Douglas Hofstadter - I Am a Strange Loop

Page 83 their category systems became arbitrarily extensible… I defend this point of view in [Hofstadter 2001]. For more on human categories, see [Sander], [Margolis], [Minsky 1986], [Schank], [Aitchison], [Fauconnier], [Hofstadter 1997], and [Gentner et al. ].

Page 85 memories of episodes can be triggered… See [Kanerva], [Schank], and [Sander].

Page 86 That deep and tangled self-model is what “I”-ness is all about… See [Dennett 1991], [Metzinger], [Horney 1942], [Horney 1945], [Wheelis], [Nørretranders], and [Kent].

Page 89 Abstraction piled on abstraction… Should anyone care to get a taste of this, try reading [Ash and Gross] all the way to the end. It’s a bit like ordering “Indian hot” in an authentic Indian restaurant — you’ll wonder why you ever did.

Page 91 radicals, such as Évariste Galois… The great Galois was indeed a young radical, which led to his absurdly tragic death in a duel on his twenty-first birthday, but the phrase “solution by radicals” really refers to the taking of n th roots, called “radicals”. For a shallow, a medium, and a deep dip into Galois’ immortal, radical insights into hidden mathematical structures, see [Livio], [Bewersdorff ], and [Stewart], respectively.

Page 95 there is a special type of abstract structure or pattern… “Real Patterns” in [Dennett 1998] argues powerfully for the reality of abstract patterns, based on John Conway’s cellular automaton known as the “Game of Life”. The Game of Life itself is presented ideally in [Gardner], and its relevance to biological life is spelled out in [Poundstone].

Page 102 I am sorry to say, now hackneyed… I have long loved Escher’s art, but as time has passed, I have found myself drawn ever more to his early non-paradoxical landscapes, in which I see hints everywhere of his sense of the magic residing in ordinary scenes. See [Hofstadter 2002], an article written for a celebration of Escher’s 100th birthday.

Page 103 Is there, then, any genuine strange loop — a paradoxical structure that… Three excellent books on paradoxes are [Falletta], [Hughes and Brecht], and [Casati and Varzi 2006].

Page 104 an Oxford librarian named G. G. Berry… Only two individuals are thanked by the (nearly) self-sufficient authors of Principia Mathematica, and G. G. Berry is one of them.

Page 108 Chaitin and others went on… See [Chaitin], packed with stunning, strange results.

Page 113 written in PM notation as… I have here borrowed Gödel’s simplified version of PM notation instead of taking the symbols directly from the horses’ mouths, for those would have been too hard to digest. (Look at page 123 and you’ll see what I mean.)

Page 114 the sum of two squares… See [Hardy and Wright] and [Niven and Zuckerman].

Page 114 the sum of two primes… See [Wells 2005], an exquisite garden of delights.

Page 116 The passionate quest after order in an apparent disorder is what lights their fires… See [Ulam], [Ash and Gross], [Wells 2005], [Gardner], [Bewersdorff ], and [Livio].

Page 117 Nothing happens “by accident” in the world of mathematics… See [Davies].

Page 118 Paul Erdös once made the droll remark… Erdös, a devout matheist, often spoke of proofs from “The Book”, an imagined tome containing God’s perfect proofs of all great truths. For my own vision of “matheism”, see Chapter 1 of [Hofstadter and FARG].

Page 119 Variations on a Theme by Euclid… See [Chaitin].

Page 120 God does not play dice… See [Hoffmann], one of the best books I have ever read.

Page 121 many textbooks of number theory prove this theorem… See [Hardy and Wright] and [Niven and Zuckerman].

Page 122 About a decade into the twentieth century… The history of the push to formalize mathematics and logic is well recounted in [DeLong], [Kneebone], and [Wilder].

Page 122 a young boy was growing up in the town of Brünn… See [Goldstein] and [Yourgrau].

Page 125 Fibonacci …explored what are now known as the “Fibonacci numbers”… See [Huntley].

Page 125 This almost-but-not-quite-circular fashion… See [Péter] and [Hennie].

Page 126 a vast team of mathematicians… A recent book that purports to convey the crux of the elusive ideas of this team is [Ash and Gross]. I admire their chutzpah in trying to communicate these ideas to a wide public, but I suspect it is an impossible task.

Page 126 a trio of mathematicians… These are Yann Bugeaud, Maurice Mignotte, and Samir Siksek. It turns out that to prove that 144 is the only square in the Fibonacci sequence (other than 1) does not require highly abstract ideas, although it is still quite subtle. This was accomplished in 1964 by John H. E. Cohn.

Page 128 Gödel’s analogy was very tight… The essence and the meaning of Gödel’s work are well presented in many books, including [Nagel and Newman], [DeLong], [Smullyan 1961], [Jeffrey], [Boolos and Jeffrey], [Goodstein], [Goldstein], [Smullyan 1978], [Smullyan 1992], [Wilder], [Kneebone], [Wolf], [Shanker], and [Hofstadter 1979].

Page 129 developed piecemeal over many centuries… See [Nagel and Newman], [Wilder], [Kneebone], [Wolf ], [DeLong], [Goodstein], [Jeffrey], and [Boolos and Jeffrey].

Page 135 Anything you can do, I can do better!… My dear friend Dan Dennett once wrote (in a lovely book review of [Hofstadter and FARG], reprinted in [Dennett 1998]) the following sentence: “‘Anything you can do I can do meta’ is one of Doug’s mottoes, and of course he applies it, recursively, to everything he does.”

Well, Dan’s droll sentence gives the impression that Doug himself came up with this “motto” and actually went around saying it (for why else would Dan have put it in quote marks?). In fact, I had never said any such thing nor thought any such thought, and Dan was just “going me one meta”, in his own inimitable way. To my surprise, though, this “motto” started making the rounds and people quoted it back to me as if I really had thought it up and really believed it. I soon got tired of this because, although Dan’s motto is clever and funny, it does not match my self-image. In any case, this note is just my little attempt to squelch the rumor that the above-displayed motto is a genuine Hofstadter sentence, although I suspect my attempt will not have much effect.

Page 137 suppose you wanted to know if statement X is true or false… The dream of a mechanical method for reliably placing statements into two bins — ‘true’ and ‘false’ — is known as the quest for a decision procedure. The absolute nonexistence of a decision procedure for truth (or for provability) is discussed in [DeLong], [Boolos and Jeffrey], [Jeffrey], [Hennie], [Davis 1965], [Wolf], and [Hofstadter 1979].

Page 139 No formula can literally contain… [Nagel and Newman] presents this idea very clearly, as does [Smullyan 1961]. See also [Hofstadter 1982].

Page 139 an elegant linguistic analogy… See [Quine] for the original idea (which is actually a variation of Gödel’s idea (which is itself a variation of an idea of Jules Richard (which is a variation of an idea of Georg Cantor (which is a variation of an idea of Euclid (with help from Epimenides))))), and [Hofstadter 1979] for a variation on Quine’s theme.

Page 147 “…and Related Systems (I)”… Gödel put a roman numeral at the end of the title of his article because he feared he had not spelled out sufficiently clearly some of his ideas, and expected he would have to produce a sequel. However, his paper quickly received high praise from John von Neumann and other respected figures, catapulting the unknown Gödel to a position of great fame in a short time, even though it took most of the mathematical community decades to absorb the meaning of his results.