I Am a Strange Loop - Douglas R. Hofstadter [69]
There is also a kind of “junior version” of Berry’s paradox that was invented a few decades after it, and which runs like this. Some integers are interesting. 0 is interesting because 0 times any number gives 0. 1 is interesting because 1 times any number leaves that number unchanged. 2 is interesting because it is the smallest even number, and 3 is interesting because it is the number of sides of the simplest two-dimensional polygon (a triangle). 4 is interesting because it is the first composite number. 5 is interesting because (among many other things) it is the number of regular polyhedra in three dimensions. 6 is interesting because it is three factorial (3×2×1) and also the triangular number of three (3+2+1). I could go on with this enumeration, but you get the point. The question is, when do we run into the first uninteresting number? Perhaps it is 62? Or 1729? Well, no matter what it is, that is certainly an interesting property for a number to have! So 62 (or whatever your candidate number might have been) turns out to be interesting, after all — interesting because it is uninteresting. And thus the idea of “the smallest uninteresting integer” backfires on itself in a manner clearly echoing the backfiring of Berry’s definition of b.
This is the kind of twisting-back of language that turned Bertrand Russell’s sensitive stomach, as we well know, and yet, to his credit, it was none other than B. Russell who first publicized G. G. Berry’s paradoxical number b. In his article about it in 1906, Gödel’s birthyear (four syllables!), Russell did his best to deflect the paradox’s sting by claiming that it was an illusion arising from a naïve misuse of the word “describable” in the context of mathematics. That notion, claimed Russell, had to be parceled out into an infinite hierarchy of different types of describability — descriptions at level 0, which could refer only to notions of pure arithmetic; descriptions at level 1, which could use arithmetic but could also refer to descriptions at level 0; descriptions at level 2, which could refer to arithmetic and also to descriptions at levels 0 and 1; and so forth and so on. And so the idea of “describability” without restriction to some specific hierarchical level was a chimera, declared Russell, believing he had discovered a profound new truth. And with this brand-new type of theory (the brand-new theory of types), he claimed to have immunized the precious, delicate world of rigorous reasoning against the ugly, stomach-turning plague of Berry-Berry.
Blurriness Buries Berry
While I agree with Russell that something fishy is going on in Berry’s paradox, I don’t agree about what it is. The weakness that I focus in on is the fact that English is a hopelessly imprecise medium for expressing mathematical statements; its words and phrases are far too vague. What may seem precise at first turns out to be fraught with ambiguity. For example, the expression “nine cubed plus forty-eight, all times ten cubed plus one”, which earlier I exhibited as a description of 777,777, is in fact ambiguous — it might, for instance, be interpreted as meaning 777 times 1000, with 1 tacked on at the end, resulting in 777,001.
But that little ambiguity is just the tip of the iceberg. The truth of the matter is that it is far from clear what kinds of English expressions count as descriptions of a number. Consider the following phrases, which purport to be descriptions of specific integers:
• the number of distinct languages ever spoken on earth
• the number of heavenly bodies in the Solar System
• the number of distinct four-by-four magic squares
• the number of interesting integers less than 100
What is wrong with them? Well, they all involve ill-defined notions.
What, for instance, is meant by a “language”? Is sign language a language? Is it “spoken”? Is there a sharp cutoff between languages and dialects? How many “distinct languages” lay along the pathway from Latin to