Front-To-Back Log of the Highlights from the Book

Front-to-Back log of the highlights from the book:

1. Friedrich der Grosse came to power in 1740 when Johann Sebastian Bach (died 1750) was a household name

2. Page 6 – The Royal Theme

3. Page 17 – Epimenides Paradox = “I am lying”

4. Page 18 – Gödel’s statement in number theory: “This statement of number theory does not have any proof in the system of Principia Mathematica”

5. Page 31 – Zeno’s Paradox shows motion is impossible

6. Page 33 – MU Puzzle

a. Rule 1: xI -> xIU = if a string ends in an ‘I’, you can append a ‘U’

b. Rule 2: Mx -> Mxx = if a string starts with an ‘M’, you can append the whole string less the starting ‘M’

c. Rule 3: xIIIy -> xUy = if a string contains ‘III’, you can replace it with ‘U’

d. Rule 4: xUUy -> xy = if a string contains ‘UU’, you can remove it

e. Start string: MI

f. Goal string: MU

7. Page 58 – Euclid’s proof that there are infinite prime numbers

a. Principle of the proof is that for any number, there is a prime number which is larger than it, which means there cannot be a finite number of prime numbers, because Euclid demonstrates that if the number were finite, there is a prime number which is larger than the largest prime, thereby by contradiction showing that the concept of a largest prime (and therefore, of a finite number of primes) is invalid

a.i. Let N be the largest prime number

a.ii. Consider N!+1, i.e. 1 x 2 x 3 x … N-2 x N-1 x N + 1

a.iii. This number cannot be divisible by any number from 2 to N, because it has a remainder of 1

a.iv. Therefore, this number either has prime factors greater than N or is itself prime. In either case, there is a prime number greater than N

a.v. Therefore, the original assumption that there exists a largest prime number is logically untrue

a.vi. Therefore, there cannot be a finite list of prime numbers, since that leads to the concept of a largest prime number. 8. Page 64 – Approach to developing a formal system for defining all prime numbers using purely typographical rules

a. ‘p-q system’ for representing addition (Page 47 – ‘Isomorphisms Induce Meaning’)

a.i. Axiom: xp-qx-, where ‘x’ is a string of hyphens

a.ii. Rule: xpyqz -> xpy-qz-, where ‘x’, ‘y’ and ‘z’ are strings of hyphens

b. ‘t-q system’ for representing multiplication (Page 64 – ‘The tq- System’)

b.i. Axiom: xt-qx, where ‘x’ is a string of hyphens

b.ii. Rule: xtyqz -> xty-qzx, where ‘x’, ‘y’ and ‘z’ are strings of hyphens

c. Page 65 – Capturing Compositeness

c.i. Rule: if xtyqz is a theorem, then Cz is a theorem, where ‘x’, ‘y’ and ‘z’ are strings of hyphens

d. We can then say that Px is a theorem if Cx is not a theorem, but this is not very satisfactory

e. Page 73 – Primes as Figure Rather than Ground

e.i. To represent non-divisibility, xDNDy will represent that x will not divide evenly into y

e.ii. Axiom: xyDNDx, where ‘x’ and ‘y’ are strings of hyphens

e.iii. Rule: xDNDy -> xDNDxy, where ‘x’ and ‘y’ are strings of hyphens

e.iv. To represent being divisor free, xDFy will represent that x has no factors up to and including y

e.v. Rule: --DNDx -> xDF--, where ‘x’ is a string of hyphens, i.e. if 2 does not divide into x, then x is divisor-free up to 2

e.vi. Rule: zDFx and x-DNDz -> zDFx-, where ‘x’ and ‘z’ are strings of hyphens, i.e. if z has no factors up to x and x+1 does not divide into z, then z has no factors up to x+1

e.vii. Rule: z-DFz -> Pz-, where ‘z’ is a string of hyphens, i.e. if z+1 has no factors up to z, then z+1 is prime

e.viii. Axiom: P--, because 2 is prime

9. Page 152 – Hoftstadter’s Law

a. It always takes longer when you think, even when you take into account Hofstadter’s Law b. Inspired by the lack of progress in Artificial Intelligence compared to predictions early in the discipline’s history

10. Page 153 – Haiku in ‘Canon by Intervallic Augmentation’

a. Such compressed poems / with seventeen syllables / can’t have much meaning

b. Meaning lies as much / in the mind of the reader / as in the haiku

11. Page 158 – The Location of Meaning

a. In this chapter Hofstadter introduces one of the fundamental questions posed by the book, can meaning be intrinsic to a message, or does it always require some context, e.g. cultural, technological to interpret the message and extract its meaning?

b. Page 171 – Meaning is Intrinsic if Intelligence is Natural

c. The location of meaning boils down to the more fundamental question of whether intelligence, as we understand it, is natural or not; if natural, then we ought to be able to construct a message which requires only natural intelligence to extract its meaning; if not, we have no natural way of constructing a message which does not require at least a human context to be able to extract the meaning reliably

12. Page 177 – ‘Chromatic Fantasy, and Feud’ along with Page 181 – ‘The Propositional Calculus’ provide a nice introduction to making statements in Propositional Calculus

13. Page 204 – Typographical Number Theory

a. In this chapter Hofstadter introduces the formal system, TNT, which attempts to capture number theory in string manipulation rules with a syntax based on propositional logic.

b. Examples

Sentence in TNT Interpretation in Number Theory 3a:(SS0*a)=6 6 is even, i.e. there is some value ‘a’ for which 2 multiplied by a equals 6. Va:(a*0)=0 Any number times zero equals zero, i.e. for any value ‘a’, a multiplied by zero equals zero Va:(S0*a)=a Any number times one equals itself, i.e. for any value ‘a’, a multiplied by one equals a Va:Vb:(a+b)=(b+a) The addition of two numbers is the same result regardless of the order they are added, i.e. for any values ‘a’ and ‘b’, a+b equals b+a

14. Page 246 (Mumon and Gödel)

a. Introduction to Zen Buddhist philosophy leading to the contradictions we see in Zen koans and Escher drawings and Gödel’s Theorem b. Revisit the MU-puzzle from Chapter 2 and finally prove ‘MU’ is not a theorem of that system, both through informal, logical reasoning and also through a more formal approach using Gödel-numbering, a technique which allows us to map any formal system to Typographical Number Theory