God Is a Strange Loop, Part 2

Douglas Hofstadter's I Am a Strange Loop

In God Is a Strange Loop, Part 1, I discussed my conjecture as to how the ideas worked out by Douglas Hofstadter in his recent book I Am a Strange Loop, notions about the genesis of the "I" in each human brain, might be extrapolated into an understanding of God as a sort of "world mind" or "world soul."

The human "I" is an emergent phenomenon, Hofstadter shows. Like all the "symbols" which the brain gives rise to, it springs forth from the workings of the lower-level components of the brain, the neurons and the signals they ceaselessly exchange. The "I" exists at the pinnacle of the human brain's rich symbol system. It potentiates self-awareness. It makes possible our conscious, subjective experience. It is the self. It is the soul.

Stripped to its barest essentials, the "I" represents the brain's ability to think about — to make and evaluate assertions about — its own thoughts. For example, if I think to myself, "I never think about pink elephants," the presence in my symbol system of an "I" symbol makes that thought possible.

But what makes the "I" symbol possible? After all, it is not intuitively obvious that a computer — a mere machine, even if it were to be programmed with all the artificial intelligence in the world — would, or could, generate an "I." Then again, it is not intuitively clear that it couldn't.

For what makes an "I" possible, Hofstadter says, has to do with what Austrian mathematician Kurt Gödel proved in 1931 with respect to any and all formal systems of logic. If the systems are at least powerful enough to derive the mathematical theory of numbers, there are, quite shockingly, some truths about numbers — and about themselves as systems — that they simply cannot derive.


To prove this, Gödel showed that all formal, rigorous, strictly mechanical systems of logical derivation — and by extension, all computers, even though computers hadn't yet been invented — can be "arithmetized." They can have their internal statements — their formulas, their candidate theorems — turned into numbers. The numbers are intrinsically subject to the laws of computation, which are actually theorems of number theory.

Gödel accomplished his feat by "arithmetizing" a formal system: one that had been designed to derive these very laws of computation as theorems of number theory. Specifically, it was the formal system developed by Bertrand Russell and Alfred North Whitehead in their series of books called Principia Mathematica. For example — and thankfully — Russell and Whitehead's PM system was capable of proving such theorems as (in symbols we recognize, not those used in PM) "2+2=4".

Gödel cleverly did all his "arithmetizing" in such a way as to be able to show that one possible candidate theorem in PM is the formula which can be loosely rendered as, "This very formula is unprovable in the PM system"!


The well-formed PM formula for which that sentence is just one possible English translation could just as well be translated more tersely as "I am unprovable" (with "in the PM system" being tacitly understood).

Thus, any general formal system of theorem derivation, as long at it is no less powerful than the number-theoretical PM system, is implicitly mirrored by an equally mechanical system of numerical computation. The fact that computations precisely mirror the formal system which derives the laws for doing those computations is key.

Moreover, Gödel showed that, in the PM-mirroring set of computations he devised, the so-called "Gödel numbers" that serve to "arithmetize" the well-formed formulas of the theorem-deriving system can be calculated independently of that system. You can take any counting number from the set 1, 2, 3, ... and determine whether the string of PM symbols it codes for is well-formed or not, according to the rules of PM.

For example, 72900 can be factored into 2² times 3⁶ times 5². The base numbers 2, 3, and 5 are successive prime numbers that can't be further factored. The exponents 2, 6, and 2 stand for, respectively, "0", "=", and "0" again. Thus, 72900 can be converted into "0=0", which is a well-formed formula in PM.

But 576 is not the Gödel number of a well-formed formula, or "wff." Its "prime factorization" reveals it to be equal to 2⁶ times 3². Hence, since 2 stands for "0" and 3 stands for "=", the exponents taken in sequence — 6, then 2 — translate the number 576 into the PM formula "0=". "0=" is a symbol string that is, shall we say, "ungrammatical" in PM.


To get back to Gödel's main goal: it was to prove that any such system as PM will have as one of its well-formed formulas the assertion "I (i.e., this formula) am unprovable."

Hence, any such system will have, in effect, an "I". If it didn't, all formulas of that general form would be sheer nonsense — which they aren't. They are as well-formed and as "grammatical" as "0=0".

The human brain, moreover, is presumably a general, powerful, PM-equivalent computational device. By "PM-equivalent" I mean that at some level of its inner operation the brain does exactly what PM or any other formal system does. It starts with some basic, unassailable rules of theorem derivation, a set of basic symbols, and some basic axioms which any fool can see are true, and proceeds to work out a huge set of further truths, beyond the axioms.

Accordingly, the operation of the brain itself mirrors PM, and every system like it. Furthermore, it cannot fail to do so in a Gödelian way, such that the brain's operation is also mirrored (however abstrusely) by numbers and the way numbers can be calculated, based on other numbers.

Like whatever formal system of logical derivation the brain may mirror in its Gödelian way, it is inherently truth-seeking. It wants to extend what it already knows — its axioms, plus those theorems it has already proven, so to speak — to derive in a strict, rule-bound way, more truths. It wants to know the truth, the whole truth, and nothing but the truth.

The whole truth, of course, includes what we know about ourselves: knowledge associated with our personal, individual "I" symbols. Given what Gödel proved about all formal systems and their arithmetical mirror images, it is not surprising that we can — indeed, we need to be able to — speak of ourselves in the first person, as an "I".


Gödel proved that any such system is "incomplete." To a mathematician, a computational system involving turning numbers into other numbers in lawful ways is incomplete if there is no way to tell for sure whether any given number actually belongs to a given well-defined set of numbers.

For example, take the set of prime numbers. A prime number is one that cannot be factored into smaller numbers which, when multiplied together, yield the original number. 11 is a prime number because the only multiplication of integers that yields it is 11 x 1. 12 is non-prime, or composite, because 4 x 3 = 12.

The set of prime numbers is infinite, it has been proved; there is no such thing as the highest prime number. Yet, because it is fairly easy to compute whether any candidate integer is prime, the set of prime numbers is considered complete.

In Gödel's proof of his so-called "Incompleteness Theorem," the well-defined set in question is not the set of primes, but instead the set of all Gödel numbers that represent theorems in PM. Gödel proved, crucially, that this particular set is incomplete.

In particular, the "I am unprovable" formula, expressed in terms of PM symbols, like all other formulas necessarily has a Gödel number: an inconveniently huge integer, unfortunately, which Gödel gave instructions for slimming down and plugging into the PM version of the "I am unprovable" formula, right in place of the "I". That hard-to-calculate number can be abbreviated g. g, in addition to forming a small part of the way this formula is expressed in PM, also is the Gödel number of the "I am unprovable" formula as a whole.

That, in a nutshell, is why g can have the pronoun "I" substituted for it in an English translation of the "I am unprovable" formula! The same formula could equally well be translated, "The formula whose Gödel number is g — which just so happens to be this very formula — is unprovable." Use of "I" makes the formula terser and a lot easier to state.


The fact that g is both a term in the formula and the Gödel number of the formula is what Hofstadter means by a "strange loop." Formal systems with no more than a modicum of theorem-generating power — just enough to derive the laws of number theory, in fact, and no more — all have strange loops. There is no way for such a system to be designed to avoid this kind of strange-loopiness, Gödel proved. There is no way for it to avoid having the ability to formulate first-person truths, truths which necessarily begin (in effect) with the pronoun "I".

Of course, it can likewise formulate first-person falsehoods. But the ability to formulate a falsehood — first-person or otherwise — is not the same as the ability to derive or prove that falsehood, as if it were somehow the truth.

The "I am unprovable" formula — Hofstadter dubs it "KG" in honor of the initials of its discoverer, Kurt Gödel — might conceivably be false. But if KG were false, then it would be provable, since "provable" is the logical opposite of "unprovable." But KG, a statement that says "I am unprovable," simply cannot be provable ... or there would be a contradiction lurking within the bounds of the PM system.

Hofstadter shows that this situation quite simply isn't allowable, for "if any false statement, no matter how obscure or recondite it was, were possible in PM, then every conceivable arithmetical statement, whether true or false, would become provable, and the whole grand edifice would come tumbling down in a pitiful shambles. In short, the provability of even one falsity would mean that PM had nothing to do with arithmetical truth at all" (pp. 163-164).

That would be unthinkable. The only other choice Gödel left us would seem, accordingly, to be inescapable. If they are not to be deemed inconsistent, then PM and all other formal systems like it with an "I" or strange loop lurking inside them have to be logically "incomplete."

Again, "incompleteness" means there are truths about the systems themselves that — although the systems can produce well-formed, "grammatical" formulas that express these truths — cannot be proven. One such well-formed formula is that whose English rendition is "I am unprovable." And there are an infinite number of other ones as well.


My "God Is a Strange Loop" conjecture is analogous to Hofstadter's belief that an "I" symbol is an emergent property of the human brain. Because the brain pretty much has to be is a formal, mechanical system of truth derivation with a numerical, computational Gödelian mirror image, it pretty much has to generate an "I".

My GISL conjecture is that the world as a whole is a formal, mechanical system of truth derivation with a numerical, computational Gödelian mirror image. Consequently, it too inescapably possesses a strange loop, an "I". The "I" of the world system is God.

God is the (absolutely necessary) emergent property of the world system who can meaningfully say, per the Old Testament, "I am that I am" (Exodus 3:14). By that odd construction, I think, God is describing himself as the quintessence of "I"-ness. God is the "I" greater than which no other "I" could conceivably be. (Perhaps the burning bush is analogous to the fact that trying to understand the logical "strange loop" at the heart of each and every "I"-ness can drive one totally bonkers!)


It is interesting that Hofstadter's argument about the relationship of every "I" to Gödelian strange-loopiness requires, at bottom, what the author calls "one article of faith" (p. 163). Namely, the requisite article of faith is the one referred to earlier in this post: the belief that the formal, mechanical, computational system from which the "I" emerges cannot contain an inconsistency.

All such systems have to be either inconsistent or incomplete, Gödel proved. Given the choice, the latter option simply has to be ruled out on its face. Otherwise, the system becomes incoherent and incapable of distinguishing truth from falsity.

It is not hard to find the same basic assumption about God carried explicitly or implicitly in the various theologies, be they liberal or conservative, of the Judeo-Christian faith communities. It just makes no sense whatsoever to talk of a God who makes no sense.

It makes no sense to think of God as the creator of the world, and of the cosmic order in the universe, if God were somehow even capable of making no sense. Coherence — a starting point of truth that forms the basis for any hope we humans may have for seeking and finding the whole truth — is simply taken as an article of faith in the Judeo-Christian worldview.

Accordingly, I assume it must also be taken as an article of faith in any theology deriving from my "God Is a Strange Loop" conjecture. Embracing such a commitment means that it may be possible to harmonize my "GISL theology" (as soon as I can work out what it is in full detail) with standard Judeo-Christian beliefs. But more on that in my next post ...