I Am a Strange Loop (75 page)
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.
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 Gödel, 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 Gödelian trap would succeed in catching it…
For an amusing interpretation of the infinite repeatability of Gödel’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].
Page 170
In that most delightful though most unlikely of scenarios…
[DeLong], [Goodstein], and [Chaitin] discuss non-Gödelian formulas that are undecidable for Gödelian reasons.
Page 172
No reliable prim/saucy distinguisher can exist…
See [DeLong], [Boolos and Jeffrey] , [Jeffrey], [Goodstein], [Hennie], [Wolf ], and [Hofstadter 1979] for discussions of many limitative results such as this one (which is Church’s theorem).
Page 172
It was logician Alfred Tarski who put one of the last nails…
See [Smullyan 1992] and [Hofstadter 1979] for discussions of Tarski’s deep result. In the latter, there is a novel approach to the classical liar paradox (“This sentence is not true”) using Tarski’s ideas, with the substrate taken to be the human brain instead of an axiomatic system.
Page 172
what appears to be a kind of upside-down causality…
See [Andersen] for a detailed technical discussion of downward causality. Less technical discussions are found in [Pattee] and [Simon]. See also Chapters 11 and 20 in [Hofstadter and Dennett], and especially the Reflections. [Laughlin] gives fascinating arguments for the thesis that in physics, the macroscopic arena is more fundamental or “deeper” than the microscopic.
Page 174
leaving just a high-level picture of information-manipulating processes…
See [Monod], [Berg and Singer], [Judson], and Chapter 27 of [Hofstadter 1985].
Page 177
symbols in our respective brains…
See [Hofstadter 1979], especially the dialogue “Prelude… Ant Fugue” and Chapters 11 and 12, for a careful discussion of this notion.
Page 178
the forbidding and inaccessible level of quarks and gluons…
See [Weinberg 1992] and [Pais 1986] for attempts at explanations of these incredibly abstruse notions.
Page 178
the only slightly more accessible level of genes…
See [Monod], [Berg and Singer], [Judson], and Chapter 27 (“
T
he
G
enetic
C
ode:
A
rbitrary?”) in [Hofstadter 1985].
Page 179
we…best understand our own actions as…
See [Dennett 1987] and [Dennett 1998].
Page 181
embellished by a fantastic folio of alternative versions…
[Steiner 1975] has a rich and provocative discussion of “alternity”, and the dialogue “Contrafactus” in [Hofstadter 1979] features an amusing scenario involving “subjunctive instant replays”. See also [Kahneman and Miller] and Chapter 12 of [Hofstadter 1985] for further musings on the incessantly flickering presence of counterfactuals in the subconscious human mind. [Hofstadter and FARG] describes a family of computational models of human thought processes in which making constant forays into alternity is a key architectural feature.
Page 182
housing a loop of self-representation…
See [Morden], [Kent], and [Metzinger].
Page 186
as the years pass, the “I” converges and stabilizes itself…
See [Dennett 1992].
Page 188
we cannot help attributing reality to our “I” and to those of other people…
See [Kent], [Dennett 1992], [Brinck], [Metzinger], [Perry], and [Hofstadter and Dennett].
Page 189
I was most impressed when I read about “Stanley”, a robot vehicle…
See [Davis 2006].
Page 193
just a big spongy bulb of inanimate molecules…
I suppose almost any book on the brain will convince one of this, but [Penfield and Roberts] did it to me as a teen-ager.
Page 194
pioneering roboticist and provocative writer Hans Moravec…
For some of Moravec’s more provocative speculations about the near-term future of humanity, see [Moravec].
Page 194
from the organic chemistry of carbon…
See Chapter 22 in [Hofstadter and Dennett], in which John Searle talks about “the right stuff”, which underwrites what he terms “the semantic causal powers of the brain”, a rather nice-sounding but murky term by which Searle means that when a human brain, such as his own or, say, that of poet Dylan Thomas, makes its owner come out with words, those words don’t just
seem
to stand for something, they really
do
stand for something. Unfortunately, in the case of poet Thomas, most of his output, though it sounds rather nice, is so full of murk that one has to wonder what sort of “stuff” could possibly make up the brain behind it.
Page 199
its symbol-count might well exceed “Graham’s constant”…
See [Wells 1986].
Page 208
For those who enjoy the taboo thrills of non-wellfounded sets…
See [Barwise and Moss].
Page 209
the deeper and richer an organism’s categorization equipment is…
See [Hofstadter 2001].
Page 233
a devilishly clever bon mot by David Moser…
One evening not long after we were married, Carol and I invited some friends over for an Indian dinner at our house in Ann Arbor. Melanie Mitchell and David Moser, well aware of Carol’s terrific Indian cooking, were delighted to come. It turned out, however, that at the last minute, our oldest guests, in their eighties, called up to tell us that they couldn’t handle very spicy foods, which unfortunately torpedoed Carol’s cooking plans. Somehow, though, she turned around on a dime and prepared a completely different yet truly delicious repast. A couple of hours after dinner was over, after a very lively discussion, most of our guests took off, leaving just David, Melanie, Carol, and me. We chatted on for a while, and finally, as they were about to hit the road, Carol casually reminded them of what she had originally intended to fix and told them why she hadn’t been able to follow through on her promise. Quick as a wink, David, feigning great indignation, burst out, “Why, you Indian-dinner givers, you!”
Page 233
her personal gemma (to borrow Stanislaw Lem’s term …)…
See “Non Serviam” in [Hofstadter and Dennett], which is a virtuosic philosophical fantasy masquerading as a book review (of a book that, needless to say, is merely a figment of Lem’s imagination).
Page 239
someone trying to grapple with quantum-mechanical reality…
[Pais 1986], [Pais 1992], and [Pullman] portray the transition period between the Bohr atom and quantum mechanics, while [Jauch] and [Greenstein and Zajonc] chart remaining mysteries.
Page 239
it might be tempting for some readers to conclude that in the wake of Carol’s death…
See Chapter 15 of [Hofstadter 1997], another place where I discuss many of these ideas.
Page 242
meaning of the term “universal machine”…
See [Hennie] and [Boolos and Jeffrey].