Miscellaneous Comments
pages 61--63: The dialog Sonata for Unaccompanied Achilles describes a phone conversation between Achilles and the Tortoise. The dialog only records Achilles' side of the conversation, from which we are meant to infer the Tortoise's side of the conversation. This device serves to introduce the theme of the following chapter Figure and Ground: Achilles' side of the conversation constitutes the figure of the dialog, and the Tortoise's side constitutes the ground. Achilles asks the Tortoise the following riddle: What's a word that begins with the letters "HE" and also ends with "HE"?It is implied that the Tortoise's first (incorrect) answer to the riddle is "hehe", and that his second (correct) answer is "headache". (The Tortoise is able to find the answer quickly because he himself is suffering from a headache.) The riddle also illustrates the theme of Figure and Ground: the letters "he" at the beginning and end of "headache" constitute the figure, and the letters "adach" in the middle of "headache" form the ground.
pages 75--81: The dialog Contracrostipunctus is an acrostic; if we take the first letter of each line of dialog, the following message is obtained: HOFSTADTER'S CONTRACROSTIPUNCTUS ACROSTICALLY BACKWARDS SPELLS 'J.S. BACH'(Spaces have been added where necessary.) Now spell this message "acrostically backwards"; that is, take the first letter of each word in the message, starting with the last word and moving backwards. The result is J. S. BACH(Spaces and periods have been added where necessary.)
Page 193: On page 193, Hofstadter refers back to the Carroll dialog and translates part of it into the language of propositional calculus: Here Achilles and the Tortoise are talking in English about propositional logic. One can understand the the difference between Achilles' and the Tortoise's interpretations in terms of the use/mention distinction. Achilles is asserting (1) '((A^B)>Z)' and '(A^B)' imply '((((A^B)>Z)^(A^B))>Z)'.This is a statement in the English (the metalanguage) about the propositional calculus (the object language), in which the words 'and' and 'implies' are being used. The Tortoise, on the other hand, interprets this assertion as (2) '((((A^B)>Z)^(A^B))>Z)'.This is a statement in the object language (propositional calculus), in which the symbols '^' and '>' are being mentioned. The fact that (2) follows from (1) is a consequence of the deduction theorem, which states that if we can prove 'B' by assuming 'A', then we can prove '(A>B)'. The deduction theorem is equivalent to the Hofstadter's fantasy rule.
page 402: In the dialog Aria with Diverse Variations, the Tortoise mentions that a friend of his is writing a book called Copper, Silver, Gold: an Indestructible Metallic Alloy. Achilles replies that he is confused by the title, and he and the Tortoise consider the following alternatives:
If we view the three three-word titles as a three-by-three matrix, and take the first letter of each word along the diagonal, the result is "GEB". This is clearly meant to evoke Cantor's diagonal construction, which is discussed in the following chapter Bloop and Floop and Gloop. The imaginary book Copper, Silver, Gold: an Indestructible Metallic Alloy is actually listed in the bibliography to GEB: Gebsdadter, Egbert B. Copper, Silver, Gold: an Indestructible Metallic Alloy. Perth: Acidic Books, 1979. A formidable hodge-podge, turgid and confused--yet remarkably similar to the present work. Professor Gebstadter's Shandean digressions include some excellent examples of indirect self-reference. Of particular interest is a reference in its well-annotated bibliography to an isomorphic, but imaginary, book.
page 435: At one point in the dialog Air on G's String, the following exchange occurs: Note that the phrase ""to quine a phrase", to quine a phrase" can be obtained by quining the phrase "to quine a phrase". |