God Is a Strange Loop, Part 3

Douglas Hofstadter's I Am a Strange Loop

In God Is a Strange Loop, Part 2, I enlarged upon my conjecture as to how the ideas worked out by Douglas Hofstadter in his recent book I Am a Strange Loop might play into a belief in God.

Hofstadter lays out an argument about how a human brain generates its symbolic "I" entity as a sense of self, a seat of consciousness, and even a soul. Details as to how the Hofstadter argument is couched can be read in the previous post and also in its predecessor, God Is a Strange Loop, Part 1. At this point, I would simply like to summarize the argument in a way that I dreamed up overnight.

Imagine, if you would, a T-shirt with the following message emblazoned across its front:

My Gödel number is not prim.

That's right ... that final word is "prim," not "prime." It applies to a number that, when suitably decoded using a standard recipe of algebraic computations, turns into a formula. This formula — one possible formula is simply "0=0" — consists of a string of symbols defined in a system of axioms and rules to be used for deriving the theorems of mathematical number theory.

Number theory is the theory underlying all the computations we (or our machines) do every day, including the computations by which the number that we wish to decode into a formula of this so-called "axiomatic system" is actually decoded. One theorem of the system that can derive mathematical number theory — or, actually, one axiom — is, of course, that zero equals zero.

If a number is "prim," it decodes into a symbol-string formula that our axiomatic system of number theory can derive, or "prove." But a number that is not "prim" — Hofstadter calls it a "saucy" number — decodes into a symbol string that is not provable within the axiomatic system.

By reversing the decoding process, one can encode any arbitrary string of symbols of the axiomatic system (which is called PM, after Principia Mathematica, a three-volume work by Bertrand Russell and Alfred North Whitehead — whence comes "prim"). The encoding process involves another computation, and it spits out a single number, called the string's "Gödel number." That name comes from Kurt Gödel, the Austrian mathematician/logician who, in 1931, figured out how to turn symbol strings of PM into numbers, and vice versa.

If the Gödel number of a string is prim, it means the symbol string is a well-formed formula of PM, and not just some hodgepodge of PM symbols thrown together at random. Furthermore, that particular well-formed formula, or "wff," happens to be one that PM can derive/prove — hence the "primness" of its Gödel number.

A wff such as "ss0+ss0=sssss0" is not provable, thankfully, because it means "2+2=5". ("ss0" stands for "the successor of the successor of zero," or 2; "sssss0" stands for the number which is fifth in the successor-to-zero sequence, or 5.)


Accordingly, when my hypothetical T-shirt says "My Gödel number is not prim," it's implying three things:

1. "I" have a Gödel number, which means "I" am in some sense like a PM symbol string
   
2. "My" symbol string is a wff, and not "symbol salad" — for if it were not a well-formed formula, the question of whether its Gödel number is prim would be totally irrelevant
   
3. The wff which "my" symbol string composes is, alas, not one whose Gödel number is prim
   

From those three implications can be derived a fourth, which is stenciled on the T-shirt's back:

I am not provable (therefore I am).

Any wff of PM which is not provable (not able to be derived by applying PM's rules to its axioms and previously derived theorems) is not a theorem of PM, and therefore presumably untrue. But, nominally, the theorems of PM are about numbers, not about theorems of PM. The theorem which Gödel derived whose English translation is "I am not provable" is an important exception. It's one which turns out to be among an infinite number of such exceptions ... but who's counting?

Even though "I am not provable" isn't obviously about a number, it has a Gödel number. For it to say, in effect, "My Gödel number is not prim" is merely to restate the selfsame "I am not provable" claim in different terms.


As an exception to the general rule that PM formulas are manifestly about numbers, this "My Gödel number is not prim" formula (which Hofstadter names KG, after Gödel's initials) has the same form as the non-exceptional "72900 is not prim". The latter formula happens to be false, since 72900 is the Gödel number of the PM theorem to the effect that "0=0". But "576 is not prim" is true, since 576 is the Gödel number of the PM formula "0=", which is not even well-formed, much less provable.

The "My Gödel number is not prim" formula, KG, is constructed like "72900 is not prim" or "576 is not prim", but with a twist. The hard-coded "72900" or "576" is replaced with a reference to the Gödel number of the very symbol string composing the KG formula itself!

When I speak of "a reference to" the Gödel number of KG itself, I mean the numerical or mathematical equivalent to our first-person pronoun "I" (or its variants such as "me" or "my").

This is why the KG formula can be translated as "I am not provable."


Now for an explanation of why I tacked "(therefore I am)" onto the end of the second T-shirt message.

Whether KG is expressed as "I am not provable" or as "My Gödel number is not prim," it simply cannot be false. It must be true. For if it were false, then it would be, in fact, provable ... and its Gödel number would accordingly be prim. But if provable, KG would be asserting a lie. Such a contradiction is inadmissible to any axiomatic system, since it would be like the one rotten apple that spoils the bunch. If any false assertion could be proved, then every false assertion could likewise be proved ... and the original intent of the axiomatic system to separate truths from falsehoods would crumble into dust.

So KG is true ... but not provable. Ergo, every axiomatic system (not just PM) lacks the ability to prove not just this but an infinite number of truths about itself.

Also, every axiomatic system that is at least as powerful as PM (i.e., it can derive the laws of standard number theory) is entitled to construct an internal symbolic reference to what amounts to an "I". The formulas that contain this symbolic "I" can typically not be proven ... yet the emergent phenomenon for which the English-language shorthand symbol is "I" has to be admitted to, in some sense, be in existence. It emerges, willy-nilly, from the fact that any PM-equivalent axiomatic system contains an infinitude of well-formed formulas — provable or not — each of whose Gödel number is that of the formula as a whole, and is also (by means of a proxy reference) a part of the formula.

Whenever that happens, the axiomatic system is entitled to tack on "(therefore I am)" to the tail end of its KG formula.


Hofstadter shows that the human brain is at some level precisely such an axiomatic system. It is a mechanical, computational machine, which is to say exactly the same thing about it. Yet, for the reason just given, it has an "I", and it is entitled to tack on "(therefore I am)" to the tail end of its KG formula and every other formula — every other thought — in which its "I" symbol necessarily appears.

Thus does each one of us possess legitimate certainty that he or she exists — never mind that none of our thoughts to this effect can be proven by rule-based derivations. If my "I" exists, it does so independently of provability.

When I imagine that "I" have on a T-shirt that says "My Gödel number is not prim" on its front and "I am unprovable (therefore I am)" on its back, what that really means is that such a T-shirt adorns my body — not my mind or soul. My "I" emerges from matter but is itself immaterial.

In my "God Is a Strange Loop" theology, I take that image and reapply it to the universe as a whole as if it were a body with a brain from which emerges an immaterial "I". The universe-as-a-body would, of course, need a "size cosmic" T-shirt, not the mere "size large" that I wear. Still, it makes sense to me to at least conjecture that God is the immaterial "I" of which the cosmic T-shirt speaks.