I Am a Strange Loop

my intense curiosity was my sense that what was really being explored by Godel, as well as by many people he had inspired, was the mystery of the human mind and the mechanisms of human thinking. So many questions seemed to have been suddenly and sharply brought into light by Godel’s 1931 article — questions such as…

What happens inside mathematicians’ heads when they do their most creative work? Is it always just rule- bound symbol manipulation, deriving theorems from a fixed set of axioms? What is the nature of human thought in general? Is what goes on inside our heads just a deterministic physical process? If so, are we all, no matter how idiosyncratic and sparkly, nothing but slaves to rigid laws governing the invisible particles out of which our brains are built? Could creativity ever emerge from a set of rigid rules governing minuscule objects or patterns of numbers? Could a rulegoverned machine be as creative as a human? Could a programmed machine come up with ideas not programmed into it in advance? Could a machine make its own decisions? Have its own opinions? Be confused? Know it was confused? Be unsure whether it was confused? Believe it had free will? Believe it didn’t have free will? Be conscious? Doubt it was conscious? Have a self, a soul, an “I”? Believe that its fervent belief in its “I” was only an illusion, but an unavoidable illusion?

Idealistic Dreams about Metamathematics

Back in those heady days of my youth, every time I entered a university bookstore (and that was as often as possible), I would instantly swoop down on the mathematics section and scour all the books that had to do with symbolic logic and the nature of symbols and meaning. Thus I bought book after book on these topics, such as Rudolf Carnap’s famous but forbidding The Logical Syntax of Language and Richard Martin’s Truth and Denotation, not to mention countless texts of symbolic logic. Whereas I very carefully read a few such textbooks, the tomes by Carnap and Martin just sat there on my shelf, taunting and teasing me, always seeming just out of reach. They were dense, almost impenetrably so — but I kept on thinking that if only someday, some grand day, I could finally read them and fully fathom them, then at last I would have penetrated to the core of the mysteries of thinking, meaning, creativity, and consciousness. As I look back now, that sounds ridiculously naive (firstly to imagine this to be an attainable goal, and secondly to believe that those books in particular contained all the secrets), but at the time I was a true believer!

When I was sixteen, I had the unusual experience of teaching symbolic logic at Stanford Elementary School (my own elementary-school alma mater), using a brand-new text by the philosopher and educator Patrick Suppes, who happened to live down the street from our family, and whose classic Introduction to Logic had been one of my most reliable guides. Suppes was conducting an experiment to see if patterns of strict logical inference could be inculcated in children in the same way as arithmetic could, and the school’s principal, who knew me well from my years there, one day bumped into me in the school’s rotunda, and asked me if I would like to teach the sixth-grade class (which included my sister Laura) symbolic logic three times a week for a whole year. I fairly jumped at the chance, and all year long I thoroughly enjoyed it, even if a few of the kids now and then gave me a hard time (rubber bands in the eye, etc.). I taught my class the use of many rules of inference, including the mellifluous modus tollendo tollens and the impressive-sounding “hypothetical syllogism”, and all the while I was honing my skills not only as a novice logician but also as a teacher.

What drove all this — my core inner passion — was a burning desire to see unveiled the secrets of human mentation, to come to understand how it could be that trillions of silent, synchronized scintillations taking place every second inside a human skull enable a person to think, to perceive, to remember, to imagine, to create, and to feel. At more or less the same time, I was reading books on the brain, studying several foreign languages, exploring exotic writing systems from various countries, inventing ways to get a computer to generate grammatically complicated and quasi-coherent sentences in English and in other languages, and taking a marvelously stimulating psychology course. All these diverse paths were focused on the dense nebula of questions about the relationship between mind and mechanism, between mentality and mechanicity.

Intricately woven together, then, in my adolescent mind were the study of pattern (mathematics) and the study of paradoxes (metamathematics). I was somehow convinced that all the mysterious secrets with which I was obsessed would become crystal-clear to me once I had deeply mastered these two intertwined disciplines. Although over the course of the next couple of decades I lost essentially all of my faith in the notion that these disciplines contained (even implicitly) the answers to all these questions, one thing I never lost was my intuitive hunch that around the core of the eternal riddle “What am I?”, there swirled the ethereal vortex of Godel’s elaborately constructed loop.

It is for that reason that in this book, although I am being driven principally by questions about consciousness and self, I will have to devote some pages to the background needed for a (very rough) understanding of Godel’s ideas — and in particular, this means number theory and logic. There won’t be heavy doses of either one, to be sure, but I do have to paint at least a coarse-grained picture of what these fields are basically about; otherwise, we won’t have any way to proceed. So please fasten your seat belt, dear reader. We’re going to be experiencing a bit of weather for the next two chapters.

Post Scriptum

After completing this chapter to my satisfaction, I recalled that I owned two books about “interesting numbers” — The Penguin Dictionary of Curious and Interesting Numbers by David Wells, an author on mathematics whom I greatly admire, and Les nombres remarquables by Francois Le Lionnais, one of the two founders of the famous French literary movement Oulipo. I dimly recalled that both of these books listed their “interesting numbers” in order of size, so I decided to check them out to see which was the lowest integer that each of them left out.

As I suspected, both authors made rather heroic efforts to include all the integers that exist, but inevitably, human knowledge being finite and human beings being mortal, each volume sooner or later started having gaps. Wells’ first gap appeared at 43, while Le Lionnais held out a little bit longer, until 49. I personally was not too surprised by 43, but I found 49 surprising; after all, it’s a square, which suggests at least a speck of interest. On the other hand, I admit that squareness gets a bit boring after you’ve already run into it several times, so I could partially understand why that property alone did not suffice to qualify 49 for inclusion in Le Lionnais’s final list. Wells lists several intriguing properties of 49 (but not the fact that it’s a square), and conversely, Le Lionnais points out some very surprising properties of 43.

So then I decided to find the lowest integer that both books considered to be utterly devoid of interest, and this turned out to be 62. For what it’s worth, that will be my age when this book appears in print. Could it be that 62 is interesting, after all?

CHAPTER 9

Pattern and Provability

Principia Mathematica and its Theorems

IN THE early twentieth century, Bertrand Russell, spurred by the maxim “Find and study paradoxes; design and build great ramparts to keep them out!” (my words, not his), resolved that in Principia Mathematica, his new barricaded fortress of mathematical reasoning, no set could ever contain itself, and no sentence could ever turn around and talk about itself. These parallel bans were intended to save Principia Mathematica from the trap that more naive theories had fallen into. But something truly strange turned up when Kurt Godel looked closely at what I will call