Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

person, depending on the circumstances. These various interpreters pull

out different meanings, establish different connections, and generally

evaluate all deep aspects differently. So it seems that this notion of beauty is

extremely hard to pin down. It is for this reason that I chose to link beauty,

in the Magnificrab, with truth, which we have seen is also one of the most

intangible notions in all of metamathematics.

The Neural Substrate of the Epimenides Paradox

I would like to conclude this Chapter with some ideas about that central

problem of truth, the Epimenides paradox. I think the Tarski reproduc-

tion of the Epimenides paradox inside TNT points the way to a deeper

understanding of the nature of the Epimenides paradox in English. What

Tarski found was that his version of the paradox has two distinct levels to it.

On one level, it is a sentence about itself which would be true if it were false,

and false if it were true. On the other level-which I like to call the

arithmetical substrate-it is a sentence about integers which is true if and only if

false.

Now for some reason this latter bothers people a lot more than the

former. Some people simply shrug off the former as "meaningless", be-

cause of its self-referentiality. But you can't shrug off paradoxical state-

ments about integers. Statements about integers simply cannot be both true

and false.

Now my feeling is that the Tarski transformation of the Epimenides

paradox teaches us to look for a substrate in the English-language version. In

the arithmetical version, the upper level of meaning is supported by the

lower arithmetical level. Perhaps analogously, the self-referential sentence

which we perceive ("This sentence is false") is only the top level of a

dual-level entity. What would be the lower level, then? Well, what is the

mechanism that language rides on? The brain. Therefore one ought to

look for a neural substrate to the Epimenides paradox-a lower level of

physical events which clash with each other. That is, two ~vents which by

their nature cannot occur simultaneously. If this physical substrate exists,

then the reason we cannot make heads or tails of the Epimenides sentence

is that our brains are trying to do an impossible task.

Now what would be the nature of the conflicting physical events?

Presumably when you hear the Epimenides sentence, your brain sets up

some "coding" of the sentence-an internal configuration of interacting

symbols. Then it tries to classify the sentence as "true" or "false". This

classifying act must involve an attempt to force several symbols to interact

in a particular way. (Presumably this happens when any sentence is proces-

sed.) Now if it happens that the act of classification would physically disrupt

the coding of the sentence-something which would ordinarily never

happen-then one is in trouble, for it is tantamount to trying to force a

record player to play its self-breaking record. We have described the

conflict in physical terms, but not in neural terms. If this analysis is right so

584 Church, Turing, Tarski, and Others