1a Renee was looking out the window when Mrs. Rivas approached.
"Leaving after only a week? Hardly a real stay at all. Lord knows I won"t be leaving for a long time."
Renee forced a polite smile. "I"m sure it won"t be long for you." Mrs. Rivas was the manipulator in the ward; everyone knew that her attempts were merely gestures, but the aides wearily paid attention to her lest she succeed accidentally.
"Ha. They wish I"d leave. You know what kind of liability they face if you die while you"re on status?"
"Yes, I know."
"That"s all they"re worried about, you can tell. Always their liability-"
Renee tuned out and returned her attention to the window, watching a contrail extrude itself across the sky.
"Mrs. Norwood?" a nurse called. "Your husband"s here."
Renee gave Mrs. Rivas another polite smile and left.
1b Carl signed his name yet another time, and finally the nurses took away the forms for processing.
He remembered when he had brought Renee in to be admitted, and thought of all the stock questions at the first interview. He had answered them all stoically.
"Yes, she"s a professor of mathematics. You can find her in Who"s Who Who"s Who."
"No, I"m in biology."
And: "I had left behind a box of slides that I needed."
"No, she couldn"t have known."
And, just as expected: "Yes, I have. It was about twenty years ago, when I was a grad student."
"No, I tried jumping."
"No, Renee and I didn"t know each other then."
And on and on.
Now they were convinced that he was competent and supportive, and were ready to release Renee into an outpatient treatment program.
Looking back, Carl was surprised in an abstracted way. Except for one moment, there hadn"t been any sense of deja vu at any time during the entire ordeal. All the time he was dealing with the hospital, the doctors, the nurses: the only accompanying sensation was one of numbness, of sheer tedious rote.
2.
There is a well-known "proof" that demonstrates that one equals two. It begins with some definitions: "Let a = 1; let b = 1." It ends with the conclusion "a = 2a," that is, one equals two. Hidden inconspicuously in the middle is a division by zero, and at that point the proof has stepped off the brink, making all rules null and void. Permitting division by zero allows one to prove not only that one and two are equal, but that any two numbers at all- real or imaginary, rational or irrational- are equal.
2a As soon as she and Carl got home, Renee went to the desk in her study and began turning all the papers facedown, blindly sweeping them together into a pile; she winced whenever a corner of a page faced up during her shuffling. She considered burning the pages, but that would be merely symbolic now. She"d accomplish as much by simply never glancing at them.
The doctors would probably describe it as obsessive behavior. Renee frowned, reminded of the indignity of being a patient under such fools. She remembered being on suicide status, in the locked ward, under the supposedly round-the-clock observation of the aides. And the interviews with the doctors, who were so condescending, so obvious. She was no manipulator like Mrs. Rivas, but it really was easy. Simply say "I realize I"m not well yet, but I do feel better," and you"d be considered almost ready for release.
2b Carl watched Renee from the doorway for a moment, before he pa.s.sed down the hallway. He remembered the day, fully two decades past, when he himself had been released. His parents had picked him up, and on the trip back his mother had made some inane comment about how glad everyone would be to see him, and he was just barely able to restrain himself from shaking her arm off his shoulders.
He had done for Renee what he would have appreciated during his period under observation. He had come to visit every day, even though she refused to see him at first, so that he wouldn"t be absent when she did want to see him. Sometimes they talked, and sometimes they simply walked around the grounds. He could find nothing wrong in what he did, and he knew that she appreciated it.
Yet, despite all his efforts, he felt no more than a sense of duty towards her.
3.
In the Principia Mathematica Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove "1 + 1 = 2."
3a As a child of seven, while investigating the house of a relative, Renee had been spellbound at discovering the perfect squares in the smooth marble tiles of the floor. A single one, two rows of two, three rows of three, four rows of four: the tiles fit together in a square square. Of course. No matter which side you looked at it from, it came out the same. And more than that, each square was bigger than the last by an odd number of tiles odd number of tiles. It was an epiphany. The conclusion was necessary: it had a rightness to it, confirmed by the smooth, cool feel of the tiles. And the way the tiles were fitted together, with such incredibly fine lines where they met; she had shivered at the precision.
Later on there came other realizations, other achievements. The astonishing doctoral dissertation at twenty-three, the series of acclaimed papers; people compared her to Von Neumann, universities wooed her. She had never paid any of it much attention. What she did pay attention to was that same sense of rightness, possessed by every theorem she learned, as insistent as the tiles" physicality, and as exact as their fit.
