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But even if we do fail to understand ourselves, there need not be any Godelian |
"twist" behind it; it could be simply an accident of fate that our brains are too weak to |
understand themselves. Think of the lowly giraffe, for instance, whose brain is obviously |
far below the level required for self-understanding-yet it is remarkably similar to our own |
brain. In fact, the brains of giraffes, elephants, baboons-even the brains of tortoises or |
unknown beings who are far smarter than we are-probably all operate on basically the |
same set of principles. Giraffes may lie far below the threshold of intelligence necessary |
to understand how those principles fit together to produce the qualities of mind; humans |
may lie closer to that threshold perhaps just barely below it, perhaps even above it. The |
point is that there may be no fundamental (i.e., Godelian) reason why those qualities are |
incomprehensible; they may be completely clear to more intelligent beings. |
Undecidability Is Inseparable from a High-Level Viewpoint |
Barring this pessimistic notion of the accidental inexplicability of the brain, what insights |
might Godel’s proof offer us about explanations of our minds/brains? Godel’s proof |
offers the notion that a high-level view of a system may contain explanatory power which |
simply is absent on the lower levels. By this I mean the following. Suppose someone |
gave you G, Godel’s undecidable string, as a string of TNT. Also suppose you knew |
nothing of Godel-numbering. The question you are supposed to answer is: "Why isn't |
this string a theorem of TNT?" Now you are used to such questions; for instance, if you |
had been asked that question about SO=0, you would have a ready explanation: "Its |
negation, ~S0=0, is a theorem ." This, together with your knowledge that TNT is |
consistent, provides an explanation of why the given string is a nontheorem. This is what |
I call an explanation "on the TNT-level". Notice how different it is from the explanation |
of why MU is not a theorem of the MlU-system: the former comes from the M-mode, the |
latter only from the I-mode. |
Now what about G? The TNT-level explanation which worked for 50=0 does not |
work for G, because - G is not a theorem. The person who has no overview of TNT will |
be baffled as to why he can't make G according to the rules, because as an arithmetical |
proposition, it apparently has nothing wrong with it. In fact, when G is turned into a |
universally quantified string, every instance gotten from G by substituting numerals for |
the variables can be derived. The only way to explain G's nontheoremhood is to discover |
the notion of Godel-numbering and view TNT on an entirely different level. It is not that |
it is just difficult and complicated to write out the explanation on the TNT-level; it is |
impossible. Such an explanation simply does not exist. There is, on the high level, a kind |
of explanatory power which simply is lacking, in principle, on the TNT-level. G's |
nontheoremhood is, so to speak, an intrinsically high-level fact. It is my suspicion that |
this is the case for all undecidable propositions; that is to say: every undecidable |
proposition is actually a Godel sentence, asserting its own nontheoremhood in some |
system via some code. |
Consciousness as an Intrinsically High-Level Phenomenon |
Looked at this way, Godel’s proof suggests-though by no means does it prove!-that there |
could be some high-level way of viewing the mind/brain, involving concepts which do |
not appear on lower levels, and that this level might have explanatory power that does not |
exist-not even in principle-on lower levels. It would mean that some facts could be |
explained on the high level quite easily, but not on lower levels at all. No matter how |
long and cumbersome a low-level statement were made, it would not explain the |
phenomena in question. It is the analogue to the fact that, if you make derivation after |
derivation in TNT, no matter how long and cumbersome you make them, you will never |
come up with one for G-despite the fact that on a higher level, you can see that G is true. |
What might such high-level concepts be? It has been proposed for eons, by |
various holistically or "soulistically" inclined scientists and humanists, that consciousness |
is a phenomenon that escapes explanation in terms of brain-components; so here is a |
candidate, at least. There is also the ever-puzzling notion of free will. So perhaps these |
qualities could be "emergent" in the sense of requiring explanations which cannot be |
furnished by the physiology alone. But it is important to realize that if we are being |
guided by Godel’s proof in making such bold hypotheses, we must carry the |
analogy through thoroughly. In particular, it is vital to recall tnat is s nontheoremhood |
does have an explanation-it is not a total mystery! The explanation- hinges on |
understanding not just one level at a time, but the way in which one level mirrors its |
metalevel, and the consequences of this mirroring. If our analogy is to hold, then, |
"emergent" phenomena would become explicable in terms of a relationship between, |
different levels in mental systems., |
Strange Loops as the Crux of Consciousness |
My belief is that the explanations of "emergent" phenomena in our brains-for instance, |
ideas, hopes, images, analogies, and finally consciousness and free will-are based on a |
kind of Strange Loop, an interaction between levels in which the top level reaches back |
down towards the bottom level and influences it, while at the same time being itself |
determined by the bottom level. In other words, a self-reinforcing "resonance" between |
different levels-quite like the Henkin sentence which, by merely asserting its own |
provability, actually becomes provable. The self comes into being at the moment it has |
the power to reflect itself. |
This should not be taken as an antireductionist position. It just implies that a |
reductionistic explanation of a mind, in order to be comprehensible , must bring in "soft" |
concepts such as levels, mappings, and meanings. In principle, I have no doubt that a |
totally reductionistic but incomprehensible explanation of the brain exists; the problem is |
how to translate it into a language we ourselves can fathom. Surely we don't want a |
description in terms of positions and momenta of particles; we want a description which |
relates neural activity to "signals" (intermediate-level phenomena)-and which relates |
signals, in turn, to "symbols" and "subsystems", including the presumed-to-exist "self¬ |
symbol". This act of translation from low-level physical hardware to high-level |
psychological software is analogous to the translation of number-theoretical statements |
into metamathematical statements. Recall that the level-crossing which takes place at this |
exact translation point is what creates Godel's incompleteness and the self-proving |
character of Henkin's sentence. I postulate that a similar level-crossing is what creates our |
nearly unanalyzable feelings of self. |
In order to deal with the full richness of the brain/mind system, we will have to be |
able to slip between levels comfortably. Moreover, we will have to admit various types of |
"causality": ways in which an event at one level of description can "cause" events at other |
levels to happen. Sometimes event A will be said to "cause" event B simply for the |