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.
What a ridiculous woman you are, she chided herself. Was Godel 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 damned theorem made sense; in its own perverted way, it felt right. She understood it, knew why it was true, believed it.
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.”