Recently I began to write a review of Hofstadter's Godel, Escher, Bach, until I realized that the book defied summary more than all the other books I had previously said "defied summary." Thus, I gave up on reviewing the book after not too long. I present my discarded review below just in case it motivates someone else to pick up this masterful tome and let it enrich their life.
Of Hofstadter's GEB, Eliezer once wrote:
This is simply the best and most beautiful book ever written by the human species...
I'm not alone in this opinion, by the way. For one thing, Gödel, Escher, Bach won a Pulitzer Prize. Or just pick a random scientist and ask ver what vis favorite book is, and 1 out of 5 will say: "Gödel, Escher, Bach". No other book even comes close.
It is saddening to contemplate that every day, 150,000 humans die without reading what is indisputably one of the greatest achievements of our species. Don't let it happen to you.
Sure, if you're just an average person, you might not understand everything in this book - but when you're done reading, you won't be an average person any more.
It's easy to see GEB's effect on Eliezer's writing: the "concrete, then abstract" pattern, the koans, the puzzles, the conversational coverage of technical concepts in math and computer science... it's all here in spades in GEB.
What GEB Is
In the preface to the 20th anniversary edition, Hofstadter clarifies what GEB is and is not. It is not about how reality is "a system of interconnected braids." It is not about how "math, art, and music are really all the same thing at their core." Instead, says Hofstadter:
GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter... GEB approaches [this question] by slowly building up an analogy that likens inanimate molecules to meaningless symbols, and further likens selves... to certain special swirly, twisty, vortex-like, and meaningful patterns that arise only in particular types of systems of meaningless symbols. It is these strange, twisty patterns that the book spends so much time on, because they are little known, little appreciated, counterintuitive, and quite filled with mystery [that] I call... "strange loops"...
...the Godelian strange loop that arises in formal systems in mathematics... is a loop that allows such systems to "perceive itself," to talk about itself, to become "self-aware," and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self.
...the shift of focus from material components [of the human mind] to abstract patterns allows the [surprising] leap from inanimate to animate, from nonsemantic to semantic, from meaningless to meaningful, to take place. But how does that happen? After all, not all jumps from matter to pattern give rise to consciousness or soul or self... What kind of pattern is it, then, that is the telltale mark of a self? GEB's answer is: a strange loop.
The irony is that the first strange loop ever found... was found in a system tailor-made to keep loopiness out... Bertrand Russell and Alfred North Whitehead's famous treatise Principia Mathematica...
...For the French, the enemy was Germany; for Russell, it was self-reference. Russell believed that for a mathematical system to be able to talk about itself in any way whatsoever was the kiss of death, for self-reference would... necessarily open the door to self-contradiction...
Kurt Godel realized that... self-reference not only had lurked from Day One in Principia Mathematica, but in fact plagued poor PM in a totally unremovable manner. Moreover, as Godel made brutally clear, this thorough riddling of the system by self-reference was not due to some weakness in PM, but quite to the contrary, it was due to its strength. Any similar system would have exactly the same "defect."
[Godel had discovered that] any formal system designed to spew forth truths about "mere" numbers would also wind up spewing forth truths... about its own properties, and would thereby become "self-aware," in a manner of speaking.
[But] strange loops are an abstract structure that crop up in various media and in varying degrees of richness.
A Musico-Logical Offering
Hofstadter opens with the story of J.S. Bach's Musical Offering for King Frederick, which contains a particular canon that sneakily shifts from one key to another before its apparent conclusion, and when this modulation is repeated 6 times, the piece ends up at the original key but one octave higher. This is our first example of a "Strange Loop":
The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some heirarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.)
Other examples occur in the drawings of M.C. Escher, for example this famous one.
The liar's paradox (e.g. "This statement is false") is a one-step Strange Loop. Related to this is a Strange Loop found in the proof for Godel's Incompleteness Theorem, which states, roughly:
All consistent axiomatic formulations of number theory include undecidable propositions.
Before Godel, Russell and Whitehead tried to banish Strange Loops from set theory in Principia Mathematica. But Godel's theorem showed
...not only that there were irreparable "holes" in the axiomatic system proposed by Russell and Whitehead, but more generally, that no axiomatic system whatsoever could produce all number-theoretical truths, unless it were an inconsistent system!
The goal of the book is to explain these Strange Loops in more detail, and how they may explain how animate beings arise from inanimate matter.
Meaning and Form in Mathematics
After a tutorial on formal systems, Hofstadter argues that
...symbols of a formal system, though initially without meaning, cannot avoid taking on "meaning" of sorts, at least if an isomorphism is found.
The vast majority of interpretations for a formal system are meaningless, but if an isomorphism can be found between the formal system and some piece of reality, that isomorphism provides the symbols their "meaning."
But you may discover multiple isomorphisms, and thus the symbols of a formal system may have multiple meanings. It makes no sense to ask, "But which one is the meaning of the string?":
An interpretation will be meaningful to the extent that it accurately reflects some isomorphism to the real world.