Book Reports · Spirituality

An Eternal Golden Brain: Goedel, Escher, Bach

Easily one of the most complex, challenging, and interesting books I have run across, I first read Douglas Hofstadter’s “Goedel, Escher, Bach” in high school and have recently revisiting it. Below are some highlights:

  1. p. 9: Isomorphism: a transformation that preserves all the information of the original
  2. p. 19: “It had long been assumed that geometry was what Euclid had codified, and that, although there might be small flaws in Euclid’s presentation, they were unimportant and any real progress in geometry would be achieved by extending Euclid. This idea was shattered by the roughly simultaneous discovery of non-Euclidean geometry by several people – a discovery that shocked the mathematics community, because it deeply challenged the idea that mathematics studies the real world.”
  3. p. 26, on the subject of artificial intelligence: “No one knows where the borderline between non-intelligent behavior and intelligent behavior lies; in face, to suggest that a sharp borderline exists is probably silly. But essential abilities for intelligence are certainly:
    1. to respond to situations very flexibly;
    2. to take advantage of fortuitous circumstances;
    3. to make sense out of ambiguous or contradictory messages;
    4. to recognize the relative importance of different elements of a situation;
    5. to find similarities between situations despite differences which may separate them;
    6. to draw distinctions between situations despite similarities which may link them
    7. to synthesize new concepts by taking old concepts and putting them together in new ways;
    8. to come up with ideas which are novel”
  4. p. 72: “There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any formal system.”
  5. p. 100: “…it is now well established – as a matter of fact as a direct consequence of Goedel’s Theorem – that number theory is a bifurcated theory, with standard and nonstandard versions. Unlike the situation in geometry, however, the number of ‘brands’ of number theory is infinite, which makes the situation of number theory considerably more complex.”
  6. p. 100: “For practical purposes, all number theories are the same. In other words, if bridge building depended on number theory (which in a sense it does), the fact that there are different number theories would not matter, since in the aspects relevant to the real world, all number theories overlap.”
  7. p. 101: “Goedel’s Incompleteness Theorem says that any system which is ‘sufficiently powerful’ is, by virtue of its power, incomplete, in the sense that there are well-formed strings which express true statements of number theory, but which are not theorems. (There are truths belonging to number theory which are not provable within the system.)”
  8. p. 159: “Avery’s experiments shows that, of all the biological molecules, only DNA transmits hereditary properties. One can modify other molecules in an organism, such as proteins, but such modifications will not be transmitted to later generations. However, when DNA is modified, all successive generations inherit the modified DNA.”
  9. p. 160: “…’genetic meaning’ – that is, information about phenotype structure – is spread all through the small parts of a molecule of DNA, although nobody understands the language yet. (Warning: Understanding this ‘language’ would not at all be the same as cracking the Genetic Code, something which took place in the early 1960’s. The Genetic Code tells how to translate short portions of DNA into various amino acids. Thus, cracking the Genetic Code is comparable to figuring out the phonetic values of the letters of a foreign alphabet, without figuring out the grammar of the language or the meanings of any of the words.”
  10. p. 166: “In these examples of decipherment of out-of-context messages, we can separate out fairly clearly three levels of information: (1) the frame message; (2) the outer message; (3) the inner message”
  11. p. 170: “…our intelligence is not disembodied, but is instantiated in physical objects: our brains. Their structure is due to the long process of evolution, and their operations are governed by the laws of physics. Since they are physical entities, our brains run without being told how to run.
  12. p. 172: “In our chauvinism, we would call any being with a brain sufficiently much like our own ‘intelligent’, and refuse to recognize other types of objects as intelligent.”

To be continued!

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