Stories of Your Life (11 page)

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.

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 assumed 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.

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 assumptions 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.

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 assumed 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
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.

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.

Renee had known what Fabrisi would say before he opened his mouth.

"That was the damnedest 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.

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.

In 1931, Kurt Godel 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.

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 masterstroke
, 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."

He frowned. “I don't understand."

"I've discovered a formalism that lets you equate any number with any other number. That page there proves that one and two are equal. Pick any two numbers you like; I can prove those equal as well."

Carl seemed to be trying to remember something. “It's a division by zero, right?"

"No. There are no illegal operations, no poorly defined terms, no independent axioms that are implicitly assumed, nothing. The proof employs absolutely nothing that's forbidden."

Carl shook his head. “Wait a minute. Obviously one and two aren't the same."

"But formally they are: the proof's in your hand. Everything I've used is within what's accepted as absolutely indisputable."

"But you've got a contradiction here."

"That's right. Arithmetic as a formal system is inconsistent."

"You can't find your mistake, is that what you mean?"

"
No
, you're not listening. You think I'm just frustrated because of something like that? There is no mistake in the proof."

"You're saying there's something wrong within what's accepted?"

"Exactly."

"Are you—” He stopped, but too late. She glared at him. Of course she was sure. He thought about what she was implying.

"Do you see?” asked Renee. “I've just disproved most of mathematics: it's all meaningless now."

She was getting agitated, almost distraught; Carl chose his words carefully. “How can you say that? Math still works. The scientific and economic worlds aren't suddenly going to collapse from this realization."

"That's because the mathematics they're using is just a gimmick. It's a mnemonic trick, like counting on your knuckles to figure out which months have thirty-one days."

"That's not the same."

"Why isn't it? Now mathematics has absolutely
nothing
to do with reality. Never mind concepts like imaginaries or infinitesimals. Now goddamn integer addition has nothing to do with counting on your fingers. One and one will always get you two on your fingers, but on paper I can give you an infinite number of answers, and they're all equally valid, which means they're all equally invalid. I can write the most elegant theorem you've ever seen, and it won't mean any more than a nonsense equation.” She gave a bitter laugh. “The positivists used to say all mathematics is a tautology. They had it all wrong: it's a contradiction."

Carl tried a different approach. “Hold on. You just mentioned imaginary numbers. Why is this any worse than what went on with those? Mathematicians once believed they were meaningless, but now they're accepted as basic. This is the same situation."

"It's
not
the same. The solution there was to simply expand the context, and that won't do any good here. Imaginary numbers added something new to mathematics, but my formalism is redefining what's already there."

"But if you change the context, put it in a different light—"

She rolled her eyes. “No! This follows from the axioms as surely as addition does; there's no way around it. You can take my word for it."

In 1936, Gerhard Gentzen provided a proof of the consistency of arithmetic, but to do it he needed to use a controversial technique known as transfinite induction. This technique is not among the usual methods of proof, and it hardly seemed appropriate for guaranteeing the consistency of arithmetic. What Gentzen had done was prove the obvious by assuming the doubtful.

Callahan had called from Berkeley, but could offer no rescue. He said he would continue to examine her work, but it seemed that she had hit upon something fundamental and disturbing. He wanted to know about her plans for publication of her formalism, because if it did contain an error that neither of them could find, others in the mathematics community would surely be able to.

Renee had barely been able to hear him speaking, and mumbled that she would get back to him. Lately she had been having difficulty talking to people, especially since the argument with Carl; the other members of the department had taken to avoiding her. Her concentration was gone, and last night she had had a nightmare about discovering a formalism that let her translate arbitrary concepts into mathematical expressions: then she had proven that life and death were equivalent.

That was something that frightened her: the possibility that she was losing her mind. She was certainly losing her clarity of thought, and that came pretty close.