• Choose a category
• All
• Classic-Fiction

By Root 1616 0

is to turn the classic tragicomedy starring Alfred North Whitehead and Bertrand Russell (jointly alias the Alfbert) and Kurt Gödel (alias the Klüdgerot) 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 Gödelian meanings. This is the diametric opposite of what one would naïvely expect, since PM notation was invented expressly to write down statements about numbers and their properties, certainly not to write down Gödelian 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 (Gödel’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 Gödel’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.)

Gödel 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). Gödel 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, Gödel 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.