by Carl Sagan
The book said something else: π was called a “transcendental” number. There was no equation with ordinary numbers in it that could give you π unless it was infinitely long. She had already taught herself a little algebra and understood what this meant. And π wasn’t the only transcendental number. In fact there was an infinity of transcendental numbers. More than that, there were infinitely more transcendental numbers than ordinary numbers, even though pi was the only one of them she had ever heard of. In more ways than one, π was tied to infinity.
She had caught a glimpse of something majestic. Hiding between all the ordinary numbers was an infinity of transcendental numbers whose presence you would never have guessed unless you looked deeply into mathematics. Every now and then one of them, like π, would pop up unexpectedly in everyday life. But most of them—an infinite number of them, she reminded herself—were hiding, minding their own business, almost certainly unglimpsed by the irritable Mr. Weisbrod.
• • •
She saw through John Staughton from the first. How her mother could ever contemplate marrying him—never mind that it was only two years after her father’s death—was an impenetrable mystery. He was nice enough looking, and he could pretend, when he put his mind to it, that he really cared about you. But he was a martinet. He made students come over weekends to weed and garden at the new house they had moved into, and then made fun of them after they left. He told Ellie that she was just beginning high school and was not to look twice at any of his bright young men. He was puffed up with imaginary self-importance. She was sure that as a professor he secretly despised her dead father, who had been only a shopkeeper. Staughton had made it clear that an interest in radio and electronics was unseemly for a girl, that it would not catch her a husband, that understanding physics was for her a foolish and aberrational notion. “Pretentious,” he called it. She just didn’t have the ability. This was an objective fact that she might as well get used to. He was telling her this for her own good. She’d thank him for it in later life. He was, after all, an associate professor of physics. He knew what it took. These homilies would always infuriate her, even though she had never before—despite Staughton’s refusal to believe it—considered a career in science.
He was not a gentle man, as her father had been, and he had no idea what a sense of humor was. When anyone assumed that she was Staughton’s daughter, she would be outraged. Her mother and stepfather never suggested that she change her name to Staughton; they knew what her response would be.
Occasionally there was a little warmth in the man, as when, in her hospital room just after her tonsillectomy, he had brought her a splendid kaleidoscope.
“When are they going to do the operation,” she had asked, a little sleepily.
“They’ve already done it,” Staughton had answered. “You’re going to be fine.” She found it disquieting that whole blocks of time could be stolen without her knowledge, and blamed him. She knew at the time it was childish.
That her mother could truly love him was inconceivable. She must have remarried out of loneliness, out of weakness. She needed someone to take care of her. Ellie vowed she would never accept a position of dependence. Ellie’s father had died, her mother had grown distant, and Ellie felt herself exiled to the house of a tyrant. There was no one to call her Presh anymore.
She longed to escape.
“‘Bridgeport?’ said I.
“‘Camelot,’ said he.”
CHAPTER 2
Coherent Light
Since I first gained the use of reason my inclination toward learning has been so violent and strong that neither the scoldings of other people…nor my own reflections…have been able to stop me from following this natural impulse that God gave me. He alone must know why; and He knows too that I have begged Him to take the light of my understanding, leaving only enough for me to keep His law, for anything else is excessive in a woman, according to some people. And others say it is even harmful.
—JUANA INES DE LA CRUZ
Reply to the Bishop of Puebla (1691), who had attacked her scholarly work as inappropriate for her sex
I wish to propose for the reader’s favourable consideration a doctrine which may, I fear, appear wildly paradoxical and subversive. The doctrine in question is this: that it is undesirable to believe a proposition when there is no ground whatever for supposing it true. I must, of course, admit that if such an opinion became common it would completely transform our social life and our political system; since both are at present faultless, this must weigh against it.
—BERTRAND RUSSELL
Skeptical Essays, I (1928)
Surrounding the blue-white star in its equatorial plane was a vast ring of orbiting debris—rocks and ice, metals and organics—reddish at the periphery and bluish closer to the star. The world-sized polyhedron plummeted through a gap in the rings and emerged out the other side. In the ring plane, it had been intermittently shadowed by icy boulders and tumbling mountains. But now, carried along its trajectory toward a point above the opposite pole of the star, the sunlight gleamed off its millions of bowl-shaped appendages. If you looked very carefully you might have seen one of them make a slight pointing adjustment. You would not have seen the burst of radio waves washing out from it into the depths of space.
FOR ALL the tenure of humans on Earth, the night sky had been a companion and an inspiration. The stars were comforting. They seemed to demonstrate that the heavens were created for the benefit and instruction of humans. This pathetic conceit became the conventional wisdom worldwide. No culture was free of it. Some people found in the skies an aperture to the religious sensibility. Many were awestruck and humbled by the glory and scale of the cosmos. Others were stimulated to the most extravagant flights of fancy.
