Division by Zero
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 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.
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 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.”
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?”
