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Douglas Hofstadter's I Am a Strange Loop

For the last few weeks I have been exploring in this blog Douglas Hofstadter's new book, I Am a Strange Loop. The book argues that the "I" or self is a symbol that naturally arises in each human brain in the same way as self-reference ineluctably arises in logical-mathematical systems used for the derivation of theorems that are proved by applying rules of inference. The theorems such systems generate are analogous to our thoughts, and the individaul symbols which are strung together to form these formulas and theorems are analogous to the symbols, categories, and meanings that emerge from the neural interactions of our cranial gray matter.

My response to this view of the self has taken me on something of a roller-coaster ride. At first, in a series of posts that challenged Strange Loop's casual assumption that the self is the same as the conscious soul, I sketched out an alternate view of human self-awareness in which our capacity for conscious experience echoes God's own such capacity. In that alternate view, which I labeled "Genesis By Experience," or GBE, I held that mind is distinct from matter in the same way as God is distinct from the physical world.

GBE was thus a dualistic worldview in which the fact of being seen by an observant, conscious mentality — God's — is what confers existence on us and everything around us. I noted that quantum physics seems to show something similar: that possibly, just possibly, when we observe one of two equiprobable quantum events, we confer existence on the one event and consign the other to oblivion. This existence-conferring act of observation on our part applies to quantum alternatives that to our ordinary way of understanding had to be chosen in the past. Yet by observing them in the present, ex post facto, we "fix" them in existence and expunge their twins' existence from reality retrospectively.

Not being all that comfortable with the intrinsic mind-matter/God-world dualism of my ideas, I then looked for and thought I spotted a way in which it could be eliminated without sacrificing the notion that there is a God. I thought God might in fact be the "I" of the world, a symbolic category that could emerge from the mechanical physical workings of the universe in much the same way as the individual "I" emerges from each of our brains.


I was merrily constructing a "God Is a Strange Loop" (or GISL) theology along just those lines — lines suggested to my mind by Hofstadter even though he himself owns to no God — when I experienced a philosophical cave-in. I noticed that the "strange loop" in Hofstadter's crucial discussion of Principia Mathematica — the logical-mathematical system wherein self-referentiality obtrudes despite all its designers' efforts to exclude it — requires of us a commitment to the notion that such a system cannot be logically inconsistent.

The alternative possibility is that the system might very well be inconsistent, which simply means it can prove theorems that aren't true. To allow inconsistency in such a system is to turn coherence into incoherence, which accordingly turns the world which we hope to model by such a mathematical system into an unknowable farce.

There's a pattern here. An external mind is required to choose between a pair of choices. One choice is that the system is an incoherent, logically inconsistent, useless mess. The other choice is for the system to be logically consistent ... which, as Hofstadter shows by presenting to us lay readers Gödel's Incompleteness Theorem, has the side effect of making it logically incomplete. Logical incompleteness means there are system-internal truths which the system itself is powerless to prove. Again, an external observer is required to "see" the veracity of those truths.


The requisite external observer is usually one that has a mind, is conscious, and has a sense of self — typically, it is one of us humans. But if we were to envision the universe-as-a-whole as basically a repository of truths — some of which are unprovable in a Gödelian sense, some not — who then would the external observer be?

Here, obviously, is where I think the God/world dualism comes into play. The cosmos-external observer is God. In so saying, I am simply claiming that the pattern of needing an external observer to choose between two mutually exclusive alternatives, incoherence and incompleteness, applies to the cosmos as a whole, whose external observer is God, but it does not apply to God himself. By virtue of the mind-matter/God-world dualism, the need to confer existence on God — via an external-observational imposition of causal coherence upon God — simply doesn't arise. God's very being is made, as it were, of causal coherence.


In short, I am back to a dualistic view of reality: God is distinct from the world which God creates and sustains, by virtue of his ongoing, conscious act of "seeing" or observing the world and all in it. The distinction between each conscious, observing human mind and the body/brain that carries it around echoes that very God/world dichotomy.

The "I" or sense of self may well arise within the human brain exactly as Hofstadter proposes — but it is not the same thing as consciousness per se, and it is not the same as the soul.

David J. Chalmers's The Conscious Mind

I accordingly say consciousness arises, just as David J. Chalmers argues in The Conscious Mind: In Search of a Fundamental Theory, as subjective "phenomena" within the mind: "raw sense experience" lacking all causal efficacy. We quasi-empathetically know something of "what it is like to be red," whenever we observe a red tricycle. But having (or lacking) such knowledge makes zero difference to our behavior, or to its causal impact on the external world.

Moreover, I say, the soul comes from God. Specifically, it is God's seeing us that confers an immortal soul upon each of us.

In chapter 15, "Entwinement," Hofstadter constructs an elaborate thought experiment about "Twinwirld," a world that is just like ours except that almost all babies are born as identical twins. The linguistic and cultural customs of Twinwirld are such that each pair of twins develops a single, unitary sense of self — and thus, in Hofstadter's estimation, a single consciousness, and just one soul.

Any hesitation Hofstadter's reader might have in crediting the single-souledness of Twinwirld's ubiquitous twins is supposed to vanish when Hofstadter tweaks the rules, and each dual "pairson" becomes a conjoined ("Siamese") twinset. I couldn't get through this part of the book without wondering why the conjoined Twinwirld twins couldn't be surgically separated, thus creating two persons, two minds, two consciousnesses ... and two souls.

Nor could I find an answer to the nagging question, what happens if one half of a Twinwirld twinset dies and the other survives? The question may not matter if there is no afterlife, as Hofstadter seems to believe. But if there is ... and if there is only one soul per twinset ... then what?


It seems to be quite true, as Hofstadter points out in his "Post Scriptum re Twinwirld" section, that in at least one rare case of earthly twindom — that of Greta and Freda, the Chaplin twins of York, England — two twins can indeed develop a single "self," for all practical purposes. But, I would add, this is not necessarily the same thing as having a single consciousness, which by Chalmers' reckoning cannot be either proved or disproved for Greta and Freda, or for any other pair of twins.

Nor is it the same thing, by my reckoning, as having a single soul. If, God forbid, Greta dies and Freda lives (assuming they're both still with us), I believe Greta's soul would find a heavenly abode while bereft Freda's remained, waiting patiently, here on earth.


So Hofstadter and I now seem to be at daggers drawn. Peeking ahead at chapters still to come, I can see that he intends to take up the topic of dualism, presumably to kayo it summarily. Needless to say, when I get there I will be highly skeptical.

God Is a Strange Loop, Part 5

Douglas Hofstadter's I Am a Strange Loop

In God Is a Strange Loop, Part 4, I suggested that God is an emergent property of the universe. Specifically, God is the "I" who emerges from that which he bestows truth upon.

A lot of prior discussion led up to that conclusion. In that previous post and in its three predecessors from this, my "God Is a Strange Loop" series, I leveraged ideas laid out by Douglas Hofstadter in his book I Am a Strange Loop, ideas concerning how a human brain produces a self-referential "I" symbol, into the basis for a theology.

The gist of Hofstadter's notion of the "I" is that it is analogous to the self-referential, externally knowable-as-true propositions lurking in supposedly purely mechanical systems that do nothing but extend a handful of basic axioms into a cornucopia of provable theorems. They do so by means of applying one rule after another, each rule aptly chosen from a small set of rules of inference.

Hofstadter shows that emergent self-referentiality in such "axiomatic systems," as they are called, mirrors the same phenomenon in the brain, which at some level is just an elaborate machine. Hence a brain is just like an axiomatic system in its basic operation, with the system's "theorems" being replaced by the brain's "thoughts."

But certain of the system's self-referential, manifestly true propositions, such as "I am not provable," are not capable of being derived in any systematic, bottom-up way. We see the truth of them, but only from above, as it were. The system itself is blind to their truth.


In the series' previous post, I called this top-down phenomenon "truth bestowal," while Hofstadter calls it by the less grandiose name "downward causation." In that earlier post I glossed over the fact that these two phrases are not really perfect synonyms. For Hofstadter, "downward causation" is only "real" to a limited, nuanced degree, while to me, "truth bestowal" is much more "real" than that — it implies there is a God.

In short, exactly here is where you can begin to see a sliver of daylight coming between my worldview and Hofstadter's.


For Hofstadter, the "reality" of many things that we know about, such as, for example, a rainbow, is due to how our brains evolved. We simply see the world as fundamentally organized into high-level objects, causal patterns, and meaningful categories. A rainbow is a real object, to our brain's way of looking at things, even if it is just a collection of water droplets refracting sunlight into organized hues of color.

To take another example, even though the image on a TV screen is just a bunch of colored pixels, we see in it faces and trees, blue skies and white clouds and the like. Those identifiable, meaningful splotches and blobs in the image are real to us, as long as they behave coherently in the ways we associate with the symbols we have long since built up in our brains for ... well, for faces, trees, blue skies, clouds, and so on.

The categories we viewers impose on the TV image have no causal effect on how the scene develops. But for the creators of the TV program, it is a different story. Imagine the program is a cartoon. The cartoonist imposes order and meaning on the moving picture according to some preconceived mental notion of how the characters and objects represented by the moving blobs and splotches ought to behave. From this perspective, Mickey Mouse or Homer Simpson is much more real to the cartoonist than just a bunch of screen pixels. Their behavior is imbued by the creator with order and meaning.

We, too, impose order and meaning upon our own behavior. We do this in accordance with our idea of how we as individuals — as the special persons we each refer to as "I" — ought to behave. Our brain's internal "I" symbol has "causal potency," says Hofstadter. If our "I" calls for us to, for instance, shake somebody's hand, it arranges for our body to make the proper muscular movements in order to accomplish that. In other words, the ephemeral high-level symbol called our "I" somehow "pushes stuff around" at the lower, physical levels of bodily organization, so that they do its bidding.


I haven't finished reading I Am a Strange Loop yet, but as of chapter 14, it is not clear exactly how Hofstadter thinks the "I" pushes stuff around at lower, more physical levels.

I gather he thinks that the top-down symbols (including the "I") by which a brain comprehends its body and its surrounding world it exists in are genuinely real, but with an asterisk. The asterisk points to, in effect, a footnote which says, "These high-level things of the mind are 'real' only to the extent that they map what's really going on at the lower level of neurons and the chemicals they squirt back and forth at one another millions of times each second."

I think they are "realer" than that. To show why, I'm hoping to concoct an argument that I derive from Hofstadter's own presentation concerning the nature of the "I". It is an argument that answers a question that Hofstadter himself seems to sidestep. Unfortunately, however, it is also an argument that undermines one of my own basic assumptions in this series of posts.


The argument I would like to concoct runs something like this. Hofstadter shows that an axiomatic system that is ostensibly about the truths of number theory — "4 is not a prime number," for example — contains unsuspected self-referential propositions that can be interpreted on another level entirely.

For instance, take the proposition "I, this very proposition, am not provable." This proposition actually exists, Hofstadter shows, as a well-formed formula of the axiomatic system called PM, after Russell and Whitehead's masterwork Principia Mathematica. The proposition is called, by Hofstadter, KG, after the initials of Kurt Gödel, its discoverer.

Of course, KG has a lower-level, number-theoretical meaning within PM as well — one which, as its higher-level meaning suggests, cannot be derived within PM through the application of its rules of inference to other true propositions.

If KG indeed cannot be proven from the bottom up within PM, as "4 is not a prime number" can, then how can we know it is true? As Hofstadter shows, we can arrive at our certainty of KG's truth by virtue of recognizing that if KG were false — if "this" very proposition were in fact provable — an inconsistency would exist within the axiomatic system as a whole.


Such an inconsistency would be like the one rotten apple that spoils the bunch, for once a lie can be proven within any axiomatic system of mathematical logic, every lie can be proven. Hofstadter shows that much quite well, I think.

But what Hofstadter fails to show, I think, is for what reason KG is true. That question is, I would say, distinct from that of why we prefer to reject the alternative proposition that KG is false. We prefer to reject that alternative proposition in order to avoid turning all of mathematical logic into a cocked hat. But what gives us the right to impose that preference of ours on objective "reality"?


The "KG is true" proposition is, as Hofstadter correctly shows, one whose truth must come from outside PM, since it certainly does not come from within. Ergo, the "I" which emerges from within the bowels of the axiomatic system PM, and which makes it possible to construct well-formed formulas like "I am unprovable," is just as blind as the system as a whole to the truth of KG!

Recognition of that fact does not make me terribly happy. The reason is that I have been trying to show that God could be the "I" which emerges from the world-as-a-whole — if we look at the entire world as a entity that is fully analogous to a mechanical system such as PM — in just the same way as each of our brains is also a PM-equivalent machine.

In other words, if the organ inside the cranium of each of us is, at its lowest levels of operation, just as much of a theorem-deriver (or thought-deriver) as PM is, then Hofstadter is right: an "I" can be expected to arise within it, just as one does in Russell and Whitehead's axiomatic system PM, once Kurt Gödel gets through with it. I have simply been trying to extend that notion to consider the world as a whole as if it, too, is (or has) a PM-equivalent brain.

If the world-as-a-whole somehow can be assumed to be "conscious" and to have an emergent "self" or "I" arising from within it, then perhaps the proper name of that "I" is God. This, at least, is what I have been trying to claim in the present series of posts.


But, I now see, my hopeful claim won't work. Why not? Because any "I" that arises from the operation of the world-as-a-whole as if it were some kind of PM-equivalent axiomatic system will necessarily be a "blind I." It will necessarily be blind, that is, to the truth of any propositions about the world-as-a-system that take the form of KG, "I am unprovable."

Hence, adopting if only for the sake of argument the notion that such a "world I" is capable of being generated at all, as a necessarily "blind I" it would fail to provide an essential reason why the world-as-a-system is not riddled through and through with logical inconsistency.

For, if the "world's KG" is true, the system-internal "world I" is unavoidably blind to the reason why.

That leaves us right where we were before: without a vantage point outside the world system from which "the truth of the 'world's KG'" — the truth that is coherence itself — can be bestowed and known.


There must be a reason why the world "prefers" truth and coherence to lies and logical inconsistency. And — unless you turn a "blind I" to the usual presumption that facts have reasons — that reason must come from outside the world. The basis of all logical coherence, a.k.a. "the truth of the 'world's KG'," cannot be derived from within the world-as-a-system itself.

Which suggests that God does not emerge, as I had been hoping to demonstrate, as a sort of "world I." And we are back to a dualistic scenario in which the "mind" of God is wholly separate from the "body" of the world. This is where I was as of the last post in my "Genesis by Experience" series, which I abandoned out of distaste for the mind-body dualism. Instead of continuing to develop that "GBE" philosophy, I embarked on this, my "God Is a Strange Loop" series. Now I fear I will have to go back to the dualism of GBE.

More later ...

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 ...

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.

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 ...

God Is a Strange Loop, Part 1

Having laid out what I intended to be a definitive theistic metaphysics in a series of five posts winding up with Genesis by Experience, Part 5, I now proceed blithely to contradict myself, if only to a degree.

"Genesis by experience" was the name I gave my philosophy of the existence of the physical world, as well as of the conscious mind we each harbor and the immortal soul our religions tell us we, each of us, have. In my Part 5 post about GBE metaphysics, I brought up the fact that GBE proposes a dualism between mind and the physical world.

Specifically, it says that God's mind, in that it partakes of the same sort of conscious, subjective experience that our minds echo, confers coherence on the world and thus guarantees to us, God's creatures, the steady causal regularity that we marvel at in our scientific inquiries.

I extrapolated this idea from certain counterintuitive, even paradoxical results known to quantum physics, wherein the incoherence, uncertainty, and incompleteness that pervade quantum phenomena are replaced by a coherent knowability whenever observation enters the picture. At a macroscopic level, quantum uncertainty disappears. I attributed this genesis of worldly coherence to the ceaseless observation of the physical world by God.

I even went so far as to state that "to exist is to be caused," where causation is the conferring of (cosmic) coherence by virtue of conscious observation. But this led me to wonder if it would be inconsistent to claim that the mind whose consciousness confers coherence/existence upon the world is itself something that "exists." If that question is answered no, then how could God be said to "exist"?

Or, if that question of whether God "exists" is answered yes — for it looks as if consciousness per se can never be directly observed — what act of conscious observation, by what mind, could be said to confer "existence" on God?


I felt uneasy with both horns of that dilemma, so I quickly shunted it aside and went on to discuss the topic of intentionality as it applies to our minds and to God's. But the dilemma nagged at me overnight.

Douglas Hofstadter's I Am a Strange Loop

Douglas Hofstadter's Gödel, Escher, Bach

I woke up this morning to the realization that ideas put forth by Douglas Hofstadter in his most recent book, I Am a Strange Loop, and in its predecessor, Gödel, Escher, Bach: An Eternal Golden Braid, can bridge the dilemma's two horns.

And so I'm undertaking the series of posts of which this one is the precursor of what will undoubtedly be many more yet to come. The general thrust of this series will be to modify GBE to remove the intrinsic mind-matter dualism that, as it stands now, GBE seems to require.

So modified, my GBE metaphysics will (I hope) become what I'll dub my "God Is a Strange Loop" theology — for short, GISL.


It will take me quite a while to lay out my GISL insight in all its glory. In fact, I'm going to sneak up on doing so gradually, because the subject is, I hate to say it, very much like the Hofstadter books in being conceptually challenging to the max.

Before I even begin stealthily approaching any sort of definitive statement of the inner kernel of my insight, I want to present a broad sketch of what I'm going for. The general idea of Hofstadter's books is to ask how the human brain gives rise to a mind that houses within itself a symbolized self or "I." Hofstadter's answer is that it happens by virtue of an intrinsically self-referential "strange loop."


Imagine the mind as, first of all, a machine, a computer — a vessel of "artificial" intelligence. Like all computing machines, it works just like a "formal system" works. That is, using a set of rules, it shunts symbols around to make well-formed symbol strings called formulas.

One formal system that mathematicians know is number theory, in which "0=0" ("zero equals zero") is a formula, because it is a well-formed symbol string. It is also a theorem (actually, an axiom, needing no proof) because it is true. On the other hand, "2+2=5", though well-formed, is false and hence is not a theorem. It cannot be proven — the rules of number-theoretical theorem derivation simply cannot be used to derive it.

About a century ago, Alfred North Whitehead and Bertrand Russell wrote a triptych of tomes called Principia Mathematica that tried to ground number theory in yet another formal system, set theory. It was their hope to do so in a fashion that guaranteed the PM system to be free of self-reference, because if a formal system's theorems can refer in any way to the system qua system, all hell breaks loose.


By all hell breaking loose is meant that what Hofstadter calls the Mathematician's Credo gets violated. According to the credo, every single true statement concerning the "stuff" the formal system deals with — numbers, in the case of number theory — corresponds to a derivable theorem within the system, while no false statement does so. If the first of those two assumptions is violated, the system is intrinsically incomplete. If the second is violated, the system is intrinsically inconsistent. Russell and Whitehead were on a mission to ground all of mathematics, via number theory, in set theory, in such a way as to avoid both incompleteness and inconsistency.

In 1931, along came 25-year-old Austrian mathematician-logician Kurt Gödel and upset Russell and Whitehead's apple cart. Gödel cleverly showed that there were, concealed within PM, well-formed symbol strings that, if false, rendered PM intrinsically inconsistent, while, if true, rendered it incomplete.

One such well-formed symbol string was the one which, loosely translated into English, reads "I am unprovable" (see I Am a Strange Loop, p. 138).


That bald statement needs some hasty elucidation. First of all, the "I" which is its first word refers to the statement itself (!). Second, "unprovable" means "unable to be derived within the PM system by means of applying the system's official rules of theorem derivation to well-formed symbol strings that have already been derived and/or to well-formed symbol strings which are taken to be axioms that don't need proof." So a statement which has the same import would be: This very wff — a "wff" is a well-formed formula within the system-at-hand — is not derivable within the system.

Standing back a way, one may twig to the fact that Gödel found a way to make wffs of PM talk about, not numbers like zero or two or 79,406, but wffs of PM! How he did takes Hofstadter two chapters to spell out Basically, Gödel figured out how to turn every conceivable symbol string of PM — well-formed or not, true or not — into a unique number. He devised a straightforward way of computing whether any given symbol string's "Gödel number" was in the set of numbers corresponding to all the wffs of PM. He demonstrated that the Gödel number of the "I am unprovable" symbol string was in the set. Hence, the PM-internal equivalent of "I am unprovable" is a wff.

Given that it is a wff, it cannot simply be thrown upon the trash heap as not even rising to the level of being a formula. So it must be either true (i.e., derivable) or false (i.e., not derivable).

Now — and this part of my discourse has to have a "to the best of my understanding" slapped conspicuously across it — it has not yet been determined which of these two possibilities is correct. For somewhat abstruse mathematical reasons, it is much, much harder to compute whether the number corresponding to a Gödel-arithmetized symbol string is in the set of numbers representing derivable theorems than it is to compute whether the number is in the wff set.

But never mind. It remains the case that the "I am unprovable" formula poses a double-barreled threat to the Mathematician's Credo. For if it does happen to be true, then there is at least one true statement of PM-extensible number theory that is — by its own self-implication! — unable to be derived within PM. In that case, PM would be intrinsically incomplete.

Or, if the "I am unprovable" formula happens to be false, then the formula is provable, and PM contains a contradiction.

Accordingly, PM is either incomplete or inconsistent.


Bertrand Russell could never accept that, according to Hofstadter. The hope of Russell and his co-author, Whitehead, was to constrain PM very carefully, in terms of its permissible symbols and rules, so that it was guaranteed to be both complete and consistent. To to that, Russell and Whitehead banned self-reference entirely, or so they thought. By not allowing sets, the theory of which was to serve as the rock-solid foundation for number theory, to "contain themselves," Messrs. R and W expected to forestall self-reference from ever cropping up within the PM system, no matter how elaborate it got.

What Gödel proved is that there simply is no way to banish self-reference from PM or any other equally rich formal system that mechanically shunts symbols around according to a fixed set of rules, thereby to produce scads of symbol strings that all qualify as being true.

But, says Hofstadter, the human brain is just such a symbol manipulator.

Accordingly, there is in effect no way that the brain — assuming it is elaborate enough, as the human brain surely is — could fail to be able to entertain as its own well-formed symbol strings such statements as (loosely translated into English) "I am this" or "I am not that."


Furthermore, that human capacity for self-referential "I"-ness has the ability to stand outside itself. It has what I am calling "externality."

A good example comes from Hofstadter's discussion of how Bertrand Russell judged his own Principia Mathematica system. Russell knew that at one level it was correct to say that PM was just a system that could derive certain wffs, but not other wffs. It was mechanical. The wffs had no "meaning." Neither, for that matter, had the derivable theorems. The latter were in some sense "true," but nonetheless they had no "meaning."

Yet at another level Russell was concerned lest, for instance, the wff which stood for "2+2=5" should turn out to be a derivable theorem. In the back of his head, Russell knew that the wff which stood for "2+2=5" did have a sort of "meaning" — by virtue, that is, of its being able to be "mapped" to "two plus two is five." Such a mapping — or, in technical lingo, such an "isomorphism" — indeed confers meaning on the wff by way of the wff's analogy with "two plus two is five." If the "2+2=5" wff turned out to be true within the system, whereas the analogous "two plus two is five" statement is clearly false outside the system, that would be a crushing blow.

Russell was, in fact, trying to "play God" by employing his own mind — a self-referential, "I"-generating formal system, in Hofstadter's view — to stand outside PM, another self-referential, "I"-generating formal system, and see how well the latter maps to a preconceived, PM-external standard of truth.


Thus, externality. The human mind, although it is itself a self-referential, "I"-generating formal system, is capable of standing outside other self-referential, "I"-generating formal systems of its own contrivance and judging the "meanings" of those systems' theorems against an external standard.

But here comes one of Hofstadter's "strange loops." The human mind can likewise judge itself — its own self, its "I" — and compare it to a preconceived, seemingly external standard of "truth." Indeed, I would hope my own self, my "I," to be as thoroughly self-consistent as Russell and Whitehead manifestly hoped their Principia Mathematica system to be. I would hope none of my self's "meanings" — none of the well-formed symbol strings that its rule system is capable of deriving; none of the thoughts which, upon due consideration, I believe to be "true" — turn out to be "false."

Stated by way of analogy with my religious understandings, I would hope that none of my beliefs contradict God's truth — for then I would be in sin.


One of the key facts about having a mind, accordingly, is that (assuming the mind arises from a sufficiently complex physical substrate) it turns out to be capable of standing outside itself and judging the truth value of its own "derivable theorems." It is an engine of theorem derivation, yes, but even as such, it is intrinsically incomplete. There is more to truth than it can mechanically derive — which doesn't faze it in the least!

The Gödelian alternative does faze it, though: that it is not (in a mathematical-logical sense) incomplete, but inconsistent.

Perhaps it is this preference for incompleteness over incoherence that underlies it's search for God — inasmuch as God, surely, possesses the one mind which is complete in the mathematical-logical sense. Or perhaps God is a stand-in for the mind's own externality: its ability to stand outside its own mechanical symbol-shunting process and attempt to judge how true the results of that process are.


My idea about God himself being a strange loop takes that thinking a step further. What if the whole world is conscious? What if, just as the complexity of a human brain is such that it is perforce a self-referential, "I"-generating formal system that can stand outside its own mechanical symbol shunting, the complexity of the world as a whole generates a "mind of its own."

This mind would, by strict analogy with ours, have the ability to transcend its own logical mechanics. Also by analogy with our own minds, it would be conscious — meaning that it would be able to confer coherence — nay, even "existence" — on what it consciously observes. That which it consciously observes would be ... what? Why, it would be the very physical substrate that gives rise to it: the world!

Furthermore, this "world mind" would be intentional, meaning that it is capable of goal-seeking behavior. The difference would be that, whereas our goal-seekingness can choose to manipulate things external to that which gives rise to it — things outside its body, that is — the "body" of the "world mind" would comprise everything in the physical world.

In ordinary religious language, the "goals" sought by the "world mind" would qualify as "God's will," and the process of bringing those goals about would represent "providence."

For, in my God-Is-a-Strange-Loop theology, GISL for short, the mind of God is this "world mind"!

More in my next post ...