God Is a Strange Loop, Part 4

Douglas Hofstadter's I Am a Strange Loop

In God Is a Strange Loop, Part 3, I continued showing the ideas worked out by Douglas Hofstadter in his recent book I Am a Strange Loop might play into a belief in God. It and its two predecessors, Part 2 and Part 1, dealt with "strange loops" of systems of mathematical logic wherein the systems unexpectedly turn right around and talk about themselves as axiomatic systems, while at the same time continuing to talk about whatever it was that they were intended to talk about in the first place.

An axiomatic system called Principia Mathematica, or PM, is set up to derive the laws of numerical computation, based on some simple axioms and rules of inference, as well-formed formulas consisting of strings of arcane symbols. If a well-formed formula is derivable — if it has a proof — it is called a theorem of PM.

PM has may theorems. One of them is (when translated into ordinary algebraic notation) "2+2=4". But "2+2=5" is not a theorem. While "2+2=4" is true, "2+2=5" is false. It cannot be derived, or proved.

Another theorem of PM — derivable, provable, hence true — is "There are infinitely many prime numbers." In fact, every true statement about numbers, or so it might be hoped, is mirrored by a theorem of PM.

But, no. As Austrian mathematician-logician Kurt Gödel showed, the following truth has a well-formed formula in PM which cannot be proven:

The formula that happens to have the code number g is not provable via the rules of Principia Mathematica.

I explained in my earlier posts what a "code number" —a.k.a. "Gödel number" — is. Suffice it to say that every formula of PM can be "arithmetized" to yield a single number which stands for the formula itself (!).

The above formula is stated in English translation, of course; inside PM, it appears in the form of a symbol string that is pretty much incomprehensible to the average eye. But never mind. We can still refer to the formula quite easily, amongst ourselves, by assigning it a name: KG, in honor of Kurt Gödel. Then we can ask, "What formula of PM happens to have g as its code number? And, for that matter, what number is g, and how is it computed?"

To answer the second question first, g does not really need to be computed per se! Gödel gave some "assembly instructions" for it by means of which it can be referred to within KG ... and that's all that is truly required.

The answer to the first question is that the formula which happens to have g as its code number is KG itself!

All of which means that the following is a "second-level meaning" of the original formula:

I am unprovable.

Whichever level of meaning you care to focus on, the formula in question is in fact unprovable. Which means it's true even though it can't be proven ... since if it were false, there would be a germ of inconsistency within PM that would spread to infect the whole system, rendering it useless.

Extrapolating from the above, we can see that it is not possible for mechanical systems of truth derivation to be "complete," in the sense that all truths about themselves are derivable. Oh, there are degenerate cases wherein the mechanical systems of truth derivation are so limited in their powers that they cannot even prove "2+2=4". But any system that can generate what mathematicians call number theory, in all its glory, is necessarily incomplete in a Gödelian sense.


The obvious conclusion we may draw is that truth is larger than provability. There is indeed in this world what Hofstadter calls a "true/false dichotomy." Yet he shows that

... the boundary line is so peculiar and elusive that it is not characterizable in any mathematical fashion at all. (p. 172)

Which means much of what is true has only "downward causality." All truth simply cannot be produced in a strictly mechanical fashion from the bottom up.

The formula KG discussed above can only be produced by a clever mind such as that of Kurt Gödel, working from the outside in, or from the top down. It cannot be generated in the "ordinary" way — from the bottom up, from the inside out — by applying PM's rules of mechanical inference.

KG's truth, likewise, can be known only by one who stands outside PM and peers in. Once KG has been oh-so-cleverly constructed by a great mind looking at PM from the outside, its truth is in a sense bestowed on it by the very mind of its constructor. Its original constructor was Gödel, of course, but he showed the rest of us (today, with Hofstadter's able help) how to construct this crucial formula of PM in such a way that we feel compelled to bestow truth upon it as well.

The argument which serves as our justification for bestowing truth on the unprovable is, once understood, irresistible. It crucially depends on the second-level meaning of KG, "I am unprovable," and on our seeing that a formula with such a second-level meaning simply must be true, even if unprovable ... or the entire system turns incoherent.

By analogy, truth bestowal — what Hofstadter calls "downward causation" — applies to the world as a whole, I would say. What works with respect to an axiomatic system like PM works equally for the cosmos we live in. There are things that are true about the universe that do not derive mechanically from the low-level workings of its particles and force fields. Some of its truth is bestowed from above.

But not, I would say, by a God who exists wholly outside the universe. Rather, God is an emergent property of the universe. God is the "I" who emerges from that which he bestows truth upon. That is why I call my philosophy a "God Is a Strange Loop" theology!

More later ...