Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
FIGURE 13. Liberation, by M. C. Escher (lithograph, 1955). ...
decided to try to capsulize all of number theory in an ideal system, is it really possible to do the job completely? Are numbers ...
can't be a multiple of N (because it leaves lover, when you divide by N); In other words, N! + 1, if it is divisible at all (oth ...
it by using phrases like "whatever N is", or "no matter what number N is". We could also phrase the proof over again, so that it ...
Sonata for Unaccompanied Achilles The telephone rings; Achilles picks it up. Achilles: Hello, this is Achilles. Achilles: Oh, he ...
Achilles: Wasn't it terrifying to see so many of them at the same time? Achilles: A guitar!? Of all things to be in the midst of ...
Achilles: Yes, I see all the black animals. Achilles: Yes, I also see how their "negative space"-what's left out- defines the wh ...
CHAPTER III Figure and Ground Primes vs. Composites THERE IS A strangeness to the idea that concepts can be captured by simple t ...
AXIOM SCHEMA: xt-qx is an axiom, whenever x is a hyphen-string. RULE OF INFERENCE: Suppose that x, y, and z are all hyphen-strin ...
lIIegally Characterizing Primes It is very tempting to jump from the C-type theorems directly to P-type theorems, by proposing a ...
The "holes" in this list are the nontheorems. To repeat the earlier question: Do the holes also have some "form" in common? Woul ...
FIGURE 16. Tiling of the plane using birds, by M. C. Escher (from a 1942 notebook). Our distinction is not as rigorous as one in ...
FIGURE 17. FIGURE. FIGURE Figure, by Scou E. Kim (1975). ...
you will see "FIGURE" everywhere, but "GROUND" nowhere! It is a paragon of recursive figures. In this clever drawing, there are ...
Unreachable falsehoods Well-formed formulas Strings FIGURE 18. Considerable visual symbolism is featured in this diagram of the ...
prime number of hyphens. So far, the only way we have found to represent prime numbers typographically is as a negative space. I ...
a typographical method for generating the set of nontheorems. But accord- ing to the earlier statement (which we here accept on ...
RULE: If x 0 N Oy is a theorem, then so is x 0 N Oxy. If you use the rule twice, you can generate this theorem: -----ON 0------- ...
C ontracrostipunctus Achilles has come to visit his friend and jogging companion, the Tortoise, at his home. Achilles: Heavens, ...
Achilles: Naturally, I suppose you disagreed. Tortoise: True, but he would hear nothing of my arguments. He staunchly maintained ...
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