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MissT

Forumer storico
[...]
Now Princeton was an august school and going there was a great honor, but no one got around to mentioning either of these facts to Lawrence, who had no way of knowing. This had bad and good consequences. He accepted the scholarship with a faintness of gratitude that infuriated the oat lord. On the other hand, he adjusted to Princeton easily because it was just another place . It reminded him of the nicer bits of Virginia, and there were some nice pipe organs in town, though he was not all that happy with his engineering homework of bridge-designing and sprocket-cutting problems. As always, these eventually came down to math, most of which he could handle easily. From time to time he would get stuck, though, which led him to the Fine Hall: the headquarters of the Math Department.
There was a motley assortment of fellows wandering around in Fine Hall, many sporting British or European accents. Administratively speaking, many of these fellows were not members of the Math Department at all, but a separate thing called IAS, which stood for Institute for Advanced something-or-other. But they were all in the same building and they all knew a thing or two about math, so the distinction didn’t exist for Lawrence. Quite a few of these men would pretend shyness when Lawrence sought their advice, but others were at least willing to hear him out. For example: he had come up with a way to solve a difficult sprocket tooth shape problem that, as normally solved by engineers, would require any number of perfectly reasonable but aesthetically displeasing approximations. Lawrence’s solution would provide exact results. The only drawback was that it would require a quintillion slide-rule operators a quintillion years to solve. Lawrence was working on a radically different approach that, if it worked, would bring those figures down to a trillion and a trillion respectively. Unfortunately, Lawrence was unable to interest anyone at Fine Hall in anything as prosaic as gears, until all of a sudden he made friends with an energetic British fellow, whose name he promptly forgot, but who had been doing a lot of literal sprocket-making himself lately. This fellow was trying to build, of all things, a mechanical calculating machine—specifically a machine to calculate certain values of the Riemann Zeta Function [http://en.wikipedia.org/wiki/Riemann_zeta_function]
where s is a complex number.
Lawrence found this zeta function to be no more and no less interesting than any other math problem until his new friend assured him that it was frightfully important, and that some of the best mathematicians in the world had been gnawing on it for decades. The two of them ended up staying awake until three in the morning working out the solution to Lawrence’s sprocket problem. Lawrence presented the results proudly to his engineering professor, who snidely rejected it, on grounds of practicality, and gave him a poor grade for his troubles.
Lawrence finally remembered, after several more contacts, that the name of the friendly Brit was Al something-or-other. Because Al was a passionate cyclist, he and Al went on quite a few bicycle rides through the countryside of the Garden State. As they rode around New Jersey, they talked about math, and particularly about machines for taking the dull part of math off their hands.
But Al had been thinking about this subject for longer than Lawrence, and had figured out that computing machines were much more than just labor-saving devices. He’d been working on a radically different sort of computing mechanism that would work out any arithmetic problem whatsoever, as long as you knew how to write the problem down. From a pure logic standpoint, he had already figured out everything there was to know about this (as yet hypothetical) machine, though he had yet to build one. Lawrence gathered that actually building machinery was looked on as undignified at Cambridge (England, that is, where this Al character was based) or for that matter at Fine Hall. Al was thrilled to have found, in Lawrence, someone who did not share this view.
Al delicately asked him, one day, if Lawrence would terribly mind calling him by his full and proper name, which was Alan and not Al. Lawrence apologized and said he would try very hard to keep it in mind.
One day a couple of weeks later, as the two of them sat by a running stream in the woods above the Delaware Water Gap, Alan made some kind of an outlandish proposal to Lawrence involving penises. It required a great deal of methodical explanation, which Alan delivered with lots of blushing and stuttering. He was ever so polite, and several times emphasized that he was acutely aware that not everyone in the world was interested in this sort of thing.
Lawrence decided that he was probably one of those people.
Alan seemed vastly impressed that Lawrence had paused to think about it at all and apologized for putting him out. They went directly back to a discussion of computing machines, and their friendship continued unchanged. But on their next bicycle ride—an overnight camping trip to the Pine Barrens—they were joined by a new fellow, a German named Rudy von something-or-other.
[...]
 

kikkofrio

Forumer attivo
long su h4:
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MissT

Forumer storico
Alan and Rudy’s relationship seemed closer, or at least more multilayered, than Alan and Lawrence’s. Lawrence concluded that Alan’s penis scheme must have finally found a taker.
It got Lawrence to thinking. From an evolution standpoint, what was the point of having people around who were not inclined to have offspring? There must be some good, and fairly subtle, reason for it.
The only thing he could work out was that it was groups of people—societies—rather than individual creatures, who were now trying to out-reproduce and/or kill each other, and that, in a society, there was plenty of room for someone who didn’t have kids as long as he was up to something useful.

Alan and Rudy and Lawrence rode south, anyway, looking for the Pine Barrens. After a while the towns became very far apart, and the horse farms gave way to a low stubble of feeble, spiny trees that appeared to extend all the way to Florida—blocking their view, but not the head wind. “Where are the Pine Barrens I wonder?” Lawrence asked a couple of times. He even stopped at a gas station to ask someone that question. His companions began to make fun of him.
“Vere are ze Pine Barrens?” Rudy inquired, looking about quizzically.
“I should look for something rather barren-looking, with numerous pine trees,” Alan mused.
There was no other traffic and so they had spread out across the road to pedal three abreast, with Alan in the middle.
“A forest, as Kafka would imagine it,” Rudy muttered.
By this point Lawrence had figured out that they were, in fact, in the Pine Barrens. But he didn’t know who Kafka was. “A mathematician?” he guessed.
“Zat is a scary sing to sink of,” Rudy said.
“He is a writer,” Alan said. “Lawrence, please don’t be offended that I ask you this, but: do you recognize any other people’s names at all? Other than family and close friends, I mean.”
Lawrence must have looked baffled. “I’m trying to figure out whether it all comes from in here,” Alan said, reaching out to rap his knuckles on the side of Lawrence’s head, “or do you sometimes take in new ideas from other human beings?”
“When I was a little boy, I saw angels in a church in Virginia,” Lawrence said, “but I think that they came from inside my head.”
“Very well,” Alan said.
But later Alan had another go at it. They had reached the fire lookout tower and it had been a thunderous disappointment: just an alienated staircase leading nowhere, and a small cleared area below that was glittery with shards of liquor bottles. They pitched their tent by the side of a pond that turned out to be full of rust-colored algae that stuck to the hairs on their bodies. Then there was nothing left to do but drink schnapps and talk about math.
Alan said, “Look, it’s like this: Bertrand Russell and another chap named Whitehead wrote Principia Mathematica.”
“Now I know you’re pulling my leg,” Waterhouse said. “Even I know that Sir Isaac Newton wrote that.”
“Newton wrote a different book, also called Principia Mathematica , which isn’t really about mathematics at all; it’s about what we would today call physics.”
“Then why did he call it Principia Mathematica?”
“Because the distinction between mathematics and physics wasn’t especially clear in Newton’s day—”
“Or maybe even in zis day,” Rudy said.
“—which is directly relevant to what I’m talking about,” Alan continued. “I am talking about Russell’s P.M., in which he and Whitehead started absolutely from scratch, I mean from nothing, and built it all up—all mathematics—from a small number of first principles. And why I am telling you this, Lawrence, is that—Lawrence! Pay attention!”
“Hmmm?”
“Rudy—take this stick, here—that’s right—and keep a close eye on Lawrence, and when he gets that foggy look on his face, poke him with it!”
“Zis is not an English school, you can’t do zese kind of sing.”
“I’m listening,” Lawrence said.
“What came out of P.M., which was terrifically radical, was the ability to say that all of math, really, can be expressed as a certain ordering of symbols.”
“Leibniz said it a long time before zen!” protested Rudy.
“Er, Leibniz invented the notation we use for calculus, but—”
“I’m not talking about zat!”
“And he invented matrices, but—”
“I’m not talking about zat eezer!”
“And he did some work with binary arithmetic, but—”
“Zat is completely different!”
“Well, what the hell are you talking about, then, Rudy?”
“Leibniz invented ze basic alphabet—wrote down a set of symbols, for expressing statements about logic.”
“Well, I wasn’t aware that Herr Leibniz counted formal logic among his interests, but—”
“Of course! He wanted to do what Russell and Whitehead did, except not just with mathematics, but with everything in ze whole world!”
“Well, from the fact that you are the only man on the planet, Rudy, who seems to know about this undertaking of Leibniz’s, can we assume that he failed?”
“You can assume anything that pleases your fancy, Alan,” Rudy responded, “but I am a mathematician and I do not assume anything.”
 

MissT

Forumer storico
Alan sighed woundedly, and gave Rudy a Significant Look which Waterhouse assumed meant that there would be trouble later. “If I may just make some headway, here,” he said, “all I’m really trying to get you to agree on, is that mathematics can be expressed as a series of symbols,” (he snatched the Lawrence-poking stick and began drawing things like + = 3) [square root -1] [p]; in the dirt) “and frankly I could not care less whether they happen to be Leibniz’s symbols, or Russell’s, or the hexagrams of the I Ching….”
“Leibniz was fascinated by the I Ching!” Rudy began.
“Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were, sitting in the dining car, having a nice conversation, and that train is being pulled along at a terrific clip by certain locomotives named The Bertrand Russell and Riemann and Euler and others. And our friend Lawrence is running alongside the train, trying to keep up with us—it’s not that we’re smarter than he is, necessarily, but that he’s a farmer who didn’t get a ticket. And I, Rudy, am simply reaching out through the open window here, trying to pull him onto the fucking train with us so that the three of us can have a nice little chat about mathematics without having to listen to him panting and gasping for breath the whole way.”
“All right, Alan.”
“Won’t take a minute if you will just stop interrupting.”
“But there is a locomotive too named Leibniz.”
“Is it that you don’t think I give enough credit to Germans? Because I am about to mention a fellow with an umlaut.”
“Oh, would it be Herr Türing?” Rudy said slyly.
“Herr Türing comes later. I was actually thinking of Gödel.”
“But he’s not German! He’s Austrian!”
“I’m afraid that it’s all the same now, isn’t it?”
“Ze Anschluss wasn’t my idea, you don’t have to look at me that way, I think Hitler is appalling.”
“I’ve heard of Gödel,” Waterhouse put in helpfully. “But could we back up just a sec?”
“Of course Lawrence.”
“Why bother? Why did Russell do it? Was there something wrong with math? I mean, two plus two equals four, right?”
Alan picked up two bottlecaps and set them down on the ground. “Two. One-two. Plus—” He set down two more. “Another two. One-two. Equals four. One-two-three-four.”
“What’s so bad about that?” Lawrence said.
“But Lawrence—when you really do math, in an abstract way, you’re not counting bottlecaps, are you?”
“I’m not counting anything.”
Rudy broke the following news: “Zat is a very modern position for you to take.”
“It is?”
Alan said, “There was this implicit belief, for a long time, that math was a sort of physics of bottlecaps. That any mathematical operation you could do on paper, no matter how complicated, could be reduced—in theory, anyway—to messing about with actual physical counters, such as bottlecaps, in the real world.”
“But you can’t have two point one bottlecaps.”
“All right, all right, say we use bottlecaps for integers, and for real numbers like two point one, we use physical measurements, like the length of this stick.” Alan tossed the stick down next to the bottlecaps.
“Well what about pi, then? You can’t have a stick that’s exactly pi inches long.”
“Pi is from geometry—ze same story,” Rudy put in.
“Yes, it was believed that Euclid’s geometry was really a kind of physics, that his lines and so on represented properties of the physical world. But—you know Einstein?”
“I’m not very good with names.”
“That white-haired chap with the big mustache?”
“Oh, yeah,” Lawrence said dimly, “I tried to ask him my sprocket question. He claimed he was late for an appointment or something.”
“That fellow has come up with a general relativity theory, which is sort of a practical application, not of Euclid’s, but of Riemann’s geometry—”
“The same Riemann of your zeta function?”
“Same Riemann, different subject. Now let’s not get sidetracked here Lawrence—”
[...]
 

MissT

Forumer storico
“Riemann showed you could have many many different geometries that were not the geometry of Euclid but that still made sense internally,” Rudy explained.
“All right, so back to P.M. then,” Lawrence said.
“Yes! Russell and Whitehead. It’s like this: when mathematicians began fooling around with things like the square root of negative one, and quaternions, then they were no longer dealing with things that you could translate into sticks and bottlecaps. And yet they were still getting sound results.”
“Or at least internally consistent results,” Rudy said.
“Okay. Meaning that math was more than a physics of bottlecaps.”
“It appeared that way, Lawrence, but this raised the question of was mathematics really true or was it just a game played with symbols? In other words—are we discovering Truth, or just wanking?”
“It has to be true because if you do physics with it, it all works out! I’ve heard of that general relativity thing, and I know they did experiments and figured out it was true.”
“Ze great majority of mathematics does not lend itself to experimental testing,” Rudy said.
“The whole idea of this project is to sever the ties to physics,” Alan said.
“And yet not to be vanking ourselves.”
“That’s what P.M. was trying to do?”
“Russell and Whitehead broke all mathematical concepts down into brutally simple things like sets. From there they got to integers, and so on.”
“But how can you break something like pi down into a set?”
“You can’t,” Alan said, “but you can express it as a long string of digits. Three point one four one five nine, and so on.”
“And digits are integers,” Rudy said.
“But no fair! Pi itself is not an integer!”
“But you can calculate the digits of pi, one at a time, by using certain formulas. And you can write down the formulas like so!” Alan scratched this in the dirt:
........
“I have used the Leibniz series in order to placate our friend. See, Lawrence? It is a string of symbols.”
“Okay. I see the string of symbols,” Lawrence said reluctantly.
“Can we move on? Gödel said, just a few years ago, ‘Say! If you buy into this business about mathematics being just strings of symbols, guess what?’ And he pointed out that any string of symbols—such as this very formula, here—can be translated into integers.”
“How?”
“Nothing fancy, Lawrence—it’s just simple encryption. Arbitrary. The number ‘538’ might be written down instead of this great ugly [sigma], and so on.
“Seems pretty close to wanking, now.”
“No, no. Because then Gödel sprang the trap! Formulas can act on numbers, right?”
“Sure. Like 2x.”
“Yes. You can substitute any number for x and the formula 2x will double it. But if another mathematical formula, such as this one right here, for calculating pi, can be encoded as a number, then you can have another formula act on it. Formulas acting on formulas!”
“Is that all?”
“No. Then he showed, really through a very simple argument, that if formulas really can refer to themselves, it’s possible to write one down saying ‘this statement cannot be proved.’ Which was tremendously startling to Hilbert and everyone else, who expected the opposite result.”
“Have you mentioned this Hilbert guy before?”
“No, he is new to this discussion, Lawrence.”
“Who is he?”
“A man who asks difficult questions. He asked a whole list of them once. Gödel answered one of them.”
“And Türing answered another,” Rudy said.
“Who’s that?”
“It’s me,” Alan said. “But Rudy’s joking. ‘Turing’ doesn’t really have an umlaut in it.”
“He’s going to have an umlaut in him later tonight,” Rudy said, looking at Alan in a way that, in retrospect, years later, Lawrence would understand to have been smoldering.
“Well, don’t keep me in suspense. Which one of his questions did you answer?”
“The Entscheidungsproblem,” Rudy said.
“Meaning?”
Alan explained, “Hilbert wanted to know whether any given statement could, in principle, be found true or false.”
“But after Gödel got finished, it changed,” Rudy pointed out.
“That’s true—after Gödel it became ‘Can we determine whether any given statement is provable or non-provable?’ In other words, is there some sort of mechanical process we could use to separate the provable statements from the nonprovable ones?”
“‘Mechanical process’ is supposed to be a metaphor, Alan…”
“Oh, stop it, Rudy! Lawrence and I are quite comfortable with machinery.”
“I get it,” Lawrence said.
“What do you mean, you get it?” Alan said.
“Your machine—not the zeta-function calculator, but the other one. The one we’ve been talking about building—”
“It is called Universal Turing Machine,” Rudy said.
“The whole point of that gizmo is to separate provable from nonprovable statements, isn’t it?”
“That’s why I came up with the basic idea for it,” Alan said. “So Hilbert’s question has been answered. Now I just want to actually build one so that I can beat Rudy at chess.”
“You haven’t told poor Lawrence the answer yet!” Rudy protested.
“Lawrence can figure it out,” Alan said. “It’ll give him something to do.”

[...]

“Did you solve the problem?” Alan asked him.
“Well you can turn that Universal Turing Machine of yours into any machine by changing the presets—”
“Presets?”
“Sorry, Alan, I think of your U.T.M. as being kind of like a pipe organ.”
“Oh.”
“Once you’ve done that, anyway, you can do any calculation you please, if the tape is long enough. But gosh, Alan, making a tape that’s long enough, and that you can write symbols on, and erase them, is going to be sort of tricky—Atanasoff’s capacitor drum would only work up to a certain size—you’d have to—”
“This is a digression,” Alan said gently.
“Yeah, okay, well—if you had a machine like that, then any given preset could be represented by a number—a string of symbols. And the tape that you would feed into it to start the calculation would contain another string of symbols. So it’s Gödel’s proof all over again—if any possible combination of machine and data can be represented by a string of numbers, then you can just arrange all of the possible strings of numbers into a big table, and then it turns into a Cantor diagonal type of argument, and the answer is that there must be some numbers that cannot be computed.”
“And ze Entscheidungsproblem?” Rudy reminded him.
“Proving or disproving a formula—once you’ve encrypted the formula into numbers, that is—is just a calculation on that number. So it means that the answer to the question is, no! Some formulas cannot be proved or disproved by any mechanical process! So I guess there’s some point in being human after all!”
Alan looked pleased until Lawrence said this last thing, and then his face collapsed. “Now there you go making unwarranted assumptions.”
“Don’t listen to him, Lawrence!” Rudy said. “He’s going to tell you that our brains are Turing machines.”
“Thank you, Rudy,” Alan said patiently. “Lawrence, I submit that our brains are Turing machines.”
“But you proved that there’s a whole lot of formulas that a Turing machine can’t process!”
“And you have proved it too, Lawrence.”
“But don’t you think that we can do some things that a Turing machine couldn’t?”
“Gödel agrees with you, Lawrence,” Rudy put in, “and so does Hardy.”
“Give me one example,” Alan said.
“Of a noncomputable function that a human can do, and a Turing machine can’t?”
“Yes. And don’t give me any sentimental nonsense about creativity. I believe that a Universal Turing Machine could show behaviors that we would construe as creative.”
“Well, I don’t know then… I’ll try to keep my eye out for that kind of thing in the future.’’
But later, as they were tiding back towards Princeton, he said, “What about dreams?”
“Like those angels in Virginia?”
“I guess so.”
“Just noise in the neurons, Lawrence.”
“Also I dreamed last night that a zeppelin was burning.”

[....]
 

MissT

Forumer storico
Ho un po' di robe da fare - sto tessendo dei contatti... - ma dimmi per che ora, se pomeriggio potrei farcela. Perdonami nel frattempo anzi appena entro in produzione so io come fare a farmi perdonare :)
 

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