Division by Zero
“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.”
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 assuming 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 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.
