In 1979, Douglas Hofstadter, a 27-year-old mathematician-turned-computer scientist, and an expositor of sublime originality, produced an extraordinary, Pulitzer Prize-winning bestseller entitled Gödel, Escher, Bach: An Eternal Golden Braid. Written as a series of word games, dialogues and skeins of symbolic logic (with actual mathematical symbols!), it was a tour de force presentation of “thinking about thinking”, or what, in the style of Hofstadter, I will call TAT for short. He exemplified TAT through the mathematical implications of the work of three unique geniuses: the 20th-century logician Kurt Gödel, the artist M.C. Escher, and the composer Johann Sebastian Bach.
The relevance of Bach and Escher to Hofstadter’s TAT project was not intuitively obvious, but Hofstadter’s unique gift was to bring it all into easy focus. At the height of the Baroque period, Bach managed to produce music of complexity and beauty within the confines of rules so constraining that others less brilliant could not make a fugue proceed past the fourth voice; Bach succeeded with six. Bach’s genius was to loop his music’s voices around and past each other, creating an evolving recursive pattern that sounded as magnificent as it was mathematically elegant, and could even end up making musical reference to itself. This possibility of self-reference is key.
Escher was the pre-eminent artist of visual paradox. His drawings and etchings are profound and realistic depictions of “realities” that cannot exist in three-dimensional space. Looking at any small piece of Escher’s worlds causes no trouble, but the pieces don’t coalesce properly when taken as a whole. They can even contain themselves within themselves. This visual and paradoxical “looping” provides a visualization of the complexities and contradictions of TAT.
What did the logician Kurt Gödel have to do with TAT? Plenty, as it turns out. To start at the end, so to speak, Gödel ended the dream of early 20th-century mathematics that “everything could be axiomatized”—that all of mathematics, and then all of everything, could come spilling out of an all-powerful computer once “fed” the right (and assumed to be simple) axioms. Some of these axioms were from number theory, these being necessary to understand and “prove” things from 1+1=2 up to Fermat’s Last Theorem. Beginning with these mathematical first principles, it was hoped, we could deduce the full natures of chemistry, biology or any of the natural sciences—including psychology. Indeed, perhaps only a few more such axioms would allow us to fathom the mind and consciousness itself. One could then really think about thinking, systematically and scientifically, too.
This expectation presumed a certain philosophy of mathematics, and a certain wider cosmology. Before Gödel, mathematics was taken to be both the descriptive and predictive language of physics, astronomy and cosmology. And why not, since mathematics and the physical sciences use the same machinery: the human brain, with its extraordinary abilities to conceptualize, formulate and generalize? It was further supposed that mathematics was the unambiguous language through which to communicate about the natural sciences. It was less prone to the ambiguities and misunderstandings of everyday human languages. Logically speaking, mathematics had a place for everything and put everything in its place.
If mathematics gives us a precise and exhaustive means of describing the natural world, then why not use it to discuss, and even understand, the human brain itself, which, after all, is part of that world? To do that, of course, creates a kind of loop. We would be using our brain to think about how our brains work—to “think about thinking.” And “thinking about thinking” is the quintessential human trait, the cogito ergo sum of René Descartes and the hallmark definition of philosophy itself from ancient Athens. This, anyway, is what a lot of very smart people set out to do about a hundred years ago.
Gödel doubted the possibility of this project for logical reasons, and to justify his doubts he went after one of the great minds of the first half of the 20th century, Bertrand Russell. Working with Alfred North Whitehead, Russell attempted to fully automate mathematics by banishing “the paradox of self reference”, and incidentally Hofstadter’s TAT itself. A delightful introduction to Russell, and the era of mathematics he represented, is, in his own words:
Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. . . . If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.1
M.C. Escher’s “Waterfall” [credit: © 2007 The M.C. Escher Company-Holland. All rights reserved.]
Perhaps the primal self-referential thought is “I am lying”, which can be neither true nor false if taken literally as an abstract thought. The problem lies in the fact that the sentence refers to the writer, the “I.” The Russell-Whitehead Principia Mathematica (named immodestly after Isaac Newton’s Principia) was intended to banish the “I” and create the “perfect language”, an objective language in which the very possibility of irrationality—thoughts that are neither true nor false—would be logically impossible. With the perfect language one could then set the stage for programming that notional computer that could give us, in Douglas Adams’ glib phrase, the answer to life, the universe and everything.
