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Chapter XVI: Self-Ref and Self-Rep 495

The Magn fierab, Indeed 549

Chapter XVII: Church, Turing, Tarski, and Others 559

SHRDFU, Toy of Man's Designing 586

Chapter XVIII: Artificial Intelligence: Retrospects 594

Contraf actus 633

Chapter XIX: Artificial Intelligence: Prospects 641

Sloth Canon 681

Chapter XX: Strange Foops, Or Tangled Hierarchies 684

Six-Part Ricercar 720

Notes 743

Bibliography 746

Credits 757

Index 759


Contents


VII



Overview
Part I: GEB

Introduction: A Musico-Logical Offering. The book opens with the story of Bach's Musical
Offering. Bach made an impromptu visit to King Frederick the Great of Prussia, and was
requested to improvise upon a theme presented by the King. His improvisations formed the basis
of that great work. The Musical Offering and its story form a theme upon which I "improvise"
throughout the book, thus making a sort of "Metamusical Offering". Self-reference and the
interplay between different levels in Bach are discussed: this leads to a discussion of parallel
ideas in Escher's drawings and then Godel’s Theorem. A brief presentation of the history of logic
and paradoxes is given as background for Godel’s Theorem. This leads to mechanical reasoning
and computers, and the debate about whether Artificial Intelligence is possible. I close with an
explanation of the origins of the book-particularly the why and wherefore of the Dialogues.

Three-Part Invention. Bach wrote fifteen three-part inventions. In this three-part Dialogue, the
Tortoise and Achilles-the main fictional protagonists in the Dialogues-are "invented" by Zeno (as
in fact they were, to illustrate Zeno's paradoxes of motion). Very short, it simply gives the flavor
of the Dialogues to come.

Chapter I: The MU-puzzle. A simple formal system (the MIL'-system) is presented, and the reader
is urged to work out a puzzle to gain familiarity with formal systems in general. A number of
fundamental notions are introduced: string, theorem, axiom, rule of inference, derivation, formal
system, decision procedure, working inside/outside the system.

Two-Part Invention. Bach also wrote fifteen two-part inventions. This two-part Dialogue was written
not by me, but by Lewis Carroll in 1895. Carroll borrowed Achilles and the Tortoise from Zeno,
and I in turn borrowed them from Carroll. The topic is the relation between reasoning, reasoning
about reasoning, reasoning about reasoning about reasoning, and so on. It parallels, in a way,
Zeno's paradoxes about the impossibility of motion, seeming to show, by using infinite regress,
that reasoning is impossible. It is a beautiful paradox, and is referred to several times later in the
book.

Chapter II: Meaning and Form in Mathematics. A new formal system (the pq-system) is
presented, even simpler than the MlU-system of Chapter I. Apparently meaningless at first, its
symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they
appear in. This revelation is the first important insight into meaning: its deep connection to
isomorphism. Various issues related to meaning are then discussed, such as truth, proof, symbol
manipulation, and the elusive concept, "form".

Sonata for Unaccompanied Achilles. A Dialogue which imitates the Bach Sonatas for
unaccompanied violin. In particular, Achilles is the only speaker, since it is a transcript of one
end of a telephone call, at the far end of which is the Tortoise. Their conversation concerns the
concepts of "figure" and "ground" in various


Overview


VIII



contexts- e.g., Escher's art. The Dialogue itself forms an example of the distinction, since
Achilles' lines form a "figure", and the Tortoise's lines-implicit in Achilles' lines-form a "ground".

Chapter III: Figure and Ground. The distinction between figure and ground in art is compared to
the distinction between theorems and nontheorems in formal systems. The question "Does a
figure necessarily contain the same information as its ground%" leads to the distinction between
recursively enumerable sets and recursive sets.

