Thursday, November 08, 2007

I Am a Strange Loop: Gödel's Loop

Continuing my review of I Am a Strange Loop, today I get to tackle metamathematics. Hofstadter tackles it too, and finds it rich in philosophic insight. Strangely rich, actually.

I suppose I ought to explain who Kurt Gödel is and why he is a hero of many, many nerds today (I am among those ranks). And that tale doesn't start with Gödel, so stay patient while I explain the background.

Human languages are ambiguous. In many cases, perhaps in most cases, the ambiguity is not sufficient to fundamentally confuse communication. When I say, "The tortfeasor caused the injuries of the plaintiff," that little word "caused" is ambiguous, even in context. If I brought about a dangerous situation that led to the plaintiff's injuring himself, in what sense did I cause those injuries? Yet the law recognizes that I am the cause of those injuries, whatever the chain of events and whatever the underlying physics. Meeting the legal definition of "cause," an ambiguous, fuzzy concept, does not require epistemological certainty. Of course, "cause" itself has many meanings that have little to do with the meaning in the phrase "cause and effect." One fights for a "cause," for example. Contextual clues and the fact that we can understand each other without comprehending things in exactly the same way make such ambiguities trivial.

In some cases, of course, absolute precision is vital. In physics, and even more so (infinitely more so, as it happens) in mathematics, ambiguity is lethal. 1 + 1 does not sort of equal 2; it equals 2 or it does not. But notice that even here, language must be used to describe mathematical principles. Ambiguity can creep into any field when that field is explained in the rough, broad-brush language humans use every day. To bring this to even more concreteness, take this statement: "one and one and one is three." This could mean at least two different things: "1 + 1 + 1 = 3" or "1 ^ 1 ^ 1 <-> 3" or a number of other things. Because some of these statements are true and some false, the original utterance is too ambiguous to be valuable as a mathematical or logical statement. It must be reduced to symbolic representation that is entirely unambiguous.

Bertrand Russell and Alfred North Whitehead attempted to reduce mathematical logic to pure symbolism in Principia Mathematica. By assigning specific, unambiguous meanings to certain symbols, they hoped that ambiguity and its incidents would be banished from logic. Of course, merely banishing ambiguity is not sufficient; Principia Mathematica, in order to serve any purpose, had still to be able to express logical and mathematical truths, or else is would be useless. The goal was being able to manipulate symbols in such a way that a true expression represented by one string of symbols could lead, by absolute laws of inference, to another string of symbols expressing a different truth. At one level, the operation was purely mechanical: one string becomes another string according to a law of string-building. At another level, one truth implies another truth according to a law of inference. By means of entirely unambiguous language, any truth that could be expressed in the symbolic language of Principia Mathematica could be derived according to the symbol-manipulating laws of that system; or so was the hope.

Gödel extinguished that hope. The details may be appropriate for a separate blog entry someday, but in short, he constructed a string of Principia Mathematica that said "I am not provable." If this statement were true, it would have no proof, and thus Principia Mathematica would be capable of expressing a truth with no proof, and fail in its goal. If it were false, then it would have a proof, but then a proof would exist for something not true, something that should not be provable. This was rather inconvenient.

"I am not provable" is the source of Hofstadter's supposed insight into consciousness. Gödel's sentence says something about itself; similarly, conscious things are capable of thinking about themselves. What a marvelous analogy! Hofstadter calls this self-reflection a "strange loop." Principia Mathematica has a strange loop formula that is capable of making logical claims about itself. Human beings are capable of thinking about the very thing that is thinking about thinking; our consciousness is strangely loopy.

Is it? Strings of Principia Mathematica have dual meanings - symbols mean essentially nothing when engaged in the process of pure symbol-manipulation, but they still can be seen to stand for expressions about mathematics and logic, and the rules for changing symbols into each other are exactly isomorphic to the rules for deriving one true statement from another. Thus a statement, like "I am not provable," can be seen as a string of symbols and as a statement about reasoning at the same time. But by whom? The statement itself is lifeless; it needs an external observer to connect the two meanings. And this need is present in all cases of symbolic representation. That a stop sign is a symbol for a state-sanctioned command to perform some activity with one's motor vehicle is not immediately evident from the properties of the sign itself. It's metal; it's red and white; it's meaningless. It takes something with consciousness that can recognize the isomorphism between color patterns on the sign and legal meaning for the symbols to mean anything, to be symbols at all.

Hofstadter disagrees; in fact, he explicitly claims what I am about to say next. Hofstadter thinks that certain symbols are necessarily symbols, that anything with sufficient consciousness to understand them will recognize that they have dual meanings. This is quite a leap. Some of us (ahem) believe that symbols, being essentially meaningless, have to have something added to them to possess meaning. "I am not provable" is a mess of pixels of varying shades unless you're conscious. That those pixels should have intrinsic meaning is a strange (not loopy) idea, something that I cannot admit.

Think carefully about how Gödel's sentence has two meanings. It has those two meanings not because of anything intrinsic to the symbol-string it's embodied in (and that would seem a very unlikely, almost philosophically realist thing to think), but because an external observer (with consciousness!) recognizes an isomorphism. If consciousness is a strange loop, where is that external observer? If the "I" is the strange loop, then what observes the "I" and recognizes that it exists and has certain thoughts?

Far from locating the "I" in a strange loop, Hofstadter has pushed the inquiry away into another realm. For any "strange loop" to be conscious, there must be an "I" that observes the loop, recognizes its different levels of meaning, and recognizes itself. The Gödelian analogy is wide of the mark because the sentence, while expressed as "I am not provable," cannot actually talk about itself. The "I" in that sentence is a semantic shorthand; it would be more appropriate for the sentence not to use it at all. The "I" inside us is not the "I" of that sentence.

So what is the "I"? Calling it a "strange loop" and comparing it to Gödel's sentence is worthless because the particular strange loop in that example is only strange because of a third person who is capable of recognizing the strangeness; and such a third person (a human self) is exactly what we want to reveal when we ask what the "I" is.

"I" just don't get it.


At 5:17 PM, November 08, 2007 , Blogger Nick Milne said...

today I get to tackle metamathematics.

You should have tackled mathametmetics instead. Inf. more interesting

At 5:32 PM, November 08, 2007 , Blogger Vernunft said...

the ball is loose he's at the 40, the 30, the i, the pi, touchup

At 3:48 AM, August 13, 2008 , Anonymous Anonymous said...

Why didn't you just say that you didn't understand the book in the first line? Instead you waste your readers' time by blathering on until you finally admit you have nothing cogent to say.

At 4:08 AM, August 13, 2008 , Blogger Vernunft said...

What a perfectly contentless comment! Thank you.


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