Gödel single-handedly demonstrated that the Russell-Whitehead project was doomed to failure. What he showed in his famous incompleteness theorems was that in any axiomatic mathematical system there are, of necessity, true statements that could be proved neither true nor false within that system. It was not possible to banish the “I”, or all self-referential statements, or to occupy the Archimedian reference point of pure objectivity. Thus the “program” of building comprehensive and exhaustive mathematics and science from a few simple axioms cannot be.2 But, in its place, the ability of mathematical statements to refer to their own meaning emerged. Ah, the possibility of self-reference is alive: It is to become the “I” of I Am a Strange Loop.
In computer science a (roughly) equivalent outcome is the Church-Turing notion of “non-computability.” This notion states that some perfectly correct outcomes can never be reached on a computer in any finite computation, and that no computer can ever tell us in advance which are the computable and the non-computable outcomes! Gödel was delighted with this, and grateful he lived to see it. This proved to him that the mind was not a machine, and that human intuition and intelligence were of a special order. It was not a biological computer following a fixed set of rules. The loop, the contradiction, the recursive, the self-referential were irrepressible. There is no way around it.
Ironically enough, then, starting directly from the groundwork laid by Principia Mathematica, Gödel turned logic, mathematics, computer science and perhaps much more upside down, leaving wreckage everywhere. Nonetheless, Gödel left almost everyone to continue daily work uninterrupted, as neither everyday life nor creative thinking is actually based on mathematical logic, at least not at any level we would notice. Successful physicists, mathematicians and psychologists, theologians and musicians, too, for that matter, don’t study “mathematical logic” to know what to do next. They follow experience, tradition and intuitive thought patterns suggesting how the bending and breaking of logical rules “just enough” leads to creative synthesis.
But all this goes on at what Hofstadter calls the higher level of thinkodynamics, whose basic constructs are higher level symbols, found in, created by and manipulated in the conscious brain. This is a level of thought that is un-self-aware and that must remain so in order to work. It is a realm beyond the mere “and”, “or”, and “not” of symbolic logic. Hofstadter, a defiant and proud anti-reductionist, notes:
Our existence as animals whose perception is limited to the world of everyday macroscopic objects forces us . . . to function without any reference to entities or processes at microscopic levels. . . . Magellan circumnavigated the globe, Shakespeare wrote some plays, J. S. Bach composed some cantatas, and Joan of Arc got herself burned at the stake, all for reasons, none which from their point of view, had the least thing to do with DNA, RNA, and proteins, or with carbon, oxygen, hydrogen, and nitrogen, or with photons, electrons, protons, neutrons, let alone with quarks, gluons, W and Z bosons, gravitons, and Higgs particles.
In Gödel, Escher, Bach, Hofstadter had quite a sale to make, and the vehicle for it was so interestingly, provocatively and imaginatively constructed that, in the introduction to his new book, I Am a Strange Loop, he correctly notes that many readers enjoyed the trees (with their many branches and variations) so much that they entirely missed the arboreal. Gödel, Escher, Bach was a treatise whose central goal was to explain consciousness and mind, but with endless fun and digression on the way. In I Am a Strange Loop the single focus is the mind. Hofstadter’s strange loop, quintessentially exemplified by the “I” of the title, is precisely that phenomenon of multi-level self-referentiality that enables thought. Who is that “I”, referring to exactly and precisely the “us”, within each of us, and how did it get there? And where in the brain is it? Hofstadter answers:
the “I” symbol, like all symbols in our brain, starts out pretty small and simple, but it grows and grows, eventually becoming the most important abstract structure residing in our brains. . . . It is not in some small localized spot; it is spread out all over, because it has to include so much about so much.
He argues that TAT, defined essentially as self-awareness, is the basis for understanding the conscious mind. How does self-awareness proceed? Through symbols and symbolization, the conscious mind is
a kind of perception of internal symbol-patterns, rather than the perception of outside events. Someone seems to be looking at configurations of activated symbols and perceiving their essence, thereby triggering the retrieval of other dominant symbols . . . memory packages . . . and round and round it all goes, giving rise to a lively cycle of symbolic activity . . . a smooth but completely improvised symbolic dance.
Does this really make any sense? Well, Hofstadter persuaded someone, at least, that it does. The publishers of I Am a Strange Loop state boldly that Hofstadter’s new book is “the long awaited sequel” to Gödel, Escher, Bach!
