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 assumed 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 damnedest 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
"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 assumed 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 damnedest 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
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