More on God as Genotype

In God as Genotype, I broached the idea that in Douglas Hofstadter's "religion," God is the implicit meaning, the inner message, the seed-genotype of reality's phenotype.

Douglas R. Hofstadter's Gödel, Escher, Bach

Hofstadter is the mathematician-cum-philosopher whose magnificent 1979 book Gödel, Escher, Bach: an Eternal Golden Braid is the subject of my ongoing "Strange Loops" series of posts. To date, they are:

• Gödel, Escher, Bach
• Insight Through Isomorphism and Paradox
• A Picture of God?
• God as Genotype
  

In Hofstadter's Chapter VI, "The Location of Meaning," he (without mentioning God by name) hints that the reality we each bear witness to is a message with an intrinsic meaning. If we call that meaning God, we do no violence to the truth.

Like all messages, reality or truth has three levels of "message-ness." The "frame message" signals that a meaning lies within, waiting for our intelligence to decipher it: it is not mere noise or static buzzing in our ears. Next, the "outer message" conveys, in a manner necessarily distinct from that in which the interior meaning is conveyed, clues to how to decipher the interior meaning. Finally, the "inner message" contains the interior meaning that the source of the message qua message wished to present to whoever receives and decodes the message.

As such, an inner message is like a genotype wherein is found all the requisite information as to how living cells are to proceed to build a canary, say, or how the mind is to construct the infinite Fibonacci sequence of numbers, in which every number is the sum of the previous two. The canary, or the expanded Fibonacci sequence, is the phenotype which the genotype internally codes for.

The relationship of the entire Fibonacci sequence, which is an endless stream of numbers, to their finite numeric "starter set" which acts as frame message, outer message, and inner message, all at once, is easy to see. It is straightforward and quite prosaic. The relationship of a canary or a tiger to its genotype is more exotic, tangled up as it is with the vast number of molecular reactions and biochemical pathways that serve to construct the finished product over time (if it can be said that a canary is ever finished).


The relationship of reality as a whole to its genotype, which I am calling God (and I hope Hofstadter would agree with the appellation), must be yet more exotic than that.

For one thing, a main point of Hofstadter's book is that no formal system (such as one that can manipulate logical symbol strings to generate a list of Fibonacci numbers "typographically"), if it is deemed to be internally consistent and if it is also of a generative power much more impressive than the relatively puny Fibonacci generator, is complete.

If the Fibonacci generator is expanded sufficiently, it can be turned into the entirety of what mathematicians call number theory. If that is done, the resulting formal system will be quite powerful, but it will perforce be unable to derive all the truths about number theory that the human intellect is capable of apprehending!

The Austrian mathematician Kurt Gödel demonstrated in 1931 that such systems cannot help but be incomplete. The theorems they are able to generate, and thus prove, map to true statements, but there are other true statements which do not have theorem-cognates internal to the system — to, that is, the number theory per se.

By extension, no matter what the formal system, if it is at all powerful, then "provability is a weaker notion than truth," writes Hofstadter (p. 19). Still, human intelligence can apprehend the "unreachable" truth beyond what the formal system can derive or prove.

This is a paradox: the human mind is housed in a brain which presumably operates "typographically," i.e., in the same mechanical way that any formal symbol system does its thing. Yet the mind can entertain and apprehend thoughts that properly lie outside its underlying system's power to prove.

Intelligence, accordingly, can reach supposedly unreachable truth. A man's reach exceeds his grasp, or what's a heaven for?

God as Genotype

Douglas R. Hofstadter's Gödel, Escher, Bach

Now, more about Douglas R. Hofstadter's Gödel, Escher, Bach: an Eternal Golden Braid. I started blogging about this marvelous 1979 book with Gödel, Escher, Bach and, following Insight Through Isomorphism and Paradox, I most recently added A Picture of God?.

I have now read the book's first six chapters and am given to understand the next swatch of material ratchets up the technical heat several notches. The book decidedly stretches the intellect.

One of the questions that is always with me as I peruse Gödel, Etc. is how religious is it? Hofstadter himself admits (if that is the right word) in his introductory "Words of Thanks" that

In a way, this book is a statement of my religion. I hope that this will come through to my readers, and that my enthusiasm and reverence for certain ideas will infiltrate the hearts and minds of a few people. That is the best that I could ask for (p. xxi).

