regains consciousness. First it hears shouts of excitement echoing all around, and then, when it dares to look, it beholds a startling sight; on every side, as far as its sole eye can see, masses of Kludgerot are staring with unmistakable delight at something moving, somewhere above its white head. It turns to see what this could possibly be, just in time to catch the most fleeting glimpse of a thin shape making a strange, high-pitched rustling sound as it rapidly plummets towards —
Brief Debriefing
I offer my apologies to the late Ambrose Bierce for this rather feeble imitation of the plot of his masterful short story “An Occurrence at Owl Creek Bridge”, but my intentions are good. The raison d’etre of my rather flippant allegory is to turn the classic tragicomedy starring Alfred North Whitehead and Bertrand Russell (jointly alias the Alfbert) and Kurt Godel (alias the Kludgerot) on its head, by positing bizarre creatures who cannot imagine the idea of any number-theoretical meaning in PM strings, but who nonetheless see the strings as meaningful messages — it’s just that they see only high-level Godelian meanings. This is the diametric opposite of what one would naively expect, since PM notation was invented expressly to write down statements about numbers and their properties, certainly not to write down Godelian statements about themselves!
A few remarks are in order here to prevent confusions that this allegory might otherwise engender. In the first place, the length of any PM string that speaks of its own properties (Godel’s string KG being the prototype, of course) is not merely “enormous”, as I wrote at the allegory’s outset; it is inconceivable. I have never tried to calculate how many symbols Godel’s string would consist of if it were written out in pure PM notation, because I would hardly know how to begin the calculation. I suspect that its symbol-count might well exceed “Graham’s constant”, which is usually cited as “the largest number ever to appear in a mathematical proof”, but even if not, it would certainly give it a run for its money. So the idea of anyone directly reading the strings that grow on Austranius, whether on a low level, as statements about whole numbers, or on a high level, as statements about their own edibility, is utter nonsense. (Of course, so is the idea that strings of mathematical symbols could grow in jungles on a faraway planet, as well as the idea that they could be eaten, but that’s allegoric license.)
Godel created his statement KG through a series of 46 escalating stages, in which he shows that in principle, certain notions about numbers could be written down in PM notation. A typical such notion is “the exponent of the kth prime number in the prime factorization of n”. This notion depends on prior notions defined in earlier stages, such as “exponent”, “prime number”, “kth prime number”, “prime factorization” (none of which come as “built-in notions” in PM). Godel never explicitly writes out PM expressions for such notions, because doing so would require writing down a prohibitively long chain of PM symbols. Instead, each individual notion is given a name, a kind of abbreviation, which could theoretically be expanded out into pure PM notation if need be, and which is then used in further steps. Over and over again, Godel exploits alreadydefined abbreviations in defining further abbreviations, thus carefully building a tower of increasing complexity and abstractness, working his way up to its apex, which is the notion of prim numbers.
Soaps in Sanskrit
This may sound a bit abstruse and remote, so let me suggest an analogy. Imagine the challenge of writing out a clear explanation of the meaning of the contemporary term “soap digest rack” in the ancient Indian language of Sanskrit. The key constraint is that you are restricted to using pure Sanskrit as it was in its heyday, and are not allowed to introduce even one single new word into the language.
In order to get across the meaning of “soap digest rack” in detail, you would have to explain, for starters, the notions of electricity and electromagnetic waves, of TV cameras and transmitters and TV sets, of TV shows and advertising, the notion of washing machines and rivalries between detergent companies, the idea of daily episodes of predictable hackneyed melodramas broadcast into the homes of millions of people, the image of viewers addicted to endlessly circling plots, the concept of a grocery store, of a checkout stand, of magazines, of display racks, and on and on… Each of the words “soap”, “digest”, and “rack” would wind up being expanded into a chain of ancient Sanskrit words thousands of times longer than itself. Your final text would fill up hundreds of pages in order to get across the meaning of this three-word phrase for a modern banality.
Likewise, Godel’s string KG, which we conventionally express in supercondensed form through phrases such as “I am not provable in PM”, would, if written out in pure PM notation, be monstrously long — and yet despite its formidable size, we understand precisely what it says. How is that possible? It is a result of its condensability. KG is not a random sequence of PM symbols, but a formula possessing a great deal of structure. Just as the billions of cells comprising a heart are so extremely organized that they can be summarized in the single word “pump”, so the myriad symbols in KG can be summarized in a few well-chosen English words.
To return to the Sanskrit challenge, imagine that I changed the rules, allowing you to define new Sanskrit words and to employ them in the definitions of yet further new Sanskrit words. Thus “electricity” could be defined and used in the description of TV cameras and televisions and washing machines, and “TV program” could be used in the definition of “soap opera”, and so forth. If abbreviations could thus be piled on abbreviations in an unlimited fashion, then it is likely that instead of producing a book-length Sanskrit explanation of “soap digest rack”, you would need only a few pages, perhaps even less. Of course, in all this, you would have radically changed the Sanskrit language, carrying it forwards in time a few thousand years, but that is how languages always progress. And that is also the way the human mind works — by the compounding of old ideas into new structures that become new ideas that can themselves be used in compounds, and round and round endlessly, growing ever more remote from the basic earthbound imagery that is each language’s soil.
Winding Up the Debriefing
In my allegory, both the Kludgerot and the Alfbert supposedly have the ability to read pure PM strings — strings that contain no abbreviations whatsoever. Since at one level (the level perceived by the Kludgerot) these strings talk about themselves, they are like Godel’s KG, and this means that such strings are, for want of a better term, infinitely huge (for all practical purposes, anyway). This means that any attempt to read them as statements about numbers will never yield anything comprehensible at all, and so the Alfbert’s ability, as described, is a total impossibility. But so is the Kludgerot’s, since they too are overwhelmed by an endless sea of symbols. The only hope for either the Alfbert or the Kludgerot is to notice that certain patterns are used over and over again in the sea of symbols, and to give these patterns names, thus compressing the string into something more manageable, and then carrying this process of patternfinding and compression out at the new, shorter level, and each time compressing further and further and further until finally the whole string collapses down into just one simple idea: “I am not edible” (or, translating out of the allegory, “I am not provable”).
Bertrand Russell never imagined this kind of a level-shift when he thought about the strings of PM. He was trapped by the understandable preconception that statements about whole numbers, no matter how long or complicated they might get, would always retain the familiar flavor of standard number-theoretical statements such as “There are infinitely many primes” or “There are only three pure powers in the Fibonacci sequence.” It never occurred to him that some statements could have such intricate hierarchical structures that the number-theoretical ideas they would express would no longer feel like ideas about numbers. As I observed in Chapter 11, a dog does not imagine or understand that certain large arrays of colored dots can be so structured that they are no longer just huge sets of colored dots but become pictures of people, houses, dogs, and many other things. The higher level takes perceptual precedence over the lower level, and in the process becomes the “more real” of the two. The lower level gets forgotten, lost in the shuffle.
Such an upwards level-shift is a profound perceptual change, and when it takes place in an unfamiliar, abstract setting, such as the world of strings of Principia Mathematica, it can sound very
