Minds: What Are They, Anyway?

Douglas R. Hofstadter's Gödel, Escher, Bach

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

• Gödel, Escher, Bach
• Insight Through Isomorphism and Paradox
• A Picture of God?
• God as Genotype
• More on God as Genotype
• Hints As To the Nature of Reality
• Two Dilemmas of Consciousness
• The Incompleteness of Provability

In the last, I summed up Part I of Hofstadter's book as a discourse on "incompleteness": the inability of formal logical systems — systems by means of which "theorems" can be derived, based on axioms and rules — to encompass all knowable truths about themselves. All formal systems turn out to be cognates of number theory, when the latter is formalized rigorously. Hence, incompleteness results in there being truths concerning number theory that number theory ought conceivably be able to derive, but cannot.

So the mind is capable of knowing things — about number theory, about reality — that the brain from which the mind arises simply cannot demonstrate the truth of by operations that happen at a mechanistic level of neuron interactions. The mind, rather, attaches meanings to the symbols that it manipulates — meanings that are not, strictly speaking, justified by the lower-level, mechanistic rules of symbol manipulation.


What could this notion of the transcendence of mind over (brain) matter mean? How could it be so? Such questions are brought into focus in the second part of Hofstadter's book. Early on in Part II, the author conveys the essence of what he thinks is going on, human-brain-wise, in a pair of dialogues between the whimsical personages of Achilles, the Tortoise, the Anteater, and the Crab.

Actually, it is a single dialogue called "Prelude ... Ant Fugue," split into two unequal parts: first a short "Prelude," and then a long "... Ant Fugue." The misspelling of "ant" is intentional; read on for more on that. Between the two parts of this dialogue is interposed a chapter on how computers, those paragons of mechanistic symbol manipulation, work.

I'll discuss what Hofstadter has to say about the nature of mind, vis-à-vis brain, presently. First, though, why is this question of mind-over-brain significant from the point of view of religious inquiry?

Sharon Begley's Train Your Mind, Change Your Brain

One way to answer that question is hinted at by another book I am currently reading: Sharon Begley's Train Your Mind, Change Your Brain: How a New Science Reveals Our Extraordinary Potential to Transform Ourselves. Begley, a science columnist for The Wall Street Journal, delves into the topic of "neuroplasticity," a fancy word for the ability of the brain radically to rewire itself to incorporate new capabilities.

Neuroscientists were long under the impression that the human brain is pretty much set in its capacities and abilities by the time a person reaches late childhood, not to mention full adulthood. Wholesale rededication of large areas of the brain — say, to compensate for the catastrophe of going blind as a teenager or adult — were thought impossible. Now, research is indicating otherwise.

Instead of lacking any ability to "rezone" itself to establish new functionality, the brain is actually capable of amazing feats of neuronal regeneration and signal-path rededication. This remains so well into adulthood, and even at an advanced age.


One of the capacities that this new cutting-edge science is finding we have a chance to enhance, if we take advantage of our inbuilt but until now unsuspected neuroplasticity, is our intrinsic potential for compassion and empathy. This may be the principal reason that one of the individuals looming largest in Begley's book is none other than the Dalai Lama.

Begley's book is structured around a series of annual conferences, hosted by His Holiness at his home in Dharamsala, India, that have brought together some of the world's leading scientists to exchange views with experienced practitioners of the ancient wisdom of Buddhism, so that they might discuss what their worldviews have in common. And nowhere is that commonality more evident than when it comes to the cultivation of compassion.

"In Buddhism," Begley writes (pp. 184-5), the greatest wish is 'May the suffering of all sentient beings be relieved' — the very definition of compassion." Along these lines, the research of social psychologist Phillip Shaver has shown that there are many people whose childhood experience has left them with brain circuitry that disposes toward being (in Begley's description, p. 184) "insecure, closed-minded, deluded, biased, defensive, and selfish." Even such people can successfully be primed make the switch to compassion, empathy, and altruism, Shaver has learned.

Shaver's methodology for turning a human being on to compassion involves enhancing what he calls the person's "attachment security," an aspect of the individual's mental apparatus which has typically been impaired since childhood — due possibly to, among other factors, a lack of nurturing from the person's mother.

The Buddhists have a different methodology to achieve the same end, Begley says: "compassion meditation," which she calls "one of the primary forms of mental training for monks, yogis, and other practitioners" (p. 185). Whatever the methodology, it seems to me (and in this I believe I am echoing the Dalai Lama himself) that compassion for other humans, and indeed for all sentient creatures, is the final destination of human religious awareness and practice, no matter what the person's religion happens to be.

