I Am a Strange Loop

Bertrand Russell’s attempt to bar Berry’s paradoxical construction by instituting a formalism that banned all self-referring linguistic expressions and self-containing sets was not only too hasty but quite off base. How so? Well, a friend of mine recently told me of a Russell-like ban instituted by a friend of hers, a young and idealistic mother. This woman, in a well-meaning gesture, had strictly banned all toy guns from her household. The ban worked for a while, until one day when she fixed her kindergarten-age son a peanut-butter sandwich. The lad quickly chewed it into the shape of a pistol, then lifted it up, pointed it at her, and shouted, “Bang bang! You’re dead, Mommy!” This ironic anecdote illustrates an important lesson: the medium that remains after all your rigid bans may well turn out to be flexible enough to fashion precisely the items you’ve banned.

And indeed, Russell’s dismissal of Berry had little effect, for more and more paradoxes were being invented (or unearthed) in those intellectually tumultuous days at the turn of the twentieth century. It was in the air that truly peculiar things could happen when modern cousins of various ancient paradoxes cropped up inside the rigorously logical world of numbers, a world in which nothing of the sort had ever been seen before, a pristine paradise in which no one had dreamt paradox might arise.

Although these new kinds of paradoxes felt like attacks on the beautiful, sacred world of reasoning and numbers (or rather, because of that worrisome fact), quite a few mathematicians boldly embarked upon a quest to come up with ever deeper and more troubling paradoxes — that is, a quest for ever more powerful threats to the foundations of their own discipline! This sounds like a perverse thing to do, but they believed that in the long run such a quest would be very healthy for mathematics, because it would reveal key weak spots, showing where shaky foundations had to be shored up so as to become unassailable. In short, plunging deeply into the new wave of paradoxes seemed to be a useful if not indispensable activity for anyone working on the foundations of mathematics, for the new paradoxes were opening up profound questions concerning the nature of reasoning — and thus concerning the elusive nature of thinking — and thus concerning the mysterious nature of the human mind itself.

An Autobiographical Snippet

As I mentioned in Chapter 4, at age fourteen I ran across Ernest Nagel and James R. Newman’s little gem, Gödel’s Proof, and through it I fell under the spell of the paradox-skirting ideas on which Gödel’s work was centered. One of the stranger loops connected with that period in my life was that I became acquainted with the Nagel family at just that time. Their home was in Manhattan, but they were spending the academic year 1959–60 “out west” at Stanford, and since Ernest Nagel and my father were old friends, I soon got to know the whole family. Shortly after the Nagels’ Stanford year was over, I savored the twisty pleasure of reading aloud the whole of Gödel’s Proof to my friend Sandy, their older son, in the verdant yard of their summer home in the gentle hills near Brattleboro, Vermont. Sandy was just my age, and we were both exploring mathematics with a kind of wild intoxication that only teen-agers know.

Part of what pulled me so intensely was the weird loopiness at the core of Gödel’s work. But the other half of my intense curiosity was my sense that what was really being explored by Gödel, 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 Gödel’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