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Home · Book Reports · 2019 · Godel, Escher, Bach - An Eternal Golden Braid

Published: January 24, 2019 (5 years 2 months ago.)

The book in...
One sentence:
A braiding of formal systems, biology, music, and artificial intelligence, among other things, that is presented in a laugh-out-loud next to intellectual way.

Five sentences:
The book builds in a very approachable manner, for even the most uninitiated reader, from basic formal systems to DNA and artificial intelligence. Each step is presented first with a humorous take in the form of a dialog between characters like Achilles and Tortoise and then the idea is elaborated on in a more formal manner with intellectual exercises that should engage the reader. Also the progression is made with a great many analogies where things like music, memory, DNA, and other seemingly unrelated topics are woven together to give a purchase point from many angles and thus be accessible to a great many people with a variety of backgrounds. I can say that I personally, with those 40 years of hindsight, don't give much credibility to some of the future (current) prospects of AI, but non-the-less find the book interesting and a worthwhile read for anyone interested in the topics. Overall the book is funny, informative, and a great introduction to some topics that 40+ years later we are still grappling with.

designates my notes. / designates important.


These thoughts were written over a year after I read the book and are thus not to be taken as the best representation of the work.

Before I get into the content of the book, I must say that the entire work is laced with excellent, laugh-out-loud, humor. For such a long work it is actually very easy to read and would suggest it to anyone interested in philosophy or the early days of artificial intelligence. That said, I don’t mean to imply that I agree with what is presented here, merely that it is certainly interesting.

Part 1

Right off the bat my alarm bells are already ringing when he mentions sexism and apologizes for using the word men. He goes on to talk about how men eat animals and we think they have ‘smaller’ souls than our, so we “extinguish the dim lights in the heads of these fractionally-souled beasts and to gobble down their once warm and wiggling, now chilled and stilled protoplasm with limitless gusto, and not to feel a trace of guilt while doing so.” It feels very “liberal” and goes on to address sexism in language like mankind. The author is a Stanford undergrad and did some short-lived graduate work at Berkeley.

This was a most interesting read. While I am skeptical of many of the arguments the author puts forth, I can not deny that reading this book was both intellectually stimulating and enjoyable. If nothing else, a look at the thoughts of an artificial intelligence researcher of the 1970’s was insightful given the prominence today of said field.

The book itself if constructed in a peculiar manner; each chapter is preceded by a dialog between a number of recurring characters, the primary ones being Achilles and the Tortoise, but also includes a few of their friends. These dialogs are, of course, inspired by Lewis Carroll’s ‘What the tortoise said to Achilles.’ While I was and am capable of delaying judgment based on this Carroll fandom, it can not be overstated how influential Carroll was to modern day oligarchs, not to mentioned his often unmentioned love of photographing children. If you ever wondered where his inspiration came for Alice, I’d suggest you do a little research yourself; it is more than shocking.

Back to the book. It begins by introducing the MU-puzzle, a light introduction to formal systems. I think this is a wonderful jumping off point as it pulls the reader in immediately involving them instead of (dis?) passionately preaching to them.

Next, another formal system the ‘p q -’ system is introduced. This is an extremely simple system that is isomorphic with addition. Again, I think the way that formal systems are introduced, gradually and approachable, is of great service to a lay reader.

Building on these formal systems the book then examines figure and ground, terms familiar to any artist readers. Here the work of M.C. Escher is used to show the difference between positive and negative space (figure and ground). A few images of alphabets where the letters are both figure and ground are used as examples. One is the Figure-Figure drawing where the figure and the ground are the same.

With this understanding of how the figure and the ground can define one another, the author turns back to formal systems. Can the true theorems and false theorems relate to one another in a figure/ground relationship? The ‘p q -’ system is modified to map onto multiplication and then used to build theorems for composites. The question then becomes: can we build on the negative space of composite theorem, to reveal the primes? The answer in this case is yes, but not all negative space gives rise to another system. What is not in a system (ground) will not always be the theorems of another system (figure).

This part of the book finishes up with a dialog about breaking phonographs and hidden messages in music/poetry before moving on to exploring the consistency and completeness of formal systems. It focuses on geometry: euclidean, non-euclidean, and absolute (4 postulate) geometry.

Next, in a section titled little harmonic labyrinth, the tortoise and Achilles go on nested adventures. Nested as in recursive. We are given some explanation as to what a base case is (“bottoming out”) and some well known examples (Fibonacci). Then the author looks at what a recursive program would look like if there was no base case.

Needless to say, there can be a trio of procedures which call one another, cyclically-and so on. There can be a whole family of RTN’s [recursive transitive network] which are all tangled up, calling each other and themselves like crazy. A program which has such a structure in which there is no single “highest level”, or “monitor”, is called a heterarchy (as distinguished from a hierarchy). The term is due, I believe, to Warren McCulloch, one of the first cyberneticists, and a reverent student of brains and minds.

You might well wonder whether such an intricate structure would ever show up in an experiment. Frankly, I would be the most surprised person in the world if Gplot came out of any experiment. The physicality of Gplot lies in the fact that it points the way to the proper mathematical treatment of less idealized problems of this SOI·t. In other words, Gplot is purely a contribution to theoretical physics, not a hint to experimentalists as to what to expect to see! An agnostic friend of mine once was so struck by Gplot’s infinitely many infinities that he called it “a picture of God”, which I don’t think is blasphemous at all.

Finally the topic of recursion is closed off by looking at recursive enumeration, a process where new things can emerge from old things that follow a fixed set of rules. This reminds me of Conway’s game of life.

In Canon by Intervallic Augmentation looks at messages and the levels of information contained within them. The example includes a jukebox with one record and many phonographs and discusses the frame, outer, and inner messages.

In Chromatic Fantasy, And Feud Tortoise contradicts himself and Achilles points it out, but Tortoise turns it back on him. Achilles feels bad. This leads into propositional calculus (logic) and its rules like and, or, not, and implies.

Within a system like this, if you can imagine a contradiction, the rules are in question (prudence). So (imprudence) don’t imaging contradictions.

As I take these notes, I think this is laying the foundation for being unable to prove a system from within itself.

The Crab Canon is basically same forward and backwards, absolutely delightful!

The next system explored is Typographical Number Theory, which has Propositional Calculus embedded in it.

Any system that is strong enough to prove TNT’s consistency is at least as strong as TNT itself. And so circularity is inevitable.

In A MU Offering zen koans and molecular biology are compared, which leads to discussion on Godel numbering, converting between systems, and isomorphic systems.

Part 2

Part 2 starts off with a more grounded look into things like programming languages, particularly machine and assembly language. There is a look at the hierarchy of computers with lots of comparisons to computers and brains hinting at AI. AI here is spoken as a computer that can program itself or understand the same data in two different contexts.

Examples of hierarchy, that can be thought of as levels which are sealed off from one another, are presented with simplicities in mind are given as sub-atomic, atomic, molecular, cellular, etc. This part reminds me of Koestler’s discussion on parts and whole in (I think) Act of Creation, but may have been The Ghost in the Machine.

I am also again reminded of Conway’s game of life during a discussion on epiphenomena, emergent phenomena from a system that is not specifically programmed into it.

In Ant Fugue we look at signals in an ant colony and ask if this is a kind of intelligence. Could you map any intelligence onto these signals, but not the individual ant? Is there a caste distribution of ants like a ‘brain state’? Are the building blocks are not necessary to describe the higher level?

Is it reasonable to compare ants and neurons? There are similar things happening already when we look at the isomorphic mapping from what you see into 2D electrical signals in the retina to 3D electrical signals farther back in the brain (lateral geniculate).

This gives rise to things like symbols and memory. We don’t remember things, but instead somehow electrical signals (symbols) can allow recall (memory). What about procedural vs. explicit knowledge. We know a mountain is bigger than a car not by rote memorization, but by deduction, procedurally.

Following the symbols and memory reasoning, there is an interesting comparing a trip via roads through cities to a thought and its neuropathways. The thoughts move along the roadways and the cities represent symbols. Each individual’s map will be different because we all know different things, but the bigger parts, like New York City or Chicago, will be omnipresent. This will allow us to communicate using those commonalities as landmarks.

This makes me think about how people so often “talk past” one another. If you don’t have these commonalities (in language) you won’t ever get anywhere. A simple example can be seen with the way people talk about (economic) inflation. The textbook definition would be an expanding monetary supply. This in turn will very likely lead to price increases to accommodate the large money supply. The problem is when people use the word inflation to mean rising prices. There are other reasons why prices might increase from forces that have nothing to do with actual inflation. This leads to an inability to communicate about inflation and possible price increases.

It doesn’t seem to be purposeful, but a few chapters later there is a discussion on supernatural numbers and the consequences of things like rational, irrational, negative, and imaginary numbers or the extending geometry in a non-Euclidean manner. This gives rise to the author talking about “abstract space” and how it is used in physics as if they were as real as actual reality.

This is interesting because the book was written in 1979 and it “predicted” the rise of theoretical physics. While theoretical physics isn’t brand new, I think it is safe to say there has been an explosion in the acceptance over the last 2 decades or so. I point to this with a twist on an old saying: “the math is not the territory.” I also think that what we are seeing now with people “living” online and likely soon in augmented/virtual reality, is a natural consequence of Kantian philosophy that will ultimately lead to the atomization of society to the sub-individual level. By sub-individual I mean that when people can’t even know/decide if they are boys or girls, or even humans, the atomization will be worse that if you were (or felt) merely isolated from the group, but you are effectively isolated from yourself.

Next we move back to comparing brains to programs. If we extend Godel’s theorem from TNT to TNT+G (etc), are humans as susceptible to Godel’s “trick” as any machine?

Lucas argues humans can always transcend the formal system that a computer must be bound by, but the author argues that eventually humans won’t be able to do so, at which point the machine is as powerful as a human. Both are not ‘all knowing’, but neither can know more. The author makes an analogy: some days you might be able to lift 250 pounds, others days not, but you will never lift 250 tons.

Basically the argument is that humans are simply very complex formal systems that, like their mechanical counterparts, won’t be able to step outside of themselves.

As much as humans try, we are still, for example, bound by the laws of physics, a higher (or lower) system that rules us.

This comparison continues into the molecular biology of cells, DNA, xRNA, and enzymes to computer hard/software systems like processors, programs, and interpreters.

The author claims that the substrate and the higher level symbols of the world, the mind, and the computer can be seen as partially isomorphic. This feels very close to: “as above, so below.”

This all leads up to a retro- and prospectus on artificial intelligence. How, as we build more complex and more feedback into our AI programs, they will appear to be a giant knot that we might not even understand. Could this unknowable knot give rise to “strange loops” that self-reference and might one day be called sentience? For my part, I am highly skeptical that anything like the Hollywood or Silicon Valley version of AI will ever exist. That said, I think the ability for programs that are extremely good at classification will ultimately be able to replace a lot of work that humans never imagined would be done by software.

The book ends on another humorous take on self-referentialism:

Author: I guess it all depends on how you look at it. But major or minor, I’d be most pleased to tell you how I braid the three together, Achilles. Of course, this project is not the kind of thing that one does in just one sitting-it might take a couple of dozen sessions. I’d begin by telling you the story of the Musical Offering, stressing the Endlessly Rising Canon, and-

Achilles: Oh, wonderful! I was listening with fascination to you and Mr. Crab talk about the Musical Offering and its story. From the way you two talk about it, I get the impression that the Musical Offering contains a host of formal structural tricks.

Author: After describing the Endlessly Rising Canon, I’d go on to describe formal systems and recursion, getting in some comments about figures and grounds, too. Then we’d come to self-reference and self- replication, and wind up with a discussion of hierarchical systems and the Crab’s Theme.

Further Reading

Exceptional Excepts

Typographical rules for manipulating numerals are actually arithmetical rules for operating on numbers.

This simple observation is at the heart of Godel’s method, and it will have an absolutely shattering effect. It tells us that once we have a Godel- numbering for any formal system, we can straightaway form a set of arithmetical rules which complete the Godel isomorphism. The upshot is thatwe can transfer the study of any formal system-in fact the study of all formal systems-into number theory.

By looking at things from the vast perspective of evolution, you can drain the whole colony of meaning and purpose.

The significance of the notion is shown by the following key fact: If you have a sufficiently powerful formalization of number theory, then Godel’s method is applicable, and consequently your system is incomplete. If, on the other hand, your system is not sufficiently powerful (i.e., not all primitive recursive truths are theorems), then your system is, precisely by virtue of that lack, incomplete. Here we have a reformulation of “Ganto’s Ax” in metamathematics: whatever the system does, Godel’s Ax will chop its head om Notice also how this completely parallels the high-fidelity-versus-low- fidelity battle in the Contracrostipunctus.

You fit your mathematics to the world, and not the other way around.

As the ordinals [names of infinities] get bigger and bigger, there are irregularities, and irregularities in the irregularities, and irregularities in the irregularities in the ir- regularities, etc. No single scheme, no matter how complex, can name all the ordinals. And from this, it follows that no algorithmic method can tell how to apply the method of Godel to all possible kinds of formal systems. And unless one is rather mystically inclined, therefore one must conclude that any human being simply will reach the limits of his own ability to Godelize at some point. From there on out, formal systems of that complexity, though admittedly incomplete for the Godel reason, will have as much power as that human being.

But-one should not consider TNT defective for that reason. If there is a defect anywhere, it is not in TNT, but in our expectations of what it should be able to do. Furthermore, it is helpful to realize that we are equally vulnerable to the word trick which Godel transplanted into mathematical formalisms: the Epimenides paradox. This was quite cleverly pointed out by C. H. Whitely, when he proposed the sentence “Lucas cannot consistently assert this sentence.” If you think about it, you will see that (1) it is true, and yet (2) Lucas cannot consistently assert it. So Lucas is also “incomplete” with respect to truths about the world. The way in which he mirrors the world in his brain structures prevents him from simultaneously being “consistent” and asserting that true sentence.But Lucas is no more vulnerable than any of us. He is just on a par with a sophisticated formal system.

Well, this is the kind of “heads-in-the-sand” argument which you have to be willing to stomach if you are bent on seeing men and women running ahead of computers in these intellectual battles.

one of the main theses of this book: that every aspect of thinking can be viewed as a high-level description of a system which, on a low level, is governed by simple, even formal, rules.

According to my hypothesis, then, imagery and analogical thought processes intrinsically require several layers of substrate and are therefore intrinsically non-skimmable. I believe furthermore that it is precisely at this same point that creativity starts to emerge

Table of Contents

· Preface (1999)

page x:
page xi:
page xii:
page xiii:
page xv:
page xx:
page xii:

· Introduction:A Musico-Logical Offering

page 17:

-All consistent axiomatic formulations of number theory include undecidable propositions.

Three-Part Invention

page 29:

· 01: The MU-puzzle

page 34:
start with 'MI'; try to reach 'MU'

RULE I: If you possess a string whose last letter is I, you can add on a U at
the end.

RULE II: Suppose you have Mx. Then you may add Mxx to your collection.

What I mean by this is shown below, in a few examples.
From MIU, you may get MIUIU.
From MUM, you may get MUMUM.
From MU, you may get MUU.

RULE III: If III occurs in one of the strings in your collection, you may make
a new string with U in place of III.

From UMIIIMU, you could make UMUMU.
From MIIII, you could make MIU (also MUI).
From IIMII, you can't get anywhere using this rule.
(The three I's have to be consecutive.)
From MIII, make MU.