Compassion for creatures who, like us, feel is a desideratum that is not just a religious concern. It is also the goal of what the Dalai Lama calls "secular ethics." His Holiness, whose English is apparently less than perfect, says

... the key thing is the peaceful mind. Naturally and obviously, anger, hatred, jealousy, fear, these are not helpful to develop peace of mind. Love, compassion, affection — these are the foundations of peaceful mind. But then the question, how to promote that? My approach, not through Buddhist tradition, I call secular ethics. Not talking about heaven, not of nirvana or Buddhahood, but a happy life for the world. Irrespective of whether there is next life of not. Doesn't matter. That's individual business. (p. 180)

Recognizing this to be so only reinforces the importance of sentience. To be an agent of compassion toward other sentient beings, it would seem that one must oneself be sentient. One must have a mind that is itself conscious and self-aware. Then and only then does the desideratum of compassion have real meaning.

Which brings me back to my main topic, which is the way Douglas Hofstadter characterizes the conscious mind in Gödel, Escher, Bach.


The "Fugue" portion of the cleft-in-two dialogue involving Achilles, the Tortoise, the Anteater, and the Crab is the nut of Hofstadter's argument about what sentience or consciousness is all about.

The mind is to the brain-as-a-collection-of-neurons as an ant colony is to the population of ants that make it up, Hofstadter suggests. (This is why the dialogue is called "Prelude ... Ant Fugue.") In the dialogue, the Anteater claims to be able to exchange information with his favorite ant colony, fetchingly named Aunt Hillary.

It seems that he, Dr. Anteater, knows how to interpret Aunt Hillary's thoughts by reading the ant trails set up by the to'ing and fro'ing of her constituent ants. Inasmuch as his own arrival upon the scene, which communicates itself to Aunt Hillary via his scent wafting on the air, results in a decided change of trail-making behavior among the ants within the colony, any fool can see that a sort of two-way conversation is in progress there.

In an ant colony, there are multiple castes of ants whose distributions around the colony are neither uniform nor random. The various local distributions of individual castes are, the Anteater says, analogous to pieces of knowledge stored in a brain. The overall caste distribution, then, is like a "brain state."

Moment-by-moment caste distributions within an ant colony respond to things going on in the vicinity of the colony. By virtue of teams of ants banding together and moving as one through the colony, information such as the discovery and location of new food can be disseminated.

In fact, the information superhighway of an ant colony is composed not just of teams of ants, but also of teams of teams, and teams of teams of teams. on up to quite a high level of team-within-team "nesting." Hofstadter's Anteater considers the highest-level teams to represent "symbols" and the intermediate-level teams to constitute "signals." These are concepts that apply equally well to the brain.

A symbol is analogous to, in a human language, a word that is distinguished by its ability to carry a meaning. A signal, on the other hand, is like a letter that bears no intrinsic meaning. But symbols and signals are active, while words and letters, since they are external to the brain, are passive. In order for a passive word to trigger in the brain a conceptual meaning, it has to activate an active symbol housed in the brain.

Likewise, each passive letter in a word conceivably triggers an active signal path in the brain. Thus do we recognize the difference between, say, "red" and "read," or "red" and "rod." But when the brain is confronted with words and letters in an unknown language — for example, in my case, Hebrew — the appropriate signals and symbols do not, shall we say, "light up" in the brain.

Symbols and their associated meanings are the stuff of consciousness. In a conscious brain there are active symbols that "reflect the overall state of the brain itself" (p. 328). When a mind has access to such self-referential symbols, it is conscious or self-aware. "For consciousness requires a large degree of self-consciousness," observes the Anteater.

Symbol manipulation is accordingly what a mind basically does. In particular, the human mind has access to (and only to) the active symbols that compose the highest levels of thought in the brain. Aunt Hillary is no exception here: though not human, she too is a symbol manipulator. As such, she is the "agent" by virtue of whom her internal symbols can be said to be active rather than passive.

Put another way (see p. 327) Aunt Hillary is a "full system [that] is responsible for how its [own] symbols trigger each other [such that] the state of the system gets slowly transformed, or updated [over time]." If Aunt Hillary were human, it would be her mental activity — her mind — that would provide the continuity to constrain the changes taking place in what would otherwise be her potentially chaotic, ever tumultuous "brain state."

In fact, as a self-aware agent of change, Aunt Hillary is unique in her ant-colony identity. Though she happened to be composed of the same exact population of ants as one Johant Sebastiant Fermant, her predecessor ant colony who met an untimely demise after a freak thundershower, the unique organization of the constituent ants that made J.S.F. who he was was irretrievably gone. Aunt Hillary inherited the ants but not the organization. The notion is akin to how a human body might conceivably be made of the same atoms as one which lived before and is now dead ... but the soul would be unique.

In fact, it is the implications of all this concerning the soul which I will take up in the next post in this series.

The Incompleteness of Provability

Douglas R. Hofstadter's Gödel, Escher, Bach

Douglas Hofstadter is the mathematician-cum-philosopher whose magnificent 1979 book Gödel, Escher, Bach: an Eternal Golden Braid is the subject of these "Strange Loops" posts to this blog. To date, they are:

• Gödel, Escher, Bach
• Insight Through Isomorphism and Paradox
• A Picture of God?
• God as Genotype
• More on God as Genotype
• Hints As To the Nature of Reality
• Two Dilemmas of Consciousness

In the last of those I talked about the fact that, as Hofstadter shows with great intellectual rigor, "our naive hopes of lassoing reality with our words, thoughts, and formal expressions are in vain. Moreover," I added to that, "they cannot be otherwise."

That is the lesson of the first of the two parts of Hofstadter's book. I am now in the early going of reading the second part. A bit further on, I can tell already, Hofstadter will give a yet more rigorous proof of how truth-as-knowable is ineluctably larger than the set of truths that can be proven to be true.


How strange it is, though. This intrinsic "incompleteness" — this shortfall of that which is rigorously provable, vis-à-vis the totality of all propositions that we otherwise are able to know to be true — can itself be proven with full analytic rigor!

The provability of the incompleteness of provability depends on a trick discovered by the mathematician Kurt Gödel. By means of it, any and all formal systems of logic can be morphed into theorems about ordinary, everyday numbers. Then number theory itself, Hofstadter shows, can be extended to include various theorems that refer to ... number theory itself!

That is, these "meta-theorems," like regular theorems such as A + B = B + A, qualify as statements about the arithmetic properties of "natural" numbers (the integers zero and above). After all, just because they also represent statements that say something about number theory itself doesn't mean they aren't equally interpretable as mere statements about numbers. This is because all theorems of number theory are statements about numbers, whatever else they may be statements about.


Once one has all that firmly in mind, then one can ask whether a particular meta-theorem — one that Hofstadter names G, and whose interpretation is "G [itself] is not a theorem" — is in fact a theorem of number theory.

Pay close attention: "G is not a theorem" could conceivably, yes, be a derivable theorem!

Yet, if we quite reasonably assume that number theory is free of such internal contradictions, then, according to Gödel's Incompleteness Theorem, the question of whether G is or is not a theorem is simply undecidable, at least within the confines of number theory per se. Hence number theory is, contrary to our naive expectations, incomplete with respect to the whole body of knowable truths about itself.


Because Gödel's trick can be used to turn any formal system of logic whatsoever into a body of number-theoretical constructs, all formal systems are likewise incomplete in the very same way.

At the same time, we whose minds operate outside the confines of such formal, mechanical systems of provability know perfectly well that G expresses a true statement: namely, that G is in fact not a theorem of number theory.

Hofstadter summarizes the situation thus:

A string [of the specific formal system called Typographical Number Theory] has been found; [this string, G] expresses, unambiguously, a statement about certain arithmetical properties of natural numbers; moreover, by reasoning outside the system we can determine not only that the statement is a true one, but also that the string fails to be a theorem of TNT [because if it was a theorem, TNT would contain an internal contradiction]. And thus, if we ask TNT whether the statement is true, TNT says neither yes nor no. (p. 272)

The mind thus transcends the merely mechanical or "typographical" workings that are the essence of all formal systems of logical provability. This is so despite the reasonable conjecture that at bottom, the mind is rooted in the deterministic workings of the brain, and those workings are presumably entirely "typographical."

Hofstadter uses the word "typographical" to express the idea that you can represent the workings of the brain, at a low neuronal level of operation, by the workings of any reasonably powerful formal system of logic such as his Typographical Number Theory. Such a system, however powerful it may be, does what it does by virtue of nothing but the rote manipulation of strings and symbols. It is nothing more than an (abstract) machine.

To wit, the strings of TNT and other mechanistic formal systems are made up entirely of symbols such as letters and numbers. Unambiguous rules determine which typographical manipulations are permissible within the system. By applying aptly chosen rules one at a time, starter strings called "axioms" can be turned into all manner of full-fledged theorems.

In addition to their vacuous significances within the system itself, these mechanistic theorems can have, outside the system per se, one or more true interpretations that present themselves quite readily to the mind. Still, all that is really going on within the formal system itself is that new strings are being mechanically derived on the basis of rules and old strings. Meanings and truths are extraneous to the inner workings of such formal, typographical systems.


Yet the mind deals in truth and meanings more readily than it is capable of dealing, computer-like, with symbol strings. The brain, which is at base mechanical — or so it would seem — is able to transcend the intrinsic limits of the mere mechanistic processing of symbol strings, à la formal systems.

String processing à la formal systems is presumably wholly analogous to what the brain does, and all that it does, at the low level of its neurons. But at a higher level, the symbols take on meanings that are simply unavailable at lower levels of processing.

What could this notion of the transcendence of mind over the matter of the brain mean? And how could it be so, that a mind is seemingly more than just the sum of its working parts?


Such questions are the principal topic of the second part of Hofstadter's book, at least as far as I have read into it to date. I'll return to them in the next post in this "Strange Loops" series.

Two Dilemmas of Consciousness

Douglas R. Hofstadter's Gödel, Escher, Bach

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

• Gödel, Escher, Bach
• Insight Through Isomorphism and Paradox
• A Picture of God?
• God as Genotype
• More on God as Genotype
• Hints As To the Nature of Reality

In the last of those I talked about human (and possibly, someday, machine) intelligence as able to know the truth of this unprovable statement: "I am conscious."

Consciousness is the brain's faculty of monitoring whatever other processes of awareness are going on inside it and knowing what it is like to undergo those processes. The processes are, at some level, presumably mechanical and "typographical," just like the formal systems Hofstadter so lovingly describes in his book. His Typographical Number Theory, or TNT, is one such formal system; it is an attempt to turn mathematical number theory into something completely mechanical, hopefully without flaw or omission.

Zen consciousness, on the other hand, is knowing that what we tell ourselves about our mental states, using words to describe what they are like, is necessarily a lie.

Or, rather, what we tell ourselves about reality, using expressions made of words, can be at once neither true nor false — in many cases a verbal expression's truth or falsity is ultimately undecidable, and moreover, whether it is true or false ist, to a practitioner of Zen, beside the point. Satori, or Zen consciousness, lies beyond all words and all the concepts embodied by words.

Hofstadter gives his readers the flavor of Zen awareness in Chapter IX, "Mumon and Gödel." Mumon, he says, was the thirteenth-century Zen master who wrote the Mumonkan ("The Gateless Gate"), in which many of the best kōans or Zen enigmas are to be found.

Here is a kōan from Mumon's collection (p. 251):

Shuzan held out his short staff and said: "If you call this a short staff, you oppose its reality. If you do not call it a short staff, you ignore the fact. Now what do you wish to call this?"

There is no "right" answer — that's the point! From a Zen perspective, you cannot capture the "reality" of anything — even something so prosaic as a short staff — by naming it, thus putting it in contradistinction to all which is not a short staff. Reality is entire. It cannot be subdivided and named this, this, and this. To say at any time "That is not this" is to shortchange the totality of what is real.

But there are brute facts about reality as well, and refusing to name them is equally "wrong." Naming, which is an act of separating reality into manageable chunks, is at once futile and absolutely indispensable.

"Thus," writes Hofstadter (pp. 252-3),

... words lead to some truth — some falsehood, perhaps, as well — but certainly not to all truth. Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system — no matter how powerful — cannot lead to all truths. The dilemma of mathematicians is: what else is there to rely on, but formal systems? And the dilemma of Zen people is: what else is there to rely on, but words? Mumon states the dilemma very clearly: "It cannot be expressed with words and it cannot be expressed without words."

I think it fair to say that Hofstadter, at some early point in his life as a mathematician, had implicit confidence that formal systems (or "axiomatic" systems, as they are also known) were a way around reality's obstinate refusal to be encompassed by words. If you began with a mathematical theory such as that of numbers qua numbers, and if you took that number theory out of the domain of fuzzy, slippery verbal expressions such as "the number of primes is infinite" — what exactly is meant by "infinite"? — and put it in the form of non-verbal (indeed, non-arithmetical) symbol strings that can be generated and manipulated by mechanical rules, you could (Hofstadter expected) capture truth — potentially all truth — simply by generating all the "theorems" of the formal system, one by one.

True, there would be truths that were unreachable by any particular formal system of number theory. A possible example Hofstadter gives in his Chapter III on "Figure and Ground" concerns the prime numbers. A number not divisible by any number except for itself and 1 is prime. For example, 7 is a prime number, because it has no divisors. All other numbers are composite. 6 is composite because 2 and 3 divide it.

Hofstadter describes a simple formal system, the "tq-system," which has an easy time of generating all the composite numbers. The question then is, is it fair to say that any number which it does not show to be composite is accordingly prime?