At the very moment that humans discovered the scale of the universe and found that their most unconstrained fancies were in fact dwarfed by the true dimensions of even the Milky Way Galaxy, they took steps that ensured that their descendants would be unable to see the stars at all. For a million years humans had grown up with a personal daily knowledge of the vault of heaven. I the last few thousand years they began building and emigrating to the cities. In the last few decades, a major fraction of the human population had abandoned a rustic way of life. As technology developed and the cities were polluted, the nights became starless. New generations grew to maturity wholly ignorant of the sky that had transfixed their ancestors and that had stimulated the modern age of science and technology. Without even noticing, just as astronomy entered a golden age most people cut themselves off from the sky, a cosmic isolationism that ended only with the dawn of space exploration.
• • •
Ellie would look up at Venus and imagine it was a world something like the Earth—populated by plants and animals and civilizations, but each of them different from the kinds we have here. On the outskirts of town, just after sunset, she would examine the night sky and scrutinize that unflickering bright point of light. By comparison with nearby clouds, just above her, still illuminated by the Sun, it seemed a little yellow. She tried to imagine what was going on there. She would stand on tiptoe and stare the planet down. Sometimes, she could almost convince herself that she could really see it; a swirl of yellow fog would suddenly clear, and a vast jeweled city would briefly be revealed. Air cars sped among the crystal spires. Sometimes she would imagine peering into one of those vehicles and glimpsing one of them. Or she would imagine a young one, glancing up at a bright blue point of light in its sky, standing on tiptoe and wondering about the inhabitants of Earth. It was an irresistible notion: a sultry, tropical planet brimming over with intelligent life, and just next door.
She consented to rote memorization, but knew that it was at best the hollow shell of education. She did the minimum work necessary to do well in her courses, and pursued other matters. She arranged to spend free periods and occasional hours after school in what was called “shop”—a dingy and cramped small factory established when the s
chool devoted more effort to “vocational education” than was now fashionable. “Vocational education” meant, more than anything else, working with your hands. There were lathes, drill presses, and other machine tools which she was forbidden to approach, because no matter how capable she might be, she was still “a girl.” Reluctantly, they granted her permission to pursue her own projects in the electronics area of the “shop.” She built radios more or less from scratch, and then went on to something more interesting.
She built an encrypting machine. It was rudimentary, but it worked. It could take any English-language message and transform it by a simple substitution cipher into something that looked like gibberish. Building a machine that would do the reverse—converting an encrypted message into clear when you didn’t know the substitution convention—that was much harder. You could have the machine run through all the possible substitutions (A stands for B, A stands for C, A stands for D…), or you could remember that some letters in English were used more often than others. You could get some idea of the frequency of letters by looking at the sizes of the bins for each letter of type in the print shop next door. “ETAOIN SHRDLU,” the boys in print shop would say, giving pretty closely the order of the twelve most frequently used letters in English. In decoding a long message, the letter that was most common probably stood for an E. Certain consonants tended to go together, she discovered; vowels distributed themselves more or less at random. The most common three-letter word in the language was “the.” If within a word there was a letter standing between a T and an E, it was almost certainly H. If not, you could bet on R or a vowel. She deduced other rules and spent long hours counting up the frequency of letters in various schoolbooks before she discovered that such frequency tables had already been compiled and published. Her decrypting machine was only for her own enjoyment. She did not use it to convey secret messages to friends. She was unsure to whom she might safely confide these electronic and cryptographic interests; the boys became jittery or boisterous, and the girls looked at her strangely.
• • •
Soldiers of the United States were fighting in a distant place called Vietnam. Every month, it seemed, more young men were being scooped off the street or the farm and packed off the Vietnam. The more she learned about the origins of the war, and the more she listened to the public pronouncements of national leaders, the more outraged she became. The President and the Congress were lying and killing, she thought to herself, and almost everyone else was mutely assenting. The fact that her stepfather embraced official positions on treaty obligations, dominoes, and naked Communist aggression only strengthened her resolve. She began attending meetings and rallies at the college nearby. The people she met there seemed much brighter, friendlier, more alive than her awkward and lusterless high school companions. John Staughton first cautioned her and then forbade her to spend time with college students. They would not respect her, he said. They would take advantage of her. She was pretending to a sophistication she did not have and never would. Her style of dress was deteriorating. Military fatigues were inappropriate for a girl and a travesty, a hypocrisy, for someone who claimed to oppose the American intervention in Southeast Asia.
Beyond pious exhortations to Ellie and Staughton not to “fight,” her mother participated little in these discussions. Privately she would plead with Ellie to obey her stepfather, to be “nice.” Ellie now suspected Staughton of marrying her mother for her father’s life insurance—why else? He certainly showed no signs of loving her—and he was not predisposed to be “nice.” One day, in some agitation, her mother asked her to do something for all their sakes: attend Bible class. While her father, a skeptic on revealed religions, had been alive, there was no talk of Bible class. How could her mother have married Staughton? The question welled up in her for the thousandth time. Bible class, her mother continued, would help instill the conventional virtues; but even more important, it would show Staughton that Ellie was willing to make some accommodation. Out of love and pity for her mother, she acquiesced.
So every Sunday for most of one school year Ellie went to a regular discussion group at a nearby church. It was one of the respectable Protestant denominations, untainted by disorderly evangelism. There were a few high school students, a number of adults, mainly middle-aged women, and the instructor, the minister’s wife. Ellie had never seriously read the Bible before and had been inclined to accept her father’s perhaps ungenerous judgment that it was “half barbarian history, half fairy tales.” So over the weekend preceding her first class, she read through what seemed to be the important parts of the Old Testament, trying to keep an open mind. She at once recognized that there were two different and mutually contradictory stories of Creation in the first two chapters of Genesis. She did not see how there could be light and days before the Sun was made, and had trouble figuring out exactly who it was that Cain had married. In the stories of Lot and his daughters, of Abraham and Sarah in Egypt, of the betrothal of Dinah, of Jacob and Esau, she found herself amazed. She understood that cowardice might occur in the real world—that sons might deceive and defraud an aged father, that a man might give craven consent to the seduction of his wife by the King, or even encourage the rape of his daughters. But in this holy book there was not a word of protest against such outrages. Instead, it seemed, the crimes were approved, even praised.
When class began, she was eager for a discussion of these vexing inconsistencies, for an unburdening illumination of God’s Purpose, or at least for an explanation of why these crimes were not condemned by the author or Author. But in this she was to be disappointed. The minister’s wife blandly temporized. Somehow these stories never surfaced in subsequent discussion. When Ellie inquired how it was possible for the maidservants of the daughter of Pharaoh to tell just by looking that the baby in the bullrushes was Hebrew, the teacher blushed deeply and asked Ellie not to raise unseemly questions. (The answer dawned on Ellie at that moment.)
When they came to the New Testament, Ellie’s agitation increased. Matthew and Luke traced the ancestral line of Jesus back to King David. But for Matthew there were twenty-eight generations between David and Jesus; for Luke forty-three. There were almost no names common to the two lists. How could both Matthew and Luke be the Word of God? The contradictory genealogies seemed to Ellie a transparent attempt to fit the Isaianic prophecy after the event—cooking the data, it was called in chemistry lab. She was deeply moved by the Sermon on the Mount, deeply disappointed by the admonition to render unto Caesar what is Caesar’s, and reduced to shouts and tears after the instructor twice sidestepped her questions on the meaning of “I bring not peace but the sword.” She told her despairing mother that she had done her best, but wild horses wouldn’t drag her to another Bible class.
• • •
She was lying on her bed. It was a hot summer’s night. Elvis was singing, “One night with you, that’s what I’m beggin’ for.” The boys at the high school seemed painfully immature, and it was difficult—especially with her stepfather’s strictures and curfews—to establish much of a relationship with the young college men she met at lectures and rallies. John Staughton was right, she reluctantly admitted to herself, at least about this: The young men, almost without exception, had a penchant for sexual exploitation. At the same time, they seemed much more emotionally vulnerable than she had expected. Perhaps the one caused the other.
She had half expected not to attend college, although she was determined to leave home. Staughton would not pay for her to go elsewhere, and her mother’s meek intercessions were unavailing. But Ellie had done spectacularly well on the standardized college entrance examinations and found to her surprise her teachers telling her that she was likely to be offered scholarships by well-known universities. She had guessed on a number of multiple-choice questions and considered her performance a fluke. If you know very little, only enough to exclude all but the two most likely answers, and if you then guess at ten straight questions, the is about one chance in a thou
sand, she explained to herself, that you’ll get all then correct. For twenty straight questions, the odds were one in a million. But something like a million kids probably took this test. Someone had to get lucky.
Cambridge, Massachusetts, seemed far enough away to elude John Staughton’s influence, but close enough to return from on vacation to visit her mother—who viewed the arrangement as a difficult compromise between abandoning her daughter and incrementally irritating her husband. Ellie surprised herself by choosing Harvard over the Massachusetts Institute of Technology.
She arrived for orientation period, a pretty dark-haired young woman of middling height with a lopsided smile and an eagerness to learn everything. She set out to broaden her education, to take as many courses as possible apart from her central interests in mathematics, physics, and engineering. But there was a problem with her central interests. She found it difficult to discuss physics, much less debate it, with her predominantly male classmates. At first they paid a kind of selective inattention to her remarks. There would be a slight pause, and then they would go on as if she had not spoken. Occasionally they would acknowledge her remark, even praise it, and then again continue undeflected. She was reasonably sure her remarks were not entirely foolish, and did not wish to be ignored, much less ignored and patronized alternately. Part of it—but only a part—she knew was due to the softness of her voice. So she developed a physics voice, a professional voice: clear, competent, and many decibels above conversational. With such a voice it was important to be right. She had to pick her moments. It was hard to continue long in such a voice, because she was sometimes in danger of bursting out laughing. So she found herself leaning towards quick, sometimes cutting, interventions, usually enough to capture their attention; then she could go on for a while in a more usual tone of voice. Every time she found herself in a new group she would have to fight her way through again, just to dip her oar into the discussion. The boys were uniformly unaware even that there was a problem.
