I Am A Strange Loop by Douglas Hofstadter

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I Am A Strange LoopbyDouglas Hofstadter

• Review of Chapters 7, 8, 9
• By
• Chuck Selden
• NIH Extramural Staff Training Officer
• for
• Biomedical Computing Interest Group
• February 28, 2008

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Chapter 7Epi Phenomenon

• What is perception, what is reality?
• How real is any given thing?
• You perceive matter on a scale of familiar size
• Below the surface are smaller and smaller
  components (molecules, atoms, subatomic)
• And beyond you are larger and larger scale
  objects (buildings, cities, earths surface,
  stars, galaxies)
• We observe epi phenomena. Things made of
  unseen or unrealizable parts.

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What is real?

• What you believe in? What you have experienced?
  Concepts? Illusions? Rumors?
• Ex marathoners wall Math students dream?
• Black/white vs. shades of grey
• Intellectual grounding of reality
• Analogies, abstraction, inductive reasoning
• Belief systems strength of evidence
• Mental awareness and memory ? sense of reality
• Is the Andromeda Galaxy or an earthquake in China
  as real as a hangnail?
• Is there a marble in the middle of a thick stack
  of envelopes?

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Epiphenomenon

• A large-scale illusion created by the collusion
  of many small illusory and non-illusory,
  perceived and unperceived events
• Self as an abstraction a pattern that feels
  like a self..composed of feedback loops of
  observation and expectation (repeated over and
  over again

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The Prime Mover

• Behaviors or Goals to achieve individual needs
  (desires, wants) of the moment, and long term,
  optional pursuits and activities
• The long distance run - beyond the comfort zone
  
• What is the I that pushes my body?
• The I is the Prime Mover (the will)
• Does the I really exist? Is it real?
• We embody the careenium with the simmballs and
  simms mainly hidden from view we sense a
  coarse-grained representation, with most of the
  detail thrown out.

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Chapter 8 Embarking on a Strange Loop
SafariThe Careenium Cranium as a Strange Loop

• Loop examples (self-supporting integrated
  stuctures)
• Folded lid flaps of a cardboard box
• The MC Escher lithograph Drawing hands on flat
  paper
• This latter is fake,
• an illusion

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Numbers

• GG Berry, 1904, words for numbersas few
  syllables as possible to describe the number in
  English
• Sought to find the smallest number b that
  required 30 syllables to describe
• A strange self-defining notion, and paradoxical,
  because simply stating this definition for the
  number requires fewer than 30 syllables!

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Interesting Integers

• Prime numbers products of squared numbers
• Ambiguous numbers (eg, the number of heavenly
  bodies in the solar system)
• What makes an interesting number? Hard to pin it
  down.
• Thus, need formal means to describe numbers in a
  rigorous fashion
• (number theory and logic)

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Chapter 9Pattern and Provability

• Bertrand Russell and Alfred North Whiteheads
• Principia Mathematica
• A formal system of expressions
• 1 1 2 is s0 s0 ss0
• Spent a chapter on classes of integers
• Class A is sum of two squares
• Class B is not the sum of two squares
• Hofstadter adds
• Class C is the sum of two primes
• Class D is not the sum of two primes
• All sets have infinitely many members

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Primes and Squares

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
  41, 43, 47, 53, 59, 61, 69, 71, 73, 79, 83, 89,
  97, 101
• Class A is bold, Class B is not
• Gaps between members of each class are multiples
  of 4
• Conjecture and proof - is there a pattern?
• If X is true, then X has a proof (and vice versa)
• And if X is false, then there is no proof

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Euclids Proof of the infinitude of primes

• Is there a last great prime number, P ?
• If so, then P is the last member of the prime
  club
• Lets make a huge number, Q, made by multiplying
  all primes together.
• What would Q1 be, then?
• Not divisible by anything, thus is another prime,
  but is bigger than P.
• Therefore, there is no proof, and thus no final
  prime

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Mathematicians Credo

• X is true because there is a proof of X
• X is true and so there is proof of X.
• Proofs are guarantors of truth
• Where there is a regularity, there is a reason
• Conjecture pattern of class A and B primes?
• Yes!
• Class A are always for n1 and
• Class B always for n3
• The ratio of A and B approaches 1 at infinity.

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Principia Mathematica

• Allowed all the different branches of mathematics
  to be folded in with logic, making a seamless,
  perfect unity.
• Theorems were now understood as simply being the
  bottom lines of sequences of symbol-manipulations
  whose top lines were either axioms or earlier
  theorems.
• Mathematical truth was all coming together
  elegantly.