On its face, the book may seem to be about anything that what most people would call religion. There is little explicit mention of God. Nothing that I have read so far involves taking anything on faith. Moreover, the tone and style of Hofstadter's writing is much like that of certain leading scientists well known as skeptics and atheists. The agnostic Carl Sagan (God rest his soul) probably ate this stuff up — particularly the discussion in Chapter VI about communicating with extraterrestrial intelligence.

On the other hand, I personally feel very much like sharing Hofstadter's "religion." I feel quite drawn to it, in fact. For one thing, there seems to be something so right about a religion which examines the music of Bach as a profound clue to the nature of reality.

For another thing, the ground rules of Hofstadter's religion seem to be such that there could never be a war between it and science-reason-mathematics-logic. There is only one truth, and one kind of intelligence by which to encounter it.


It is the above-mentioned Chapter VI, "The Location of Meaning," which provides me with my own clue to the nature of Hofstadter's religion. The chapter is an inquiry into the art of deciphering messages. All messages actually have three layers, though in practice the layers can be hard to tease apart. There is the "frame message," within which is the "outer message," within which is the "inner message."

The frame message is that about the message which clues us in that it is indeed a message, full of meaning, and not just some happenstance detritus floating past: it has a meaning, if we can only decipher what it is.

The outer message is that about the message by which we recognize the manner in which it must be decoded. If the sealed bottle that has washed up on our shore contains a piece of paper, and there are marks on the paper, and those marks are in Japanese, then we know that reading Japanese will be the key to deciphering the message.

The inner message is simply the meaning which the bottle-preparer intended to convey to whoever finds the bottle, opens it, and deciphers the message.


One way of understanding the relationship between the outer message and the inner meaning is by a phenotype-and-genotype analogy.

Two DNA strands, wrapped around one another in a double-helix ladder, form a genotype within each living cell. The genotype contains a "meaning" that the ongoing processes of life serve to unfold and reveal as, ultimately, a bodily phenotype: you, or me, or that canary over there. The genotype contains all the requisite information about how the canary embryo turns eventually into a canary and not a tiger. The phenotype — a yellow bird that sings — is to be considered the "outer message" to the genotype's "inner message."

By analogy, the genotype of the endless Fibonacci sequence, in which every number is the sum of the previous two numbers,

[0,] 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

could be taken as the list of the first few numbers in the sequence, up to and including (say) 34. That is, the infinity of Fibonacci numbers beyond 34 are implicit in the finite list of numbers up to and including 34. That finite list can be thought of as the genotype of the entire sequence.

Or, just the bare seed [0, 1] could be considered the genotype, once you twig to the rule for generating each next number in the series — i.e., 0 plus 1 is 1, 1 plus 1 is 2, 1 plus 2 is 3, and so on.

The entire infinity of numbers, including the genotype or seed, is like unto a phenotype. In this example of a Fibonacci sequence, it is pretty easy to see how the phenotype relates to the genotype. In biology, the relationship is more exotic. How could it be that a very long string of DNA "base pairs" codes for a canary? What meaningful role, if any, in "canary-ness" is played by the vast number of molecular transactions that serve to express the genes as the embryo evolves, one day to take flight?


I extrapolate from the above this idea about Hofstadter's God: such a God is, as it were, the genotype to reality's phenotype. Using our intelligence, we can decipher the message here: there is an implicit, inherent "meaning" to all we are witness to. God is a name, as good as any, for that implicit meaning, that inner message, that seed-genotype of reality's phenotype.

A Picture of God?

Douglas R. Hofstadter's Gödel, Escher, Bach

I've been blogging about Douglas R. Hofstadter's 1979 book Gödel, Escher, Bach: an Eternal Golden Braid, starting with Gödel, Escher, Bach and most recently with Insight Through Isomorphism and Paradox. The book does what its author in his professional life as an academician, and also presumably in his personal life as a human being, does: weaves mathematics, formal logic, cognitive psychology, computer science, artificial intelligence, music and art appreciation, linguistics, philosophy, and maybe even theology into one seamless garment of understanding.

A clue to this weaver's art comes from his discussion of "Gplot" (pp. 140ff.). If you wish to answer the scientific question, "What are the allowed energies of electrons in a crystal in a magnetic field," Gplot is a good place to begin.

Gplot, or "Hofstadter's Butterfly"

To the right is an image of Gplot, also known as "Hofstadter's butterfly." You can click on it to see a larger version. The image comes from this web page. More discussion of Hofstadter's butterfly can be found here and here.

The description Hofstadter gives for Gplot in the legend for Fig. 34 on p. 143 of his book reads:

Gplot: a recursive graph showing energy bands for electrons in an idealized crystal in a magnetic field. α, representing magnetic field strength, runs [up the vertical axis] from 0 to 1. The horizontal line segments are bands of allowed electron energies.

The graph is black where available electron energy levels exist; elsewhere it is white. The way it is constructed is interesting. You can make out a sort of butterfly pattern in white, no? If you look carefully, you can also see ever-smaller, distorted versions of the same white butterfly, flitting about all over the graph. Well, in reality the graph is nothing but butterflies surrounding butterflies surrounding butterflies ... all the way down, ad infinitum!

This is what it means for a graph to be recursive. It contains copies of itself within copies of itself within copies of itself, possible without limit.

This graph can also be generated using an algebraic function. The function is likewise recursive: it invokes itself internally. In this case, the recursion must be forced to bottom out at some level well down the algebraic ladder, or the function could never be computed.


Recursiveness and internal self-reference are big with Hofstadter, because they create and illustrate "strange loops":

The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started (p. 10).

In Hofstadter's butterfly, as you move visually "downwards" into finer and finer detail of the graph, you find the same (albeit somewhat distorted, as in a funhouse mirror) butterfly that you see at the topmost level of detail. Then, if you peer into the immediate surroundings of that butterfly, you see a profusion of yet-smaller butterflies. And so on, forever.


Strange loops embody an element of surprise, as you can see. This particular strange loop is all the more surprising because it claims to capture something fundamental about the laws of physics in our world: how electrons behave.

Do electrons really behave in such a way? Is Gplot a lifelike portrait? Maybe, maybe not:

You might well wonder whether such an intricate structure would show up in an experiment [Hofstadter writes on p. 142]. Frankly, I would be the most surprised person in the world if Gplot came out of any experiment. The physicality of Gplot lies in the fact that it points the way to the proper mathematical treatment of less idealized problems of this sort. In other words, Gplot is purely a contribution to theoretical physics, not a hint to experimentalists as to what to expect to see! An agnostic friend of mine once was so struck by Gplot's infinitely many infinities that he called it "a picture of God", which I don't think is blasphemous at all.

Thomas M. King's Enchantments: Religion and the Power of the Word

In Enchantment, Zen, and Spiritual Dialogue I began a series of posts on Thomas M. King's Enchantments: Religion and the Power of the Word. King, a Jesuit theologian and teacher, writes of the contrast between higher, ideal worlds and the world we live in. He likens the religious impulse, at its most intense, to denying the material world in favor of a realm of absolute perfection. That denial, a form of enchantment, ultimately fails, which leads a person back through a Dark Night of the Soul to true spirituality.

"A spirit," writes King (p. 158), "is an unsteady alliance of both worlds, enchantment and earth; it is the unity formed of the enchanting other and the earthen self, the Word and the World."


When Hofstadter applauds his agnostic friend's calling Gplot "a picture of God," he is being, by King's lights, spiritual. The mathematics of the graph are, in effect, a higher world. They are pristine and perfect. In this world, however, experimenters most likely would be unable to confirm that the butterflies are there. If not, the problem would perhaps be too much "noise" obscuring the "signal" that is represented by the theoretical mathematics.

The mathematics are transcendent to this "noisy" world of ours, and yet they also constitute a ladder by which we can ascend into that ethereal world of pure "signal," as it were. As we climb this ladder, however, a strange loop in it puts us, quite surprisingly, right back where we started.


An example would be consciousness. I have blogged at length about the human capability for conscious experience, starting with Quickening to Qualia, or Taking Consciousness Seriously (Q2Q I). Our minds not only process information; they also enjoy the subjective feeling of what is like to do so. This subjectivity is consciousness.

Oddly enough, it is possible to conceive of world not our own in which creatures that are in all other ways just like us do not possess consciousness. They would be just as well-adapted to negotiating the perils, pitfalls, and possibilities of their environment, but they would be zombies nonetheless. They would not be able to experience their own good or bad fortune.

Again oddly, it turns out to be impossible to prove whether such zombies, should they exist somewhere, actually possess consciousness. I cannot prove that you are conscious — or, for that matter, that I am. Yet I know that I am, in fact, conscious, and I would bet my bottom dollar that you are too.


Another way to characterize consciousness is as the ability to construct meaning. Meaning for Hofstadter is a matter of leveraging "isomorphism": matching patterns that can be rule-generated within rigorously logical formal systems to true statements about the real world. Isomorphisms are transformations of expressions — statements in natural language, "theorems" in formal systems — that preserve information.

Among the logically rigorous formal systems Hofstadter considers are the mathematics of set theory and number theory. It was shown by the Austrian mathematician Kurt Gödel in 1931 that such systems, provided they are powerful enough to justify making the attempt at all, are intrinsically unable to prove the isomorphic theorem cognates of all true statements about the real world.

"Gödel showed that provability is a weaker notion than truth," writes Hofstadter (p. 19), "no matter what axiomatic [i.e., formal] system is involved."

To learn some truths, accordingly, our minds have to be able to step outside any and all formal systems of proof. We do this all the time, in fact, when we find isomorphisms between symbol systems and truth — even if the process of comparison occurs below the level of our conscious mental operations — and then make a leap to further truths that the symbol systems cannot, strictly speaking, justify.

In jumping out of symbol systems, by making leaps of awareness and faith, we become conscious of truth that lies beyond proof. Yet even so, we are still in, and of, this real, material world. We have simply popped up and out of one tier of butterflies to land back in ... another tier of butterflies.

Insight Through Isomorphism and Paradox

Douglas R. Hofstadter's Gödel, Escher, Bach

Douglas R. Hofstadter's 1979 Gödel, Escher, Bach, an Eternal Golden Braid: A Metaphorical Fugue on Minds and Machines in the Spirit of Lewis Carroll was introduced to this blog in my last post, Gödel, Escher, Bach. Now I'd like to extend my remarks in this post, which is the second in a series that I am labeling "Strange Loops."

That odd phrase reflects Hofstadter's fascination with processes of thought that not only circle back on themselves ad infinitum, but do so in such a way as to create paradox. When Epimenides, a Cretan, said "All Cretans are liars," he introduced just such a strange loop into the human thought stream (see p. 17).

Under the assumption that Epimenides meant "All Cretans always lie," i.e., "No Cretan ever tells the truth," we extrapolate from the fact that Epimenides himself is a Cretan, and therefore presumably a perennial liar, that the truth is in fact that "All Cretans are truth-tellers," i.e., "No Cretan ever tells a lie."

But we're applying Epimenides's statement to Epimenides himself in such a way as to cast him as a lying Cretan from the get-go. Hence it's not true that no Cretan ever lies. Given that either all Cretans always lie or no Cretans ever lie — the fuzzy middle ground is excluded by the crisp rules of this particular game — then it must be the latter proposition which is correct. But that implies that Epimenides himself cannot lie, and so it must be so that he and all his fellow Cretans never lie.

But that makes Epimenides's original statement a lie ... and round and round the logic goes, neverendingly. This is the Epimenides paradox.


Maybe the problem which creates the paradox is that statements made in everyday language can be vague. We can try to strip the vagaries of human language away from this and like paradoxes by replacing statements like (a) "Epimenides is a Cretan" and (b) "Cretans always lie" and (c) "Epimenides said (b)" with symbol strings in formal systems.

Formal systems, says Hofstadter, are ways of producing nothing more than various strings of symbols. The symbols in themselves have no intrinsic meaning that might confuse us. You gin up a formal system by saying what symbols are allowed. In the first formal system Hofstadter introduces (pp. 33ff.), the only allowable symbols are M, I, and U.

The next thing you do is specify one or more starting strings (actually, to be perfectly precise, zero or more). These are axioms. The only axiom in the MIU-system is MI.

You next specify one or more rules of production, a.k.a. rules of inference. By applying one of these rules, some new string can be generated, based on a string you already have. For example, Rule 1 of the MIU-system is that any string ending in I can have U tacked on at the end. Thus, MI can be modified to MIU. Strings that can be generated by applying rules are called theorems.

Rule 2 in the MIU-system is that you may replicate whatever comes after the initial M. Thus, MIU can be altered to MIUIU, making another theorem.

Rule 3 is that III can be replaced with U anyplace it occurs. Thus, MI -> MII (by Rule 2) -> MIIII (by Rule 2, again) -> MUI or MIU (by Rule 3). MUI and MIU are also theorems in this system.

Rule 4 is that UU can be deleted wherever it occurs. Thus, MI -> MII (by Rule 2) -> MIIII (by Rule 2, again) -> MIIIIIIII (by Rule 2, yet again) -> MUIIIII (by Rule 4) -> MUUII (by Rule 4, again). MUUII is yet another theorem in this system.

This formal system is apparently incapable, given the four rules of production and the single axiom MI, of being manipulated to yield MU as one of its theorems! MU is well-formed, in that it begins with M, as all theorems in this system must, and that the remainder of the string is a mix of U's and I's (specifically, one U and zero I's). Yet MU cannot be derived in this particular formal system. (Or if it can, I have not come upon Hofstadter's solution yet.)

That may be a bit surprising: well-formed strings can sometimes fail to be theorems. It becomes not just surprising but downright significant when you consider that some formal systems have meaningful interpretations in terms of the correspondence of theorems with real-world truth.

The MIU-system, as it happens, isn't meaningfully interpretable, but one that is is Hofstadter's pq-system (pp. 46ff.). In the pq-system, the permissible symbols are p, q, and the hyphen. There are any number of axioms, one of which is -p-q--. Another is -p--q---. All of the axioms have the form xp-qx-. Here, x represents some number of hyphens: the same number of hyphens in both places where x occurs in the pattern. The pattern xp-qx- is called an axiom schema.

The only rule of production in the pq-system is that any string of the system, be it an axiom or a theorem, can be lengthened by adding exactly one hyphen to the group between p and q and exactly one hyphen to the group following the q. For example, by this rule -p----q----- can be turned into -p-----q------.

Hofstadter shows that this particular system is capable of meaningful interpretation. Basically, you can substitute the appropriate number for any quantity of successive hyphens — for example, 3 for --- — while p turns into "plus" and q becomes "equals." In that way, -p----q----- can be interpreted as "1 plus 4 equals 5."

Interpretation is the mapping of otherwise meaningless strings to statements that have a chance of being "true" externally to the system itself. In this particular interpretation of the pq-system — as having to do with real-world addition — all theorems map to true statements involving adding two numbers. Because the pq-system as a system is capable of at least one meaningful external interpretation along these lines, it differs from systems like the MIU-system which apparently have no meaningful interpretations.

The pq-system has at least one other meaningful interpretation, the one in which -p----q----- becomes "1 equals 4 taken from 5." The system itself and its two interpretations are isomorphisms. The mappings between the system and either of its interpretations, and also that between the two interpretations themselves, is perfect.

Isomorphisms are key. As Hofstadter shows (pp. 49ff.), they "induce" meaning whenever they are (somehow) discovered by human intelligence. In other words, they "create meanings in the minds of people" (p. 50). Stumbling upon a valid interpretive isomorphism between meaningless symbol strings like -p----q----- and true statements in the real world like "1 plus 4 equals 5" can be "cause for joy" and "a source of wonderment." It typically occurs as "a bolt from the blue": everything just "falls into place."


Discovering a symbol-truth isomorphism is a case of "jumping out of the system" (pp. 37ff), an act that allows you to learn something about the formal system that is not contained in the system itself.

There is nothing within the pq-system, for instance, which tells you it is isomorphic with something you already know: addition. There are no system-internal rules for the necessary interpretive mappings. Yet the likelihood is high that if you play around with the mechanics of the pq-system just a wee bit, you'll twig to the fact that it's really a stand-in for grade-school addition.

It's also likely that a bit of playing around with the MIU-system will convince you that MU simply cannot be derived, no matter what rule applications you try, in whatever order. Again, it helps to be able to jump out of the system here and notice (for instance) how you can apply Rule 2 to axiom MI to repeatedly double the number of I's following M — giving you some even number of I's.

Also, if any even number of I's following M is divisible by 6, you may divine that you can first apply Rule 1 to put a U at the end of the I's, and then you can apply Rule 3 however many times you need to to change every III into U. That will give you an odd number of U's, from which you can delete successive U-pairs by virtue of Rule 4 until you are left with just one U. Result: MU.

But, try as you might, you never seem to arrive at a number of I's following the M that is divisible by 6! Mixing in applications of Rules 1 and 4 doesn't seem to help the situation, either. There seems to be a hidden force in the system that keeps you from deriving something like, say, MIIIIII, with exactly 6 I's in a row, from which MIIIIII -> MIIIIIIU -> MUIIIU -> MUUU -> MU would finish the job for you.

Discovery of this negative hidden force may not be as satisfying as, say, noticing that the pq-system amounts to a stand-in for addition. Yet it can lead, albeit more indirectly, to recognizing the truth of a general statement, one that has to do with what mathematicians call number theory.

I do not yet know how best to couch this statement about the "hidden force" in its most general, hence useful, form. Perhaps it says something about the inability to generate every natural number — every non-negative integer, that is — by a procedure involving successive doublings starting from 1 and then some even number of divisions by 3 (which is what divisibility by 6 implies).

Or, perhaps it is something even more general than that. Call the statement at its most general Statement X. An excellent question is, is number theory itself a formal system that, locatable within itself, there is a theorem isomorphic to Statement X?

We know number theory is a formal system. So does it contain a theorem isomorphic to Statement X? I cannot answer that question yet, but I can say this: Hofstadter shows that not all true statements about a formal system such as number theory are necessarily isomorphisms of theorems within the system itself!

You can't discover all true statements about a formal system by using interpretive mappings from the theorems within the system to true external statements about the system. There are unreachable truths that lie outside the system proper. You can't reach them even if you have a "bolt from the blue" and notice a perfect isomorphism between all internal theorems and their corresponding external truths.


Provided the formal system is, by careful design, rendered immune to contradiction, inconsistency, and paradox, and provided it has a great deal of "power" (which the pq-system lacks, but which number theory as a whole possesses) it will necessarily have this characteristic of "incompleteness": some true statements about the system are not isomorphic to theorems within the system. This is basically what the mathematician Kurt Gödel demonstrated in 1931.

Now, this next part of the discussion brings back in the idea of paradoxes. I have to admit I am on shaky ground here — I don't quite understand what I am talking about yet. Perhaps as I continue to read, Hofstadter will clarify it for me. At this juncture, the best I can say is that there is some sort of weird correspondence between paradoxes such as that of Epimenides and theorems of formal systems that are "true" but unprovable within the system itself.

By "true" is meant that the theorem can be interpreted as corresponding to a demonstrably true statement in the world, real or imaginary, outside the formal system. By "unprovable," I mean that the theorem in question cannot be successfully generated within the system. It would be like MU: well-formed. Moreover, if the MIU-system were capable of external interpretation, which it isn't, it would indeed have such an interpretation, and that interpretation would be a demonstrably true statement. Yet there would be no way to get from axiom MI to nontheorem MU by the application of formal rules within the system itself.


There does seem to be (and I may be wrong about this; I feel I'm really on shaky ground here) one way around the unreachablilty of truth in formal systems of appreciable generative power, such as number theory or set theory. That way, if it exists, involves tolerating some degree of self-contradiction and paradox within the system itself.

Here is how that works out with respect to set theory: Basically, paradoxes spring from the possibility of self-reference in the theorems of the system and/or in the interpretively mapped statements derived from the system. For example, in mathematical set theory there is the idea that a set ought to be able to contain itself as a member. The set of all sets, for example, is a legal construction that contains itself. If that sort of thing is allowed, then (see p. 20) there can presumably be just two types of sets. Type A sets do not contain themselves. Type B sets do. Then, is "the set of all sets of Type A" itself of Type A, or is it of Type B? It is neither! Either choice tangles you up in paradox.

This sort of paradoxical tangle can be avoided, apparently, only by forcing set theory to dispense with sets of Type B entirely. That can make set theory entirely self-consistent ... but so much less useful and powerful as to deserve the epithet "ugly."

If you allow "self-swallowing" sets of Type B, you also admit the possibility of paradox into set theory. Now you must deal with external statements about set theory of which the truth or falsity is undecidable. "The set of all run-of-the-mill sets of Type A is itself of Type A" is one of these. You cannot decide whether it is true or not.

This fact seems to correspond to the theory-internal fact that, as a formal system, set theory cannot derive the symbol string whose external interpretive mapping is "The set of all run-of-the-mill sets of Type A is itself of Type A." Nor can it generate the symbolic equivalent of the external interpretive negation: "The set of all run-of-the-mill sets of Type A is not of Type A, i.e., it is of Type B."

Neither the positive theorem nor its negation is reachable within the confines of formal set theory. This unfortunate situation maps, in the larger ambit of external truth or falsity, to paradox.


With respect to number theory, apparently the situation is much the same as with set theory. The mathematical theory that concerns how the natural numbers — the nonnegative integers — behave is intended to be both consistent and complete. Consistency means basically that all theorems within a formalized rendition of the entirety of number theory — a sort of pq-system on steroids — are capable of being interpreted in such a way as to represent true statements such as "1 plus 1 equals 2."

Completeness means basically that all true statements of number theory correspond to theorems in the formalized rendition.

Hofstadter calls the formalized rendition "TNT," for typographical number theory — since formal systems are basically ways of "typing" particular strings of typographical symbols. Gödel found an extremely clever way to devise a typographical rendition of the statement, "TNT is consistent." Then he showed that constructing TNT so that this well-formed "sentence" is actually one of its theorems automatically makes TNT inconsistent!

Looked at another way, if TNT is consistent, then the theorem which means "TNT is consistent" cannot be part of it. On that view, TNT has to be incomplete. It cannot assert its own consistency!

It is sort of like the situation faced by Yossarian, the protagonist of Joseph Heller's Catch-22. Yossarian is a WWII bomber pilot who has flown too many perilous missions and now wants to be relieved of duty. The rules say this can happen if the pilot in question has gone insane. So Yossarian attempts to convince his superiors that that is indeed the case: he's crazy as a loon. The only catch is — and this is Catch-22 — if he's of sufficiently sound mind to campaign for a diagnosis of his own insanity, then he's obviously quite sane and cannot legitimately be relieved of duty.

Likewise, if number theory can assert it's own consistency, it's inconsistent. The only way for it to be consistent is to be unable to assert its own consistency — i.e., to be incomplete, in the sense of not being able to derive or prove every single true statement about itself. This is the mother of all paradoxes.


Just the fact that many a formal system can, upon successful interpretation, lead to paradox seems to require that you be able to (again) "jump outside the system" to see the paradox. At least, that is what I think Hofstadter is getting at. Whether you jump out to detect meaningful isomorphisms or you subsequently jump out to detect paradoxes, the ability to jump out of formal systems at all is a distinguishing mark of human intelligence.

A machine can in concept derive all the theorems of any formal system, even if for certain systems it would require an infinite amount of time to complete the derivation. Also, if a decision procedure is known by which any given symbol-string can be shown to be (or not to be) a theorem of a given system, a machine can be programmed to apply it. Machines can in some cases, as I understand it, even figure out what the decision procedure for a system happens to be.

But human intelligence goes well beyond these sorts of tasks. In fact, anything along the lines of figuring out the meaning of a theorem in a formal system is decidedly non-mechanical.


There is, moreover, a gray area in which it is not clear whether or not a mechanical process could avail. Could a machine come up with something like Euclid's Proof, for example?

Euclid proved that whatever number you pick, there exists a prime number greater than it (see pp. 58ff.) He did this by finding a pattern for a series of baby steps by means of which you can assure yourself, for every number 1, 2, 3, and on up, that that number plus 1 is either itself a prime or has a prime larger than it. His insightful discovery of the right pattern for the concatenated baby steps made for a compact and wholly convincing proof — and it may or may not be one a machine could ever discover.

That seems to be because you have to jump outside a formal system such as number theory to learn certain true statements about the system and figure out proofs for those statements.


Even if finding clever proofs, such as that of Euclid, is not absolutely machine-impossible, and even if finding correct isomorphisms is ultimately susceptible to artificial intelligence, locating and understanding implicit paradoxes seems to depend wholly on human smarts.

The human mind alone is capable of dealing meaningfully with "strange loops." The activity of neurons in the brain, itself presumably isomorphic with some rigidly formal system, can somehow miraculously jump out of the system and incorporate "bolts from the blue" into its own inner workings. It can find "unreachable" truths and detect paradoxes non-mechanically.

That, indeed, seems to be Hofstadter's overarching point: there exist officially "unreachable" truths and implicit Catch-22s within formal systems ... and the human mind, because it can deal with strange loops and paradoxes that flabbergast "intelligent" machines, can nonetheless find them.

Gödel, Escher, Bach

Douglas R. Hofstadter's Gödel, Escher, Bach

If you read a book in which the clause "provability is a weaker notion than truth" appears, you might think the author was knocking science and the mathematical/logical disciplines that might be called its atheistic co-conspirators, while at the same time upholding God as the highest preeminent-if-unprovable notion in the whole body of truth.

Douglas R. Hofstadter does indeed assert that clause about provability on page 19 of his 1979 book Gödel, Escher, Bach, an Eternal Golden Braid: A Metaphorical Fugue on Minds and Machines in the Spirit of Lewis Carroll. What he means by it is the subject of the book, but it is not clear to me after having dipped into the first chapter or so whether Hofstadter thinks it has theological implications or not.

As for me, I do think certain theological implications are justified.

What led me to this book for a rereading — I read it once, or at least started it, many years ago — is my present interest in the music of Johann Sebastian Bach. Bach wrote canons and fugues that involve dizzying self-referentiality: deceptively simple musical passages echoing, mirroring, harmonizing with and reinforcing one another to make for an endless kaleidoscope of inner experience. At least one of Bach's compositions, the "Canon per Tonos" of his Musical Offering, modulates from key to higher key so repeatedly that eventually the original key is restored!

M.C. Escher's Drawing Hands (1948)

The artist M. C. Escher did for the visual what Bach did for the aural: he showed how self-referentiality works.

Escher's Relativity (1953)

He showed how topsy-turvy and paradoxical a worldview that doesn't exclude self-reference can be. In the image at right, there is no "up" — even though "up" seems to make sense with respect to individual details of the scene, such as each individual staircase.

Escher's Möbius Strip I (1961)

Escher depicted the finite-but-unbounded Möbius strip, an apt analogue for the tonality of Bach's canon circling back repeatedly upon itself.








In 1931, an Austrian mathematician, logician, and philosopher named Kurt Gödel demonstrated that such paradoxes are not just curiosities. They illustrate the proposition that "provability is a weaker notion than truth." This was a proposition he demonstrated with respect to all formal or "axiomatic" systems of thought.

Gödel proved that mathematical set theory, in which mathematicians and logicians had invested so much faith and hope for the perfection of the human logical capability, could not contain all true propositions about itself — not if it was to remain self-consistent, i.e., non-self-contradictory.

Likewise, mathematical number theory. And, by further extension, Gödel showed that no formal system could be both self-consistent (non-self-contradictory) and complete (containing all true propositions about itself).

Some true statements can never be proved, in other words, within the frame of reference that defines how such proofs are allowed to be constructed.

This dashed the hopes of philosopher-mathematicians like Bertrand Russell and Alfred North Whitehead, who had hoped in their Principia Mathematica to banish self-contradiction from logic by banning self-referential statements. Gödel's 1931 paper "revealed not only that there were irreparable 'holes' in the axiomatic system proposed by Russell and Whitehead, but more generally, that no axiomatic system could produce all number-theoretical truths, unless it were an inconsistent system!" (p. 24).

Russell and Whitehead had tried to exorcise self-referentiality from their system by imposing strict conceptual hierarchies. Their mathematical sets could contain only raw objects and (possibly) other sets whose position in the hierarchy was lower than the containing set. Set descriptors that were in any way self-referential, no matter how indirectly, were held to be meaningless. Only run-of-the-mill sets were meaningful. For example, "the set of all run-of-the-mill sets," however much our intuition says it ought to be admissible in set theory, was simply ruled out.

Loosely speaking, Russell and Whitehead stripped out of their formal system any feature that might produce confusion about which way is "up." Their hierarchy of set types was, in effect, a staircase that couldn't loop back on itself.

Such a strategy has the effect, when transferred into the realm of ordinary language, of ruling out constructions like:

The following sentence is false.
The preceding sentence is true.

In doing so, it fails to come to grips with the fact that each of these sentences is, by itself, fine.

The Principia system thereby recuses itself from being able to help our mental faculties deal with the paradoxes, linguistic and otherwise, that the mind is perfectly capable of entertaining. In being thus pigheaded about paradox, it becomes repellent to human intuition. As Hofstadter says sourly (p. 23), "When this effort [at eliminating self-contradiction] forces you into a stupendously ugly theory, you know something is wrong." The mind wants to learn from paradox, not define it away.

My thought on this is that what is wrong with the Principia system is the nullifying of the right side of the brain. That hemisphere of the cerebrum is the locus of intuitive understanding, while the left brain purveys the kind of logic Russell and Whitehead championed.

Gödel proved, basically, that the logical left side of the brain needs the intuitive right side — it cannot generate the entirety of truth on its own.

No mechanical system of logic can suffice to establish the truth of all we intuit. If we intuit that there is a soul within each of us that belongs to God, systems of thought that would have such statements invalidated as meaningless must give way.

That, I think, is a sweet way of interpreting what Gödel showed the world.