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 Godel’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 Erdos once made the droll remark… Erdos, 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 Brunn… 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 [Peter] 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 Godel’s analogy was very tight… The essence and the meaning of Godel’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 Godel’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)”… Godel 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 Godel 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.
Page 150 respect for …the most mundane of analogies… See [Hofstadter 2001] and [Sander], as well as Chapter 24 in [Hofstadter 1985] and [Hofstadter and FARG].
Page 159 X’s play is so mega-inconsistent… This should be heard as “X’s play is omega-inconsistent”, which makes a phonetic hat-tip to the metamathematical concepts of omega- inconsistency and omega-incompleteness, discussed in many books in the Bibliography, such as [DeLong], [Nagel and Newman], [Hofstadter 1979], [Smullyan 1992], [Boolos and Jeffrey], and others. For our more modest purposes here, however, it suffices to know that this “o”-containing quip, plus the one two lines below it, is a play on words.
Page 160 Indeed, some years after Godel, such self-affirming formulas were concocted… See [Smullyan 1992], [Boolos and Jeffrey], and [Wolf].
Page 164 Why would logicians …give such good odds… See [Kneebone], [Wilder], and [Nagel and Newman], for reasons to believe strongly in the consistency of PM-like systems.
Page 165 not only although…but worse, because… For another treatment of the perverse theme of “although” turning into “because”, see Chapter 13 of [Hofstadter 1985].
Page 166 the same Godelian trap would succeed in catching it… For an amusing interpretation of the infinite repeatability of Godel’s construction as demonstrating the impossibility of artificial intelligence, see the chapter by J. R. Lucas in [Anderson], which is carefully analyzed (and hopefully refuted) in [DeLong], [Webb], and [Hofstadter 1979].
Page 167 called “the Hilbert Program”… See [DeLong], [Wolf ], [Kneebone], and [Wilder].
