joseph Veverka, Helen Simpson Vishniac, Dorothy Vitaliano, Robert Wagoner, Pete Waller, Josephine Walsh, Kent Weeks, Donald Yeomans, Stephen Yerazunis, Louise Gray Young, Harold Zirin, and the National Aeronautics and Space Administration. I am also grateful for special photographic help by Edwardo Castañeda and Bill Ray.
Reductio ad Absurdum
and the Square Root of Two
The original Pythagorean argument on the irrationality of the square root of 2 depended on a kind of argument called reductio ad absurdum, a reduction to absurdity: we assume the truth of a statement, follow its consequences and come upon a contradiction, thereby establishing its falsity. To take a modern example, consider the aphorism by the great twentieth-century physicist, Niels Bohr: “The opposite of every great idea is another great idea.” If the statement were true, its consequences might be at least a little perilous. For example, consider the opposite of the Golden Rule, or proscriptions against lying or “Thou shalt not kill.” So let us consider whether Bohr’s aphorism is itself a great idea. If so, then the converse statement, “The opposite of every great idea is not a great idea,” must also be true. Then we have reached a reductio ad absurdum. If the converse statement is false, the aphorism need not detain us long, since it stands self-confessed as not a great idea.
We present a modern version of the proof of the irrationality of the square root of 2 using a reductio ad absurdum, and simple algebra rather than the exclusively geometrical proof discovered by the Pythagoreans. The style of argument, the mode of thinking, is at least as interesting as the conclusion:
Consider a square in which the sides are 1 unit long (1 centimeter, 1 inch, 1 light-year, it does not matter). The diagonal line BC divides the square into two triangles, each containing a right angle. In such right triangles, the Pythagorean theorem holds: 12 + 12 = X2. But 12 + 12 = 1 + 1 = 2, so X2 = 2 and we write x = the square root of two. We assume is a rational number = p/q, where p and q are integers, whole numbers. They can be as big as we like and can stand for any integers we like. We can certainly require that they have no common factors. If we were to claim = 14/10, for example, we would of course cancel out the factor 2 and write p = 7 and q = 5, not p = 14, q = 10. Any common factor in numerator or denominator would be canceled out before we start. There are an infinite number of p’s and q’s we can choose. From = p/q, by squaring both sides of the equation, we find that 2 = p2/q2, or, by multiplying both sides of the equation by q2, we find
p2 = 2q2. (Equation 1)
p2 is then some number multiplied by 2. Therefore p2 is an even number. But the square of any odd number is odd (12 = 1, 32 = 9, 52 = 25, 72 = 49, etc.). So p itself must be even, and we can write p = 2s, where s is some other integer. Substituting for p in Equation (1), we find
p2 = (2s)2 = 4s2 = 2q2
Dividing both sides of the last equality by 2, we find
q2 = 2s2
Therefore q2 is also an even number, and, by the same argument as we just used for p, it follows that q is even too. But if p and q are both even, both divisible by 2, then they have not been reduced to their lowest common factor, contradicting one of our assumptions. Reductio ad absurdum. But which assumption? The argument cannot be telling us that reduction to common factors is forbidden, that 14/10 is permitted and 7/5 is not. So the initial assumption must be wrong; p and q cannot be whole numbers; and is irrational. In fact, = 1.4142135 …
What a stunning and unexpected conclusion! How elegant the proof! But the Pythagoreans felt compelled to suppress this great discovery.
The Five Pythagorean Soli
A regular polygon (Greek for “many-angled”) is a two-dimensional figure with some number, n, of equal sides. So n = 3 is an equilateral triangle, n = 4 is a square, n = 5 is a pentagon, and so on. A polyhedron (Greek for “many-sided”) is a three-dimensional figure, all of whose faces are polygons: a cube, for example, with 6 squares for faces. A simple polyhedron, or regular solid, is one with no holes in it. Fundamental to the work of the Pythagoreans and of Johannes Kepler was the fact that there can be 5 and only 5 regular solids. The easiest proof comes from a relationship discovered much later by Descartes and by Leonhard Euler which relates the number of faces, F, the number of edges, E, and the number of corners or vertices, V, of a regular solid:
V – E + F = 2 (Equation 2)
So for a cube, there are 6 faces (F = 6) and 8 vertices (V = 8), and 8 – E + 6 = 2, 14 – E = 2, and E = 12; Equation (2) predicts that the cube has 12 edges, as it does. A simple geometric proof of Equation (2) can be found in the book by Courant and Robbins in the Bibliography. From Equation (2) we can prove that there are only five regular solids: