Gödel, Escher, Bach

Douglas R. Hofstadter's Gödel, Escher, Bach

If you read a book in which the clause "provability is a weaker notion than truth" appears, you might think the author was knocking science and the mathematical/logical disciplines that might be called its atheistic co-conspirators, while at the same time upholding God as the highest preeminent-if-unprovable notion in the whole body of truth.

Douglas R. Hofstadter does indeed assert that clause about provability on page 19 of his 1979 book Gödel, Escher, Bach, an Eternal Golden Braid: A Metaphorical Fugue on Minds and Machines in the Spirit of Lewis Carroll. What he means by it is the subject of the book, but it is not clear to me after having dipped into the first chapter or so whether Hofstadter thinks it has theological implications or not.

As for me, I do think certain theological implications are justified.

What led me to this book for a rereading — I read it once, or at least started it, many years ago — is my present interest in the music of Johann Sebastian Bach. Bach wrote canons and fugues that involve dizzying self-referentiality: deceptively simple musical passages echoing, mirroring, harmonizing with and reinforcing one another to make for an endless kaleidoscope of inner experience. At least one of Bach's compositions, the "Canon per Tonos" of his Musical Offering, modulates from key to higher key so repeatedly that eventually the original key is restored!

M.C. Escher's Drawing Hands (1948)

The artist M. C. Escher did for the visual what Bach did for the aural: he showed how self-referentiality works.

Escher's Relativity (1953)

He showed how topsy-turvy and paradoxical a worldview that doesn't exclude self-reference can be. In the image at right, there is no "up" — even though "up" seems to make sense with respect to individual details of the scene, such as each individual staircase.

Escher's Möbius Strip I (1961)

Escher depicted the finite-but-unbounded Möbius strip, an apt analogue for the tonality of Bach's canon circling back repeatedly upon itself.








In 1931, an Austrian mathematician, logician, and philosopher named Kurt Gödel demonstrated that such paradoxes are not just curiosities. They illustrate the proposition that "provability is a weaker notion than truth." This was a proposition he demonstrated with respect to all formal or "axiomatic" systems of thought.

Gödel proved that mathematical set theory, in which mathematicians and logicians had invested so much faith and hope for the perfection of the human logical capability, could not contain all true propositions about itself — not if it was to remain self-consistent, i.e., non-self-contradictory.

Likewise, mathematical number theory. And, by further extension, Gödel showed that no formal system could be both self-consistent (non-self-contradictory) and complete (containing all true propositions about itself).

Some true statements can never be proved, in other words, within the frame of reference that defines how such proofs are allowed to be constructed.

This dashed the hopes of philosopher-mathematicians like Bertrand Russell and Alfred North Whitehead, who had hoped in their Principia Mathematica to banish self-contradiction from logic by banning self-referential statements. Gödel's 1931 paper "revealed not only that there were irreparable 'holes' in the axiomatic system proposed by Russell and Whitehead, but more generally, that no axiomatic system could produce all number-theoretical truths, unless it were an inconsistent system!" (p. 24).

Russell and Whitehead had tried to exorcise self-referentiality from their system by imposing strict conceptual hierarchies. Their mathematical sets could contain only raw objects and (possibly) other sets whose position in the hierarchy was lower than the containing set. Set descriptors that were in any way self-referential, no matter how indirectly, were held to be meaningless. Only run-of-the-mill sets were meaningful. For example, "the set of all run-of-the-mill sets," however much our intuition says it ought to be admissible in set theory, was simply ruled out.

Loosely speaking, Russell and Whitehead stripped out of their formal system any feature that might produce confusion about which way is "up." Their hierarchy of set types was, in effect, a staircase that couldn't loop back on itself.

Such a strategy has the effect, when transferred into the realm of ordinary language, of ruling out constructions like:

The following sentence is false.
The preceding sentence is true.

In doing so, it fails to come to grips with the fact that each of these sentences is, by itself, fine.

The Principia system thereby recuses itself from being able to help our mental faculties deal with the paradoxes, linguistic and otherwise, that the mind is perfectly capable of entertaining. In being thus pigheaded about paradox, it becomes repellent to human intuition. As Hofstadter says sourly (p. 23), "When this effort [at eliminating self-contradiction] forces you into a stupendously ugly theory, you know something is wrong." The mind wants to learn from paradox, not define it away.

My thought on this is that what is wrong with the Principia system is the nullifying of the right side of the brain. That hemisphere of the cerebrum is the locus of intuitive understanding, while the left brain purveys the kind of logic Russell and Whitehead championed.

Gödel proved, basically, that the logical left side of the brain needs the intuitive right side — it cannot generate the entirety of truth on its own.

No mechanical system of logic can suffice to establish the truth of all we intuit. If we intuit that there is a soul within each of us that belongs to God, systems of thought that would have such statements invalidated as meaningless must give way.

That, I think, is a sweet way of interpreting what Gödel showed the world.