I Am a Strange Loop

After completing this chapter to my satisfaction, I recalled that I owned two books about “interesting numbers” — The Penguin Dictionary of Curious and Interesting Numbers by David Wells, an author on mathematics whom I greatly admire, and Les nombres remarquables by François Le Lionnais, one of the two founders of the famous French literary movement Oulipo. I dimly recalled that both of these books listed their “interesting numbers” in order of size, so I decided to check them out to see which was the lowest integer that each of them left out.

As I suspected, both authors made rather heroic efforts to include all the integers that exist, but inevitably, human knowledge being finite and human beings being mortal, each volume sooner or later started having gaps. Wells’ first gap appeared at 43, while Le Lionnais held out a little bit longer, until 49. I personally was not too surprised by 43, but I found 49 surprising; after all, it’s a square, which suggests at least a speck of interest. On the other hand, I admit that squareness gets a bit boring after you’ve already run into it several times, so I could partially understand why that property alone did not suffice to qualify 49 for inclusion in Le Lionnais’s final list. Wells lists several intriguing properties of 49 (but not the fact that it’s a square), and conversely, Le Lionnais points out some very surprising properties of 43.

So then I decided to find the lowest integer that both books considered to be utterly devoid of interest, and this turned out to be 62. For what it’s worth, that will be my age when this book appears in print. Could it be that 62 is interesting, after all?

CHAPTER 9

Pattern and Provability

Principia Mathematica and its Theorems

IN THE early twentieth century, Bertrand Russell, spurred by the maxim “Find and study paradoxes; design and build great ramparts to keep them out!” (my words, not his), resolved that in Principia Mathematica, his new barricaded fortress of mathematical reasoning, no set could ever contain itself, and no sentence could ever turn around and talk about itself. These parallel bans were intended to save Principia Mathematica from the trap that more naïve theories had fallen into. But something truly strange turned up when Kurt Gödel looked closely at what I will call PM — that is, the formal system used in Principia Mathematica for reasoning about sets (and about numbers, but they came later, as they were defined in terms of sets).

Let me be a little more explicit about this distinction between Principia Mathematica and PM. The former is a set of three hefty tomes, whereas PM is a set of precise symbol-manipulation rules laid out and explored in depth in those tomes, using a rather arcane notation (see the end of this chapter). The distinction is analogous to that between Isaac Newton’s massive tome entitled Principia and the laws of mechanics that he set forth therein.

Although it took many chapters of theorems and derivations before the rather lowly fact that one plus one equals two (written in PM notation as “s0 + s0 = ss0”, where the letter “s” stands for the concept “successor of ”) was rigorously demonstrated using the strict symbol-shunting rules of PