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shore, still smudged a little bit from their swim yesterday. Lawrence started a little fire and made some tea and they woke up eventually.
    “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 riding 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.”
     
    Soon, Alan got his Ph.D. and went back to England. He wrote Lawrence a couple of letters. The last of these stated, simply, that he would not be able to write Lawrence any more letters “of substance” and that Lawrence should not take it personally. Lawrence perceived right away that Alan’s society had put him to work doing something useful—probably figuring out how to keep it from being eaten alive by certain of its neighbors. Lawrence wondered what use America would find for him .
    He went back to Iowa State, considered changing his major to mathematics, but didn’t. It was the consensus of all whom he consulted that mathematics, like pipe-organ restoration,was a fine thing, but that one needed some way to put bread on the table. He remained in engineering and did more and more poorly at it until the middle of his senior year, when the university suggested that he enter a useful line