I Am a Strange Loop

These are primal, irrational intuitions, and who knows where they come from. One might speculate that fear of any kind of feedback is just a simple, natural generalization from one’s experience with audio feedback, but I somehow doubt that the explanation is that simple. We all know that some tribes are fearful of mirrors, many societies are suspicious of cameras, certain religions prohibit making drawings of people, and so forth. Making representations of one’s own self is seen as suspicious, weird, and perhaps ultimately fatal. This suspicion of loops just runs in our human grain, it would seem. However, as with many daring activities such as hang-gliding or parachute jumping, some of us are powerfully drawn to it, while others are frightened to death by the mere thought of it.

God, Gödel, Umlauts, and Mystery

When I was fourteen years old, browsing in a bookstore, I stumbled upon a little paperback entitled “Gödel’s Proof”. I had no idea who this Gödel person was or what he (I’m sure I didn’t think “he or she” at that early age and stage of my life) might have proven, but the idea of a whole book about just one mathematical proof — any mathematical proof — intrigued me. I must also confess that what doubtlessly added a dash of spice to the dish was the word “God” blatantly lurking inside “Gödel”, as well as the mysterious-looking umlaut perched atop the center of “God”. My brain’s molecules, having been tickled in the proper fashion, sent signals down to my arms and fingers, and accordingly I picked up the umlaut-decorated book, flipped through its pages, and saw tantalizing words like “meta-mathematics”, “meta-language”, and “undecidability”. And then, to my delight, I saw that this book discussed paradoxical self-referential sentences like “I am lying” and more complicated cousins. I could see that whatever Gödel had proved wasn’t focused on numbers per se, but on reasoning itself, and that, most amazingly, numbers were being put to use in reasoning about the nature of mathematics.

Although to some readers this next may sound implausible, I remember being particularly drawn in by a long footnote about the proper use of quotation marks to distinguish between use and mention. The authors — Ernest Nagel and James R. Newman — took the two sentences “Chicago is a populous city” and “Chicago is trisyllabic” and asserted that the former is true but the latter is false, explaining that if one wishes to talk about properties of a word, one must use its name, which is the expression resulting from putting it inside quotes. Thus, the sentence “ ‘Chicago’ is trisyllabic” does not concern a city but its name, and states a truth. The authors went on to talk about the necessity of taking great care in making such distinctions inside formal reasoning, and pointed out that names themselves have names (made using quote marks), and so on, ad infinitum. So here was a book talking about how language can talk about itself talking about itself (etc.), and about how reasoning can reason about itself (etc.). I was hooked! I still didn’t have a clue what Gödel’s theorem was, but I knew I had to read this book. The molecules constituting the book had managed to get the molecules in my head to get the molecules in my hands to get the molecules in my wallet to… Well, you get the idea.

Savoring Circularity and Self-application

What seemed to me most magical, as I read through Nagel and Newman’s compelling booklet, was the way in which mathematics seemed to be doubling back on itself, engulfing itself, twisting itself up inside itself. I had always been powerfully drawn to loopy phenomena of this sort. For instance, from early childhood, I had loved the idea of closing a cardboard box by tucking its four flaps over each other in a kind of “circular” fashion — A on top of B, B on top of C, C on top of D, and then D on top of A. Such grazing of paradoxicality enchanted and fascinated me.

Also, I had always loved standing between two mirrors and seeing the implied infinitude of images as they faded off into the distance. (The photo was taken by Kellie Gutman.) A mirror mirroring a mirror — what idea could be more provocative? And I loved the picture of the Morton Salt girl holding a box of Morton Salt, with herself drawn on it, holding the box, and on and on, by implication, in ever-tinier copies, without any end, ever.