I Am a Strange Loop - Douglas R. Hofstadter [228]
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 nth 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