This is a series on the book Gödel, Escher, Bach: An eternal golden braid by Douglas Hofstadter.
Earlier diaries are here
Today, we will look at Chapter IX: Mumon and Godel, p. 246-272.
From the overview
An attempt is made to talk about the strange ideas of Zen Buddhism. The Zen monk Mumon, who gave well known commentaries on many koans, is a central figure. In a way, Zen ideas bear a metaphorical resemblance to some contemporary ideas in the philosophy of mathematics. After this "Zennery", Godel's fundamental idea of Godel numbering is introduced, and a first pass through Godel's theorem is made
Some thoughts, ideas and questions:
Page 249, top: Old Mumon said something comprehensible, he must have been greatly upset
also: If words have no power, than what are koans? But words do have power.
p 251, middle: Are words inherently dualistic? Are you sure? (Answer yes or no) .... How sure are you? Answer: Very sure, somewhat sure, a little sure, not too sure, not at all sure. Perhaps words represent categories? Maybe there are archetypes of words? Plato thought there was an ideal world and a real one.
p 254, middle: Cooking monk did it without words, by demonstration. Good win for the cook!
p 254, bottom: But then, who is asking and who is answering?
p 258: I don't really see the point of the left and right sphere - can someone enlighten me?
p. 260: What led Hofstadter to thinking of counting I's? Once you get this, the puzzle is orders of magnitude easier. But how does one get this? What happens in our brains when we solve puzzles like this? Or fail to solve them?
The key is a shift in view, from
Is MU a theorem?
Can a theorem have 0 I's?
and then to
Can the I count be a multiple of 3?
p 261: A little extra hint if you don't get this:
Rule 1 doesn't change the I count
Rule 2 doubles the I count
Rule 3 subtracts 3 from the I count
Rule 4 doesn't affect the I count.
now, we next ask how each of these affects whether the I count is divisible by 3. The key is what the remainder is on dividing by 3.
Rule 1 and 4 have no effect.
If the I-remainder (that is, the remainder after dividing the I count by 3), starts at 1, it goes to 2, then to 1, then to 2. Because the I count itself goes something like 1 2 4 8 16....
and if you divide each by 3, you get alternating 1's and 2's.
If the remainder is 2, then the same pattern holds
so, you can't go from a "not divisible by 3" to a "divisible by 3"
Rule 3: The remainder stays the same.
page 263, you only need to follow the details of the math if you are interested. The key thing is that you CAN make such rules.
p. 264: You can make statements ABOUT number theory IN number theory. Some very long number (it doesn't really matter which) will mean "17 is prime".
p 266, middle, the bit starting "the fact is..." is central to Godel's proof.