“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.”
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 classical 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 quantities 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.
Hilbert once said, “If mathematical thinking is defective, where are we to find truth and certitude?”
Would her suicide attempt brand her for the rest of her life? Renee wondered. She aligned the corners of the papers on her desk. Would people henceforth regard her, perhaps unconsciously, as flighty or unstable? She had never asked Carl if he had ever felt such anxieties, perhaps because she never held his attempt against him. It had happened many years ago, and anyone seeing him now would immediately recognize him as a whole person.
But Renee could not say the same for herself. Right now she was unable to discuss mathematics intelligibly, and she was unsure whether she ever could again. Were her colleagues to see her now, they would simply say, She's lost the knack.
Finished at her desk, Renee left her study and walked into the living room. After her formalism circulated through the academic community, it would require an overhaul of established mathematical foundations, but it would affect only a few as it had her. Most would be like Fabrisi; they would follow the proof mechanically, and be convinced by it, but no more. The only persons who would feel it nearly as keenly as she had were those who could actually grasp the contradiction, who could intuit it. Callahan was one of those; she wondered how he was handling it as the days wore on.
Renee traced a curly pattern in the dust on an end table. Before, she might have idly parameterized the curve, examined some of its characteristics. Now there seemed no point. All of her visualizations simply collapsed.
She, like many, had always thought that mathematics did not derive its meaning from the universe, but rather imposed some meaning onto the universe. Physical entities were not greater or less than one another, not similar or dissimilar; they simply were, they existed. Mathematics was totally independent, but it virtually provided a semantic meaning for those entities, supplying categories and relationships. It didn't describe any intrinsic quality, merely a possible interpretation.
But no more. Mathematics was inconsistent once it was removed from physical entities, and a formal theory was nothing if not consistent. Math was empirical, no more than that, and it held no interest for her.