The Science of Comprehension

How We Understand Abstract Concepts

Part IV

Symbol Interpreters,
Gödel's Incompleteness Theorem,
and Consciousness

The concept "Every CD needs a CD player" leads us to Gödel's Incompleteness Theorem, which in turn leads us to a possible explanation of what human consciousness is.

Gödel's Incompleteness Theorem is basically about a symbolic entity that refers to itself. And the symbols spell out, "I can not be proved." In Gödel's case the symbols are mathematical statements, and the Symbol Interpreter is the formal system of mathematics itself which processes the symbols. By demonstrating that it was possible to come up with a mathematical statement that essentially says, "I can not be proved," Gödel showed that in every well defined formal mathematical system there are statements that while true, can never be proved.¹

It's like having a CD, and a CD player, and the CD has written on it, "I can not be played on that CD player." And when you try to play the CD on the CD player, the CD player breaks.

Notice the key to breaking the CD player is to come up with a CD which talks about the system itself—the system of CD and CD player.

There is a superb book by Douglas Hofstadter, Gödel, Escher, Bach, (or "GEB")² which conceptually examines Gödel's Incompleteness Theorem by comparing the concept to drawings by Escher and music by Bach. Drawings by Escher often have that sense that the drawing is in a way referring to itself. Music by Bach also often has that self referential quality. Plus the concept of "an entity that is aware of itself" is basically what human consciousness is, so this is a very interesting book, really quite superb. There's no heavy duty math, it's all just conceptual ideas explained very well in numerous different ways. Suitable for anyone who has a keen interest in these concepts.

Hofstadter writes in the preface to the 20th-anniversary edition:

"Gödel showed that any formal system designed to spew forth truths about "mere" numbers would also wind up spewing forth truths — inadvertently but inexorably — about it's own properties, and would thereby become "self-aware", in a manner of speaking, and that such twisting-back, such looping-around, such self-enfolding, is an inevitable by-product of the system's vast power.

"The Gödelian strange loop that arises in formal systems in mathematics is a loop that allows such a system 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.

"What is so weird in this is that the formal system where these skeletal "selves" come to exist are built out of nothing but meaningless symbols. The self, such as it is, arises solely because of a special type of swirly, tangled pattern among the meaningless symbols. It's not the stuff out of which brains are made, but the patterns that can come to exist inside the stuff of a brain.

"GEB is in essence a long proposal of strange loops as a metaphor for how selfhood originates, a metaphor by which to begin to grab a hold of just what it is that makes an "I" seem, at one and the same time, so terribly real and tangible to its own possessor, and yet also so vague, so impenetrable, so deeply elusive."³

— Douglas Hofstadter

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Endnotes:

[1] For awhile it was speculated that Fermat's Last Theorem might be one of those true but unprovable equations, since no one had managed to prove or disprove it in over 300 years. Quite an achievement when Andrew Wiles finally found a way within our formal system of mathematics to reach Fermat's Last Theorem and prove it true. The 200 page proof was so complicated it took six teams of mathematicians, each working on a different part of the proof, half a year to verify that the entire proof was indeed correct. (Fermat's Enigma: The epic quest to solve the world's greatest mathematical problem, by Simon Singh. 1997)

[2] Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid. A metaphorical fugue of minds and machines in the spirit of Lewis Carroll. (The subtitle isn't very helpful in describing what the book is about. It's quite an interesting read.)

[3] From the preface to the 20th anniversary Edition : With a new preface by the author [Basic Books; 20 Anv edition (February 4, 1999). ISBN-13: 978-0465026562.]