rows of three, four rows of four: the tiles fit together in a square . Of course. No matter which side you looked at it from, it came out the same. And more than that, each square was bigger than the last by an odd number of tiles . It was an epiphany. The conclusion was necessary: it had a rightness to it, confirmed by the smooth, cool feel of the tiles. And the way the tiles were fitted together, with such incredibly fine lines where they met; she had shivered at the precision.
Later on there came other realizations, other achievements. The astonishing doctoral dissertation at twenty-three, the series of acclaimed papers; people compared her to Von Neumann, universities wooed her. She had never paid any of it much attention. What she did pay attention to was that same sense of rightness, possessed by every theorem she learned, as insistent as the tiles’ physicality, and as exact as their fit.
3b
Carl felt that the person he was today was born after his attempt, when he met Laura. After being released from the hospital, he was in no mood to see anyone, but a friend of his had managed to introduce him to Laura. He had pushed her away initially, but she had known better. She had loved him while he was hurting, and let him go once he was healed. Through knowing her Carl had learned about empathy, and he was remade.
Laura had moved on after getting her own master's degree, while he stayed at the university for his doctorate in biology. He suffered various crises and heartbreaks later on in life, but never again despair.
Carl marveled when he thought about what kind of person she was. He hadn't spoken to her since grad school; what had her life been like over the years? He wondered whom else she had loved. Early on he had recognized what kind of love it was, and what kind it wasn't, and he valued it immensely.
4
In the early nineteenth century, mathematicians began exploring geometries that differed from Euclidean geometry; these alternate geometries produced results that seemed utterly absurd, but they didn't produce logical contradictions. It was later shown that these non-Euclidean geometries were consistent relative to Euclidean geometry: they were logically consistent, as long as one assumed that Euclidean geometry was consistent.
The proof of Euclidean geometry's consistency eluded mathematicians. By the end of the nineteenth century, the best that was achieved was a proof that Euclidean geometry was consistent as long as arithmetic was consistent.
4a
At the time, when it all began, Renee had thought it little more than an annoyance. She had walked down the hall and knocked on the open door of Peter Fabrisi's office. “Pete, got a minute?"
Fabrisi pushed his chair back from his desk. “Sure, Renee, what's up?"
Renee came in, knowing what his reaction would be. She had never asked anyone in the department for advice on a problem before; it had always been the reverse. No matter. “I was wondering if you could do me a favor. You remember what I was telling you about a couple weeks back, about the formalism I was developing?"
He nodded. “The one you were rewriting axiom systems with."
"Right. Well, a few days ago I started coming up with really ridiculous conclusions, and now my formalism is contradicting itself. Could you take a look at it?"
Fabrisi's expression was as expected. “You want—sure, I'd be glad to."
"Great. The examples on the first few pages are where the problem is; the rest is just for your reference.” She handed Fabrisi a thin sheaf of papers. “I thought if I talked you through it, you'd just see the same things I do."
"You're probably right.” Fabrisi looked at the first couple pages. “I don't know how long this'll take."
"No hurry. When you get a chance, just see whether any of my assumptions seem a little dubious, anything like that. I'll still be going at it, so I'll tell you if I come up with anything. Okay?"
Fabrisi smiled. “You're just going to come in this
Later on there came other realizations, other achievements. The astonishing doctoral dissertation at twenty-three, the series of acclaimed papers; people compared her to Von Neumann, universities wooed her. She had never paid any of it much attention. What she did pay attention to was that same sense of rightness, possessed by every theorem she learned, as insistent as the tiles’ physicality, and as exact as their fit.
3b
Carl felt that the person he was today was born after his attempt, when he met Laura. After being released from the hospital, he was in no mood to see anyone, but a friend of his had managed to introduce him to Laura. He had pushed her away initially, but she had known better. She had loved him while he was hurting, and let him go once he was healed. Through knowing her Carl had learned about empathy, and he was remade.
Laura had moved on after getting her own master's degree, while he stayed at the university for his doctorate in biology. He suffered various crises and heartbreaks later on in life, but never again despair.
Carl marveled when he thought about what kind of person she was. He hadn't spoken to her since grad school; what had her life been like over the years? He wondered whom else she had loved. Early on he had recognized what kind of love it was, and what kind it wasn't, and he valued it immensely.
4
In the early nineteenth century, mathematicians began exploring geometries that differed from Euclidean geometry; these alternate geometries produced results that seemed utterly absurd, but they didn't produce logical contradictions. It was later shown that these non-Euclidean geometries were consistent relative to Euclidean geometry: they were logically consistent, as long as one assumed that Euclidean geometry was consistent.
The proof of Euclidean geometry's consistency eluded mathematicians. By the end of the nineteenth century, the best that was achieved was a proof that Euclidean geometry was consistent as long as arithmetic was consistent.
4a
At the time, when it all began, Renee had thought it little more than an annoyance. She had walked down the hall and knocked on the open door of Peter Fabrisi's office. “Pete, got a minute?"
Fabrisi pushed his chair back from his desk. “Sure, Renee, what's up?"
Renee came in, knowing what his reaction would be. She had never asked anyone in the department for advice on a problem before; it had always been the reverse. No matter. “I was wondering if you could do me a favor. You remember what I was telling you about a couple weeks back, about the formalism I was developing?"
He nodded. “The one you were rewriting axiom systems with."
"Right. Well, a few days ago I started coming up with really ridiculous conclusions, and now my formalism is contradicting itself. Could you take a look at it?"
Fabrisi's expression was as expected. “You want—sure, I'd be glad to."
"Great. The examples on the first few pages are where the problem is; the rest is just for your reference.” She handed Fabrisi a thin sheaf of papers. “I thought if I talked you through it, you'd just see the same things I do."
"You're probably right.” Fabrisi looked at the first couple pages. “I don't know how long this'll take."
"No hurry. When you get a chance, just see whether any of my assumptions seem a little dubious, anything like that. I'll still be going at it, so I'll tell you if I come up with anything. Okay?"
Fabrisi smiled. “You're just going to come in this
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