No, he answers. The "holes" in the tq-system's list of composites are only negatively defined — which isn't good enough, in a formal system. Hofstadter gives the pictorial analogy of a "figure" and its "ground." If something in the picture is not part of the object being limned by the artist, it must be part of the background. Right?

Not necessarily! In visual depictions by M.C. Escher and other artists which Hofstadter alludes to, the distinction between figure and ground is purely arbitrary. What is figure from one perspective serves just as well as ground from another.

So, is there a way to modify the tq-system so that primes become, not ground, but figure?

Turns out there is — and Hofstadter gives it on pages 73-74. Yet, he says, from the fact that figure and ground, positive and negative, are perfectly complementary in this one example, we are not entitled to generalize to a belief that figure and ground always carry precisely the same information — as do the negative of a photograph and a positive image developed from it!

Specifically, when it comes to formal systems,

There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any [other] formal system. (p. 72)

Surprise! As a budding mathematician, Hofstadter says he found this upsetting of his initial intuitions — this obstinate refusal of reality to be tamely, predictably either this or that — quite astonishing.


It is as if, in the "picture" which is reality, there is figure, there is ground, and there is a no-man's land which is neither one. Hofstadter's Fig. 18 on p. 71 gives some idea what this picture looks like. It diagrams the "relationship between various classes of TNT strings."

The diagram contains a white tree of axioms and theorems to be found in Typographical Number Theory, and also a black tree of all their negations — which, as "non-axioms"/"non-theorems," are equivalent to falsehoods in the verbal world.

Surrounding the white and black trees is a large area, black in the neighborhood of the white tree and white in the neighborhood of the black. The black area surrounding the white tree represents truths of number theory that are "unreachable" by the formal TNT system, no matter how powerful it may be spruced up to be. The white area around the black tree depicts TNT's set of "unreachable" falsehoods. Together, these two areas contain all the "sentences" of TNT whose truth or falsity simply cannot be decided within the formal system.


Undecidability is very Zen. It is even very verbal, in the sense that sentences in human languages are typically to be suspected of allowing too much interpretive wiggle room to be declared absolutely true or absolutely false. But undecidability is something a naive young mathematician might hope to eliminate in formal systems which are designed to be "isomorphic" with — to capture the same truths as — mathematical theories.

Hofstadter shows that it is possible to "transfer the study of any formal system — in fact the study of all formal systems — into number theory" (p. 264). You just substitute for the original typographical symbols specified in the formal system — whatever symbols they may be — numbers which can be manipulated arithmetically, not just typographically. When you do that, you necessarily have a system that is subject to all of the benefits and constraints of number theory.

One of these constraints, as the Austrian mathematician Kurt Gödel proved in 1931, is incompleteness — some of the "sentences" of the formal system that has, by means of this symbol-substitution process that is so aptly called "Gödel-numbering," been turned into an arithmetical "isomorph" are necessarily undecidable in their truth or falsity.

Once you twig to how Gödel-numbering unfailingly maps any formal system into the domain of number theory, and once you understand that number theory is ineluctably incomplete in terms of the decidability of some of its well-formed "sentences," you lose hope that all truth is somehow expressible. Words fall short. Numbers fall short. Well-formed strings of symbols in formal systems fall short.

And all fall short in the same way, in fact. As Mumon wrote of reality, "It cannot be expressed with words and it cannot be expressed without words."


The two dilemmas of consciousness Hofstadter speaks of — the one pertaining to Zen people and the other pertaining to mathematicians — are really one. This is why the possibility of Gödel-numbering any formal system, which Hofstadter calls "the simple observation ... at the heart of Gödel's method" of proving his Incompleteness Theorem, has "an absolutely shattering effect" (p. 264).

It is shattering because it means our naive hopes of lassoing reality with our words, thoughts, and formal expressions are in vain. Moreover, they cannot be otherwise.

If heaven is a perfect place where mind can lasso matter and make it behave, then we are barred from it, no matter how cleverly we use our brains. We have a better chance of finding heaven by "stepping outside of logic" (p. 251), or by seeking to "break the mind of logic" (p. 249). That is satori. That is Zen consciousness.

Religious Ignorance in America

A book review of Stephen Prothero's Religious Literacy: What Every American Needs to Know — and Doesn't appeared over the weekend in the Book World section of the Sunday Washington Post. The review is by Susan Jacoby, author of an excellent book on irreligion in America, Freethinkers. (I discussed Jacoby's book in several earlier posts, beginning with this one.)

The premise of Prothero's book, quite frankly, shocks me. Jacoby introduces the topic:

The United States is the most religious nation in the developed world, if religiosity is measured by belief in all things supernatural — from God and the Virgin Birth to the humbler workings of angels and demons. Americans are also the most religiously ignorant people in the Western world. Fewer than half of us can identify Genesis as the first book of the Bible, and only one third know that Jesus delivered the Sermon on the Mount.

I never would have guessed this. Nor would I have supposed that:

Approximately 75 percent of adults, according to polls cited by Prothero, mistakenly believe the Bible teaches that "God helps those who help themselves." More than 10 percent think that Noah's wife was Joan of Arc. Only half can name even one of the four Gospels, and — a finding that will surprise many — evangelical Christians are only slightly more knowledgeable than their non-evangelical counterparts.

This boggles my mind. It may not be surprising that the tiny minority of Americans who claim no religion whatever are thus ignorant of that which they deny. But that the nominally religious among us are in many ways clueless about what we supposedly believe in flabbergasts me.

According to the recent Baylor Religion Survey (see this post; the survey report itself can be downloaded here) nearly 90 percent of Americans say they belong to one broad religious tradition or another, and only about one in 20 Americans claims to be an atheist. Most of us (81.9 percent) are Protestant or Catholic Christians — at least nominally, that is. 2.5 percent of us say we are Jewish, and 5.9 percent of us identify ourselves as members of "other" broad religious traditions.


Where is the massive ignorance of religion on the part of religious believers themselves coming from? One source is, writes Jacoby, "the Second Great Awakening of the early 1800s. The fervor of America's periodic cycles of revivalism, rooted in a personal relationship with God rather than in theology handed down by learned clergy, has always had a strong anti-intellectual as well as spiritual component."

Then there is the secular, 20th-century abdication of "the Protestant-influenced 19th-century schools [which were] an important factor in maintaining the Puritan heritage of Americans as 'people of the book.'" "The once ubiquitous McGuffey readers," Jacoby points out, "rendered the Ten Commandments in such rhymes as, 'Thou no gods shall have but me/ Before no idol bend the knee.'" That synergy between literacy teaching and religious education no longer happens.

Moreover, "a bland tolerance that, while vital to pluralistic American democracy, has also discouraged our awareness of religious distinctions," says Jacoby.

Finally, over the course of the last few decades electronic media have delivered a death blow to Americans' religious literacy. "Many of the religious allusions and metaphors explained by Prothero in his glossary were once as common as the universal reference points now supplied by television."


Jacoby parts company with Prothero when it comes to what to do about religious ignorance. Prothero wants there to be "high school and college courses dealing with the historical and cultural role of religion." There is, after all, no First Amendment ban on teaching about religion, as there is on religious advocacy in schools. (By the way, Jacoby also makes this point: "According to polls conducted by the National Constitution Center, only one third of Americans can name even one of the rights guaranteed by the First Amendment.")

Even so, Jacoby is down on bringing the history of religion into school curricula:

But given the failure of so many schools to inculcate the most elementary facts about American history, it is hard to imagine that most teachers would be up to the task of explaining, say, the subtleties of biblical arguments for and against slavery. Furthermore, a curriculum that would meet with the approval of Catholic, Jewish, Muslim, Protestant and nonreligious parents would probably be a worthless set of platitudes.

Why is she so defeatist? Maybe it's in part because of this historical footnote:

In 1880, the average American still had only four years of schooling (although the figure was higher in cities than in rural areas). Yet 19th-century autodidacts, including Abraham Lincoln (who had less than a year of formal education) and Robert Green Ingersoll, the orator known as "the Great Agnostic," achieved both religious and secular literacy by reading Shakespeare and the King James Bible without any prompting from teachers.

In other words, if Americans are religiously illiterate today, the schools need accept neither the blame nor the burden of remedying the situation.


I think Jacoby's argument, if it can be dignified as such, is shallow and fallacious. There seems to be within it the tacit (and, I hope, false) assumption that religion simply cannot be taught about in schools in this country. If you try, the effort will either collapse under the weight of squabbling about whose religions will be covered and how, or it will ineluctably become a Trojan horse for religious advocacy on the part of those whose passionate fervor exceeds their sense. (How many Americans, by the way, know what the original Trojan horse was?)

I admit the danger Jacoby alludes to, but I see that danger as a symptom of the very problem under discussion. People who are not educated to be able to discuss ideas coolly and sensibly will inevitably be prone to having their buttons pushed by fulminators and bloviators of the worst kind. Some will fall for the rhetoric which says teaching about religion qua religion is anathema in a secular society. Others will fall for the rhetoric from people who say teaching about any religion other than their own is the devil's work.