RULE IV: If UU occurs inside one of your strings, you can drop it.

From UUU, get U.

No solution
page 37:
page 41:

Two-Part Invention

page 43:

· 02: Meaning and Form in Mathematics

page 46:
page 49:
page 53:

Sonata for Unaccompanied Achilles

· 03: Figure and Ground


page 81:
page 85:

· 04: Consistency, Completeness, and Geometry

page 98:
page 101:

Little Harmonic Labyrinth

· 05: Recursive Structures and Processes

page 134:
page 141:
page 142:
page 144:
page 152:

Canon by Intervallic Augmentation

· 06: The Location of Meaning

page 158:
page 172:
page 174:
page 175:

Chromatic Fantasy, And Feud

· 07: The Propositional Calculus

page 186:
page 192:
page 196:

Crab Canon

· 08: Typographical Number Theory

page 222:
page 230:

A MU Offering

· 09: Mumon and Godel

page 264:

Part 2

· Prelude

· 10: Levels of Description, and Computer Systems

page 301:
page 303:
page 308:

Ant Fugue

page 321:

page 325:
Anteater: But you see, despite appearances, the ants are not the most important
feature. Admittedly, were it not for them, the colony wouldn't exist; but
something equivalent-a brain-can exist, ant-free.  So, at least from a
high-level point of view, the ants are dispensable.

Achilles: I'm sure no ant would embrace your theory with eagerness.

Anteater: Well, I never met an ant with a high-level point of view.

Crab: What a counterintuitive picture you paint, Dr. Anteater. It seems that,
if what you say is true, in order to grasp the whole structure, you have to
describe it omitting any mention of its fundamental building blocks.

Anteater: Perhaps I can make it a little clearer by an analogy. Imagine you
have before you a Charles Dick.ens novel.

Achilles: The Pickwick Papers-will that do?

Anteater: Excellently! And now imagine trying the following game: you must find
a way of mapping letters onto ideas, so that the entire Pickwick Papers makes
sense when you read it letter by letter.

Achilles: Hmm ... You mean that every time I hit a word such as "the", I have
to think of three definite concepts, one after another, with no room for

Anteater: Exactly. They are the 't'-concept, the 'h' -concept, and the 'e'
-concept-and every time, those concepts are as they were the pre- ceding time.

Achilles: Well, it sounds like that would turn the experience of "reading" The
Pickwick Papers into an indescribably boring nightmare. It would be an exercise
in meaninglessness, no matter what concept I associated with each letter.

Anteater: Exactly. There is no natural mapping from the individual letters into
the real world. The natural mapping occurs on a higher level- between words,
and parts of the real world. If you wanted to describe the book, therefore, you
would make no mention of the letter level.

Achilles: Of course not! I'd describe the plot and the characters, and so

Anteater: So there you are. You would omit all mention of the building blocks,
even though the book exists thanks to them. They are the medium, but not the
page 334:

· 11: Brains and Thoughts

page 341:
page 344:
page 348:
page 357:
page 359:

English French German Suite

· 12: Minds and Thoughts

Aria with Diverse Variations

· 13: BlooP and FlooP and Gloop

page 406:

Air on G’s String

page 456:

Birthday Cantatatata…

· 15: Jumping Out of the System

page 475:

page 476:

page 477:
page 478:

Edifying Thoughts of a Tobacco Smoker

page 519:
page 532:
page 534:

· 16: Self-Rep and Self-Rep

page 547:

The Magnificrab, Indeed

· 17: Church, Turing, Tarski, and Others

page 558:

page 569:

page 570:
page 572:

SHRDLU, Toy of Man’s Designing

· 18: Artificial Intelligence: Retrospects

page 629:


· 19: Artificial Intelligence: Prospects

Sloth Canon

· 20: Strange Loops, Or Tangled Hierarchies

page 708:

Six-Part Ricercar

page 741: