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.