Full Spectrum 3 Part 27

"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 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 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 a.s.sumed, 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."

6b "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 G.o.dd.a.m.n 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."

7.

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 a.s.suming the doubtful.

7a 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.

What a ridiculous woman you are, she chided herself. Was G.o.del suicidal after he demonstrated his incompleteness theorem?

But that was beautiful, numinous, one of the most elegant theorems Renee had ever seen. Her own proof taunted her, ridiculed her. Like a brainteaser in a puzzle book, it said gotcha, you skipped right over the mistake, see if you can find where you screwed up; only to turn around and say, gotcha again.

She imagined Callahan would be pondering the implications that her discovery held for mathematics. So much of mathematics had no practical application; it existed solely as a formal theory, studied for its intellectual beauty. But that couldn't last; a self-contradictory theory was so pointless that most mathematicians would drop it in disgust.

What truly infuriated Renee was the way her own intuition had betrayed her. The d.a.m.ned theorem made sense; in its own perverted way, it felt right. She understood it, knew why it was true, believed it.

7b Carl smiled when he thought of her birthday.

"I can't believe you! How could you possibly have known?" She had run down the stairs, holding a sweater in her hands.

Last summer they had been in Scotland on vacation, and in one store in Edinburgh there had been a sweater that Renee had been eyeing but didn't buy. He had ordered it, and placed it in her dresser drawer for her to find that morning.

"You're just so transparent," he had teased her. They both knew that wasn't true, but he liked to tell her that.

That was two months ago. A scant two months.

Now the situation called for a change of pace. Carl went into her study, and found Renee sitting in her chair, staring out the window. "Guess what I got for us." She looked up. "What?"

"Reservations for the weekend. A suite at the Biltmore. We can relax and do absolutely nothing--"

"Please stop," Renee said. "I know what you're trying to do, Carl. You want us to do something pleasant and distracting to take my mind off this formalism. But it won't work. You don't know what kind of hold this has on me."

"Come on, come on." He tugged at her hands to get her off the chair, but she pulled away. Carl stood there for a moment, when suddenly she turned and locked eyes with him.

"You know I've been tempted to take barbiturates? I almost wish I were an idiot, so I wouldn't have to think about it."

He was taken aback. Uncertain of his bearings, he said, "Why won't you at least try to get away for a while? It couldn't hurt, and maybe it'll take your mind off this."

"It's not anything I can take my mind off of. You just don't understand."

"So explain it to me."

Renee exhaled and turned away to think for a moment. "It's like everything I see is shouting the contradiction at me," she said. "I'm equating numbers all the time now." Carl was silent. Then, with sudden comprehension, he said, "Like the cla.s.sical physicists facing quantum mechanics. As if a theory you've always believed has been superseded, and the new one makes no sense, but somehow all the evidence supports it."

"No, it's not like that at all." Her dismissal was almost contemptuous. "This has nothing to do with evidence; it's all a priori."

"How is that different? Isn't it just the evidence of your reasoning then?"

"Christ, are you joking? It's the difference between my measuring one and two to have the same value, and my intuiting it. I can't maintain the concept of distinct quant.i.ties in my mind anymore; they all feel the same to me."

"You don't mean that," he said. "No one could actually experience such a thing; it's like believing six impossible things before breakfast."

"How would you know what I can experience?"

"I'm trying to understand."

"Don't bother."

Carl's patience was gone. "All right then." He walked out of the room and canceled their reservations. They scarcely spoke after that, talking only when necessary. It was three days later that Carl forgot the box of slides he needed, and drove back to the house, and found her note on the table. Carl intuited two things in the moments following. The first came to him as he was racing through the house, wondering if she had gotten some cyanide from the chemistry department: it was the realization that, because he couldn't understand what had brought her to such an action, he couldn't feel anything for her.

The second intuition came to him as he was pounding on the bedroom door, yelling at her inside: he experienced deja vu. It was the only time the situation would feel familiar, and yet it was grotesquely reversed. He remembered being on the other side of a locked door, on the roof of a building, hearing a friend pounding on the door and yelling for him not to do it. And as he stood there outside the bedroom door, he could hear her sobbing, on the floor paralyzed with shame, exactly the same as he had been when it was him on the other side.