It is yet another case of excluding the very possibility that there is a middle ground. We do that all the time in these days of polarization. In this instance, the middle position which says people in general are capable of learning to use their brains properly is the one that gets excluded by stances like Jacoby's.

Though I have not yet read Prothero's book, I imagine that I am in agreement with him on this one, and not with Jacoby. I believe that if we Americans have become anti-intellectual to the point of cultural, historical, and religious illiteracy, then we ought to collectively roll up our sleeves and do something about it!

Hints As To the Nature of Reality

Douglas R. Hofstadter's Gödel, Escher, Bach

Douglas 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 this blog. To date, they are:

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

In the last-but-one of those posts I discussed the notion, which Hofstadter seems to suggest, that maybe God can be thought of as the implicit meaning, the inner message, the "seed-genotype" of reality, where reality can be considered to be the "phenotype" that emerges as the "seed-genotype" — God — is worked out fully over the course of time.

The analogy here is to the relationship of DNA in the nucleus of a cell to the body of an organism. The DNA in each cell is the genotype, and the body with all its various hereditary characteristics is the phenotype.

In the last post I extended that thought by suggesting (with Hofstadter, I dare to imagine) that not all truths about reality can be "derived" in a mechanical way of a sort that is appropriate to working out a formal system of logical implication. Such a formal system is Hofstadter's own "Typographical Number Theory" or "TNT." Intelligence, however, can jump out of TNT or any other formal reasoning system and twig to these "unreachable" truths in a more intuitive way.

All formal reasoning systems, or at least those of palpable power, are thus "incomplete," in the sense that there are always going to be truths that they "ought to" be able to derive, but can't. This is what the Austrian mathematician Kurt Gödel proved in 1931. The only loophole to Gödel's Incompleteness Theorem, it seems, is that a formal system that is "inconsistent" — wherein theorems of the form x and ~x, where the former means "x is true" and the latter means "x is not true," co-occur — can in fact be complete. Not a very enticing loophole, that.


The reader is apt to lose sight of notions like those two — God as the seed of reality and intelligence as perceiving truths that are not formally provable — as Hofstadter goes deeper and deeper into the mechanics of formal systems of proof. Or, rather, formal systems of derivation, since proofs are not, strictly speaking, what formal systems do. Proofs are things that we do when we are engaged with informal systems, such as mathematical number theory itself. When number theory per se is rigorously formalized, it becomes a thing like Typographical Number Theory ... and proofs necessarily give way to derivations.

See? In coping with that last paragraph, you probably lost track of the fact that it began with notions about God, reality, and truth. By the time you finished parsing the paragraph's later clauses relating to formal and informal systems, proofs, and derivations, all the stuff about God, reality, etc., had very likely evaporated from your awareness.

That is exactly the position I find myself in at this juncture in my reading of the book: Hofstadter's Chapter VIII, on Typographical Number Theory or TNT, which is dedicated to showing how to construct a formal system in order to express and derive (many of) the truths of mathematical number theory from just five axioms.

For example,

∀a:(a + 0) = a

is an axiom that expresses the meaning "For all natural numbers, represented by a, a plus zero equals a." (A natural number is any whole number, including zero, that is not negative.)

From that and four other axiomatic starter strings can be derived, by the judicious application of a few powerful yet mechanical rules, a wealth of other strings, which together form the body of theorems of TNT. Hofstadter goes to some lengths to show that (in the absence of one particular rule he brings in expressly to remedy the situation) one of the strings that cannot be derived is, surprisingly,

∀a:(0 + a) = a

which means "For all natural numbers a, zero plus a equals a."

In other words, the formal string above which contains a + 0 is not identical to that which contains 0 + a. Hence it is conceivable that one of them is a theorem of TNT and the other one isn't. Never mind that we interpret the two expressions as meaning the same thing, mathematically speaking. Interpretations are something we do outside the confines of the formal system itself. A formal system qua formal system has no such interpretive content whatsoever! All it has, at bottom, is form, and rules for the generation of new forms based on old.


The specification of the extra rule saves TNT from something called ω-incompleteness, which means the formal system cannot derive a summarizing theorem to capture an infinitude of specific examples of what the theorem ought to "say," all of which are indeed theorems that the system is able to derive. (ω, by the way, is the Greek letter "omega.")

ω-incompleteness is not the same as incompleteness-without-the-ω, which happens when the formal system cannot derive as a theorem something we otherwise "know" is true. For example, if TNT were not as well-designed as it is, it might not be able to derive the TNT string for "There are infinitely many prime numbers." As it is, this truth of number theory, long known to mathematicians as Euclid's Theorem, since Euclid first proved it, can be derived within TNT — but Hofstadter says (p. 228) it would "probably have doubled the length of the book" to have shown the derivation.

So TNT is not incomplete with respect to Euclid's Theorem, and whether it is incomplete or not with respect to other "known" truths of number theory is apparently immaterial at this stage of the discussion. (It is in fact incomplete, Gödel showed — but for now Hofstadter wants his reader to focus on the notion that it might be complete, irrespective of whether or not it is in any way ω-incomplete. Got that?)

In some cases, Hofstadter shows, it is even possible to intentionally leverage ω-incompleteness into a (desirable, if offputting) ω-inconsistency: the "obvious" summarizing theorem is negated, quite on purpose, in the formal system order in order to allow for (in Hofstadter's image which I take from p. 223) "supernatural" numbers which do not necessarily obey the "obvious" formulas.

This type of inconsistency is not really as problematic as it might seem to the lay person. It is akin to that which separates non-Euclidean geometry from the straight-up Euclidean kind we learn in school. Non-Euclidean geometry is essential to some areas of science, and so we see we must steel ourselves to, and even welcome, various kinds of ω-inconsistency in the formal systems which underpin math and science.

We will presently be told, as the percipient reader already knows, that there are real inconsistencies that inexorably plague formal systems, however, and they can indeed be seen as problematic. In fact, Kurt Gödel showed that extirpating such inconsistencies completely comes at the cost of rendering the formal system (if it purports to be as powerful as Typographical Number Theory) incomplete-without-the-ω.


When wading through material such as this, the reader can easily lose sight of the forest for the trees. This is particularly true because the forest seems (at least to me, at this juncture) to pose a bit of a paradox in its broader outlines.

One of the paradoxes stems from the fact that Hofstadter seems to be at one and the same time extolling the beauty of formal systems and disparaging them as ineluctably inconsistent and/or incomplete.

TNT affords Hofstadter an elegant, 56-line derivation of a theorem that represents the number-theoretical truth that every number "commutes" with every other, meaning that it doesn't matter in which order two numbers bracket a plus sign; their sum is the same. Hofstadter breaks the lengthy derivation up into chunks whose last lines represent points of tension and resolution in the flow of the logic.

To wit, when a preliminary, subsidiary derivation augurs the final coda in the main derivation, a sense of inevitability is set up in one's mind. The situation is much like when music sets up an expectancy of eventual resolution at the keynote.

This alternation of tension and resolution is "typical of the structure," Hofstadter writes,

... not only of formal derivations, but also of informal proofs. The mathematician's sense of tension is intimately related to his sense of beauty, and it is what makes mathematics worthwhile doing [p. 227].

He then adds:

... that in TNT itself there seems to be no reflection of these tensions. In other words TNT doesn't formalize the notions of tension and resolution ... any more than a piece of music is a book about harmony and rhythm.

And then he leaves open the question, "Could one devise a much fancier typographical [i.e., formal] system which is aware of the tensions and goals inside derivations?"


If the answer to that question is no, one is forced to conclude that human intelligence is constituted so as to transcend all forms of purely mechanical reasoning, since our minds are in fact attuned to such tensions and resolutions. The next question then is, how so? How could a brain do things beyond mechanical and typographical rote — things that no machine, no matter how elaborate we make it, could ever hope to emulate?

On the other hand, if Hofstadter's question has an affirmative answer, and machines could be built that monitor the qualia of their own mechanical transactions, then those machines — we would be forced to say, would we not? — would be conscious. They would have subjective experience, just as we do.

Yet, though conscious like us, these super-machines would presumably be as unable as we are to prove — or in their case, derive — the truth of their own consciousness!

I think that is the general shape of the forest within which formal systems like TNT represent mere trees.


The nature of our reality is, then, one in which our consciousness transcends the mere mechanical operations of formal logic. It is deeply illogical, when you come right down to it. There is no path of reasoning by which one could derive the fact that we are conscious from anything else we know about ourselves and the world we live in. Or so it seems.

Now, it may or may not be the case that machine-based artificial intelligence can aspire to emulate our human version of consciousness. Yet, even if that AI goal is someday reachable, the brute unprovability of the fact of consciousness would remain. Machine-based consciousness would be just as deeply illogical as human consciousness is!

Accordingly, as Hofstadter lingers over the beauties of derivations in formal systems like TNT, I believe I can sense an inner tension to his discourse that can only be resolved by a conclusion like the one I just mentioned. Consciousness is deeply illogical, and therefore inexplicable.

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.