Contracrostipunctus. This Dialogue is central to the book, for it contains a set of paraphrases of
Godel’s self-referential construction and of his Incompleteness Theorem. One of the paraphrases
of the Theorem says, "For each record player there is a record which it cannot play." The

Chapter XVI: Self-Ref and Self-Rep 495

The Magn fierab, Indeed 549

Chapter XVII: Church, Turing, Tarski, and Others 559

SHRDFU, Toy of Man's Designing 586

Chapter XVIII: Artificial Intelligence: Retrospects 594

Contraf actus 633

Chapter XIX: Artificial Intelligence: Prospects 641

Sloth Canon 681

Chapter XX: Strange Foops, Or Tangled Hierarchies 684

Six-Part Ricercar 720

Notes 743

Bibliography 746

Credits 757

Index 759

Contents

VII

Overview

Part I: GEB

Introduction: A Musico-Logical Offering. The book opens with the story of Bach's Musical

Offering. Bach made an impromptu visit to King Frederick the Great of Prussia, and was

requested to improvise upon a theme presented by the King. His improvisations formed the basis

of that great work. The Musical Offering and its story form a theme upon which I "improvise"

throughout the book, thus making a sort of "Metamusical Offering". Self-reference and the

interplay between different levels in Bach are discussed: this leads to a discussion of parallel

ideas in Escher's drawings and then Godel’s Theorem. A brief presentation of the history of logic

and paradoxes is given as background for Godel’s Theorem. This leads to mechanical reasoning

and computers, and the debate about whether Artificial Intelligence is possible. I close with an

explanation of the origins of the book-particularly the why and wherefore of the Dialogues.

Three-Part Invention. Bach wrote fifteen three-part inventions. In this three-part Dialogue, the

Tortoise and Achilles-the main fictional protagonists in the Dialogues-are "invented" by Zeno (as

in fact they were, to illustrate Zeno's paradoxes of motion). Very short, it simply gives the flavor

of the Dialogues to come.

Chapter I: The MU-puzzle. A simple formal system (the MIL'-system) is presented, and the reader

is urged to work out a puzzle to gain familiarity with formal systems in general. A number of

fundamental notions are introduced: string, theorem, axiom, rule of inference, derivation, formal

system, decision procedure, working inside/outside the system.

Two-Part Invention. Bach also wrote fifteen two-part inventions. This two-part Dialogue was written

not by me, but by Lewis Carroll in 1895. Carroll borrowed Achilles and the Tortoise from Zeno,

and I in turn borrowed them from Carroll. The topic is the relation between reasoning, reasoning

about reasoning, reasoning about reasoning about reasoning, and so on. It parallels, in a way,

Zeno's paradoxes about the impossibility of motion, seeming to show, by using infinite regress,

that reasoning is impossible. It is a beautiful paradox, and is referred to several times later in the

book.

Chapter II: Meaning and Form in Mathematics. A new formal system (the pq-system) is

presented, even simpler than the MlU-system of Chapter I. Apparently meaningless at first, its

symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they

appear in. This revelation is the first important insight into meaning: its deep connection to

isomorphism. Various issues related to meaning are then discussed, such as truth, proof, symbol

manipulation, and the elusive concept, "form".

Sonata for Unaccompanied Achilles. A Dialogue which imitates the Bach Sonatas for

unaccompanied violin. In particular, Achilles is the only speaker, since it is a transcript of one

end of a telephone call, at the far end of which is the Tortoise. Their conversation concerns the

concepts of "figure" and "ground" in various

Overview

VIII

contexts- e.g., Escher's art. The Dialogue itself forms an example of the distinction, since

Achilles' lines form a "figure", and the Tortoise's lines-implicit in Achilles' lines-form a "ground".

Chapter III: Figure and Ground. The distinction between figure and ground in art is compared to

the distinction between theorems and nontheorems in formal systems. The question "Does a

figure necessarily contain the same information as its ground%" leads to the distinction between

recursively enumerable sets and recursive sets.

Contracrostipunctus. This Dialogue is central to the book, for it contains a set of paraphrases of

Godel’s self-referential construction and of his Incompleteness Theorem. One of the paraphrases

of the Theorem says, "For each record player there is a record which it cannot play." The