3b Carl felt that the person he was today was born after his attempt, when he met Laura. After being released from the hospital, he was in no mood to see anyone, but a friend of his had managed to introduce him to Laura. He had pushed her away initially, but she had known better. She had loved him while he was hurting, and let him go once he was healed. Through knowing her Carl had learned about empathy, and he was remade.
Laura had moved on after getting her own master"s degree, while he stayed at the university for his doctorate in biology. He suffered various crises and heartbreaks later on in life, but never again despair.
Carl marveled when he thought about what kind of person she was. He hadn"t spoken to her since grad school; what had her life been like over the years? He wondered whom else she had loved. Early on he had recognized what kind of love it was, and what kind it wasn"t, and he valued it immensely.
4.
In the early nineteenth century, mathematicians began exploring geometries that differed from Euclidean geometry; these alternate geometries produced results that seemed utterly absurd, but they didn"t produce logical contradictions. It was later shown that these non-Euclidean geometries were consistent relative to Euclidean geometry: they were logically consistent, as long as one a.s.sumed that Euclidean geometry was consistent.
The proof of Euclidean geometry"s consistency eluded mathematicians. By the end of the nineteenth century, the best that was achieved was a proof that Euclidean geometry was consistent as long as arithmetic was consistent.
4a At the time, when it all began, Renee had thought it little more than an annoyance. She had walked down the hall and knocked on the open door of Peter Fabrisi"s office. "Pete, got a minute?"
Fabrisi pushed his chair back from his desk. "Sure, Renee, what"s up?"
Renee came in, knowing what his reaction would be. She had never asked anyone in the department for advice on a problem before; it had always been the reverse. No matter. "I was wondering if you could do me a favor. You remember what I was telling you about a couple weeks back, about the formalism I was developing?"
He nodded. "The one you were rewriting axiom systems with."
"Right. Well, a few days ago I started coming up with really ridiculous conclusions, and now my formalism is contradicting itself. Could you take a look at it?"
Fabrisi"s expression was as expected. "You want- sure, I"d be glad to."
"Great. The examples on the first few pages are where the problem is; the rest is just for your reference." She handed Fabrisi a thin sheaf of papers. "I thought if I talked you through it, you"d just see the same things I do."
"You"re probably right." Fabrisi looked at the first couple pages. "I don"t know how long this"ll take."
"No hurry. When you get a chance, just see whether any of my a.s.sumptions seem a little dubious, anything like that. I"ll still be going at it, so I"ll tell you if I come up with anything. Okay?"
Fabrisi smiled. "You"re just going to come in this afternoon and tell me you"ve found the problem."
"I doubt it: this calls for a fresh eye."
He spread his hands. "I"ll give it a shot."
"Thanks." It was unlikely that Fabrisi would fully grasp her formalism, but all she needed was someone who could check its more mechanical aspects.
4b Carl had met Renee at a party given by a colleague of his. He had been taken with her face. Hers was a remarkably plain face, and it appeared quite somber most of the time, but during the party he saw her smile twice and frown once; at those moments, her entire countenance a.s.sumed the expression as if it had never known another. Carl had been caught by surprise: he could recognize a face that smiled regularly, or a face that frowned regularly, even if it were unlined. He was curious as to how her face had developed such a close familiarity with so many expressions, and yet normally revealed nothing.
It took a long time for him to understand Renee, to read her expressions. But it had definitely been worthwhile.
Now Carl sat in his easy chair in his study, a copy of the latest issue of Marine Biology Marine Biology in his lap, and listened to the sound of Renee crumpling paper in her study across the hall. She"d been working all evening, with audibly increasing frustration, though she"d been wearing her customary poker face when last he"d looked in. in his lap, and listened to the sound of Renee crumpling paper in her study across the hall. She"d been working all evening, with audibly increasing frustration, though she"d been wearing her customary poker face when last he"d looked in.
He put the journal aside, got up from the chair, and walked over to the entrance of her study. She had a volume opened on her desk; the pages were filled with the usual hieroglyphic equations, interspersed with commentary in Russian.
She scanned some of the material, dismissed it with a barely perceptible frown, and slammed the volume closed. Carl heard her mutter the word "useless," and she returned the tome to the bookcase.
"You"re gonna give yourself high blood pressure if you keep up like this," Carl jested.
"Don"t patronize me."
Carl was startled. "I wasn"t."
Renee turned to look at him and glared. "I know when I"m capable of working productively and when I"m not."
Chilled. "Then I won"t bother you." He retreated.
"Thank you." She returned her attention to the bookshelves. Carl left, trying to decipher that glare.
5.
At the Second International Congress of Mathematics in 1900, David Hilbert listed what he considered to be the twenty-three most important unsolved problems of mathematics. The second item on his list was a request for a proof of the consistency of arithmetic. Such a proof would ensure the consistency of a great deal of higher mathematics. What this proof had to guarantee was, in essence, that one could never prove one equals two. Few mathematicians regarded this as a matter of much import.
5a Renee had known what Fabrisi would say before he opened his mouth.
"That was the d.a.m.nedest thing I"ve ever seen. You know that toy for toddlers where you fit blocks with different cross sections into the differently shaped slots? Reading your formal system is like watching someone take one block and sliding it into every single hole on the board, and making it a perfect fit every time."
"So you can"t find the error?"
He shook his head. "Not me. I"ve slipped into the same rut as you: I can only think about it one way."
Renee was no longer in a rut: she had come up with a totally different approach to the question, but it only confirmed the original contradiction. "Well, thanks for trying."
"You going to have someone else take a look at it?"
"Yes, I think I"ll send it to Callahan over at Berkeley. We"ve been corresponding since the conference last spring."
Fabrisi nodded. "I was really impressed by his last paper. Let me know if he can find it: I"m curious."
Renee would have used a stronger word than "curious" for herself.
5b Was Renee just frustrated with her work? Carl knew that she had never considered mathematics really difficult, just intellectually challenging. Could it be that for the first time she was running into problems that she could make no headway against? Or did mathematics work that way at all? Carl himself was strictly an experimentalist; he really didn"t know how Renee made new math. It sounded silly, but perhaps she was running out of ideas?
Renee was too old to be suffering from the disillusionment of a child prodigy becoming an average adult. On the other hand, many mathematicians did their best work before the age of thirty, and she might be growing anxious over whether that statistic was catching up to her, albeit several years behind schedule.
It seemed unlikely. He gave a few other possibilities cursory consideration. Could she be growing cynical about academia? Dismayed that her research had become overspecialized? Or simply weary of her work?
Carl didn"t believe that such anxieties were the cause of Renee"s behavior; he could imagine the impressions that he would pick up if that were the case, and they didn"t mesh with what he was receiving. Whatever was bothering Renee, it was something he couldn"t fathom, and that disturbed him.
6.
In 1931, Kurt G.o.del demonstrated two theorems. The first one shows, in effect, that mathematics contains statements that may be true, but are inherently unprovable. Even a formal system as simple as arithmetic permits statements that are precise, meaningful, and seem certainly true, and yet cannot be proven true by formal means.
His second theorem shows that a claim of the consistency of arithmetic is just such a statement; it cannot be proven true by any means using the axioms of arithmetic. That is, arithmetic as a formal system cannot guarantee that it will not produce results such as "1 = 2"; such contradictions may never have been encountered, but it is impossible to prove that they never will be.
6a Once again, he had come into her study. Renee looked up from her desk at Carl; he began resolutely, "Renee, it"s obvious that-"
She cut him off. "You want to know what"s bothering me? Okay, I"ll tell you." Renee got out a blank sheet of paper and sat down at her desk. "Hang on; this"ll take a minute." Carl opened his mouth again, but Renee waved him silent. She took a deep breath and began writing.
She drew a line down the center of the page, dividing it into two columns. At the head of one column she wrote the numeral "1" and for the other she wrote "2." Below them she rapidly scrawled out some symbols, and in the lines below those she expanded them into strings of other symbols. She gritted her teeth as she wrote: forming the characters felt like dragging her fingernails across a chalkboard.
About two thirds of the way down the page, Renee began reducing the long strings of symbols into successively shorter strings. And now for the master stroke And now for the master stroke, she thought. She realized she was pressing hard on the paper; she consciously relaxed her grip on the pencil. On the next line that she put down, the strings became identical. She wrote an emphatic "=" across the center line at the bottom of the page.
She handed the sheet to Carl. He looked at her, indicating incomprehension. "Look at the top." He did so. "Now look at the bottom."