It isn’t a sequel, really. I Am a Strange Loop is actually two intertwined books, one of which is a simple, down-to-earth replay of the 1979 original, for those who, for whatever reason, didn’t get the big picture the first time around. For such folks this is typically whacky Hofstadterian stream of consciousness, an onslaught of good, bad and really terrible puns and wordplay (had you noticed that Gödel is almost GOD with an umlaut? Or that Kurt is a “Turk” in disguise?), run-on thoughts and the wild analogies loved by those who love Hofstadter. But this part of I Am a Strange Loop lacks the easily overwhelming structure of Gödel, Escher, Bach. If you are not sure what “overwhelming structure” means, just take a look at the original: It is off-putting at first, but of lasting value once vanquished. I Am a Strange Loop is thus, in part, a prequel to Gödel, Escher, Bach, and if it succeeds in drawing the young or old, or the math-phobic reader into the rather more serious original—or if it leads to working up an interest in Gödel—it must be judged a success.
The “second” book is a memoir, pure but not exactly simple. Although Hofstadter says the “I” in the title is not himself, it surely is, too. This second part of I Am a Strange Loop is an exposition that amounts to an autobiography of his precocious intellectual development. Intellect, not social biography, is his mode of existence, after all. One does not expect to find, nor does one find, the actual person Kurt Gödel in Gödel, Escher, Bach. That book is about mind, logical games, artificial intelligence, TAT and patterns. In I Am a Strange Loop, however, we actually do meet Douglas Hofstadter. We meet him when his intellect crashes into a genuine real-life tragedy.
In 1993, Hofstadter’s young wife, Carol, suddenly and without warning died, leaving him and two young children. Hofstadter tells us that the resulting need was not for him to grieve, but for her to grieve. Only of course she couldn’t. Or could she? One chapter of the memoir is a collection of edited emails sent and received in the year following this tragic event: Were they not so edited, their raw agony might well make Joan Didion’s The Year of Magical Thinking seem a soothing palliative. In I Am a Strange Loop this intolerable tension is resolved by Carol becoming “a second strange loop”, albeit of lower resolution than his own strange loop, but assuredly still alive as mind in his own cranium; and able, at last, to grieve.
This is a challenging analogy, and one perhaps better described by a theologian and writer than by a computer scientist, no matter how facile. Thus Frederick Buechner:
How they do live on, those giants of our childhood, and how well they manage to take even death in their stride because although death can put an end to them right enough, it can never put an end to our relationship with them. Wherever or however else they may have come to life since, it is beyond doubt that they live still in us. Memory is more than a looking back to a time that is no longer; it is looking out into another kind of time altogether where everything that ever was continues not just to be, but to grow and change with the life that is in us still.3
Faith, too, may be a kind of loop.
Legend has it that John Locke, the British philosopher and grandfather of our Declaration of Independence, considered the following problem: Suppose a blind human being, who could easily distinguish a cube from a sphere by touch, were suddenly made to have “vision.” Would this same person now recognize the cube and the ever so different sphere, “by sight”? Locke, the empiricist, argued “no.” Centuries later, modern surgery has made this experiment a reality, and Locke’s deduction of pure reason was confirmed. Like Locke, Hofstadter argues from the same pure reason and logic, including Gödel’s more complex extensions of them.
However, since the publication of Gödel, Escher, Bach more than 25 years ago, spectacular advances in the neuro-physiology of vision, perception, speech and their computer simulations have taken place. Actual anatomical structures have been related to specific brain functions. These new findings, combined with magnetic resonance imaging—showing exactly the excitations of specific regions of brain activity—and the flow (Hofstadter’s “dance of symbols”?) of such activity in response to stimuli like vision, listening to music, or just “thinking about thinking”, suggests that pure reason will soon be confronted by hard fact—and of course vice versa. Is it possible that the “I” inside us all, as we imagine it there, is simply hard wired? Stay tuned.
Russell in Memorabilia Mathematica, ed. Robert Edouard Mortiz (Dover, 1958).
Gödel’s life and work are engagingly recounted in two recent popular biographies: Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel (W. W. Norton, 2005), and Palle Yourgrau’s A World Without Time: The Forgotten Legacy of Gödel and Einstein (Basic, 2004).
Buechner, The Sacred Journey: A Memoir of Early Days (HarperSanFrancisco, 1991).