In 1980, one year after being published, “GEB” won the Pulitzer Prize for general nonfiction and found itself on The New York Times bestseller list for several weeks running. Already then it had gained the reputation of being a book that only “smart people” read which meant, naturally, everyone had to buy a copy. Unfortunately for them, I am convinced that there is absolutely no way the majority of GEB readers actually have the patience required to understand its mathematics, and consequenly most of its other content. So what is this book about? I would say it is a technically and aesthetically impressive attempt to explain the essence of consciousness.
The book is organized into twenty chapters, with each one preceded by a dialogue between “The Tortoise and Achilles”. Through charming wordplay and sophisticated analogies, these dialogues give an intuitive and tangible feel to the material that is examined by the author in the following chapter. Bach plays a large role in the content and the physical form of these dialogues. Many of his musical works were modulations of the same musical theme and serve to illustrate how many levels of information can permeate the same object. Escher paintings are even more explicit in their direct isomorphism to the concepts that Dr. Hofstadter discusses. For instance, in chapter XV the Dragon painting by MCE shows a powerful dragon trying to defeat his two-dimensionality by ripping a hole in his own picture and sticking his head through it. Unfortunately for the dragon, since he is illustrated in the form of a wood engraving he still ends up being two-dimensional despite his efforts. This serves elegantly to depict how a powerful formal system, no matter how hard it tries to defeat its ω-incompleteness by acquiring more axioms, will always contain a theorem which can not be proven within it.
What makes this book really unique is how it approaches the concept of sentience from so many angles. Because it goes through such motley topics ranging from Zen Buddhism to the foundations of arithmetic, it might be tempting to dismiss the content as pseudo-scientific speculation, but to do so would be folly. Neuroscience has interesting insights into the brain but we need to distance ourselves from the notion that intelligence is directly related to the way we as human-beings function and interpret symbols. The author begins by examining self-reflection, or self-referencing, which seems to be something that a conscious being should be able to do. From the perspective of propositional calculus, we quickly find limitations in how accurately a system can describe itself; an arithmetic law which Kurt Gödel describes in his incompleteness theorems. This raises the question of whether our own minds might be built on simple formal rules and are “incomplete” in a similar manner? After all, neurons are quite simple in how they interact with the outside world. On a given stimulus they either fire a signal or don't.
To many, the flexibility of the human mind seems irreconcilable with the notion that it resides on rigid hardware. Though, if our self-understanding can't ever be truly complete then perhaps it's not so strange to feel that way. To make a logical example, assume you propose to someone “you can not consistently assert this sentence”. Whether the person asserts it or denies it, they would inevitably admit that they are not able to assert every single truth about themselves. Now if we make this example more concrete, more complex emergent phenomena which come from lower hierarchies of your brain, such as an intimate feeling of choice or want, would be equally inexplicable from within your own consciousness (formal systems). This fits well with the observation that we simply can’t force ourselves to describe the neural or even symbolic functions from whence something like ambition emerges from; it’s just there. So if we as humans are not able to deductively infer where these things come from, would we hold an AI at fault because a programmer could backtrack through sophisticated code to find some deterministic function at its core? The AI itself might describe a very profound feeling of free will, even though it does not understand exactly where it stems from...
While the complexity of the sentient mind is astounding, GEB does an absolutely fantastic job of discussing it. It’s not often you find a book that is both such a great pleasure to read and leaves you feeling motivated to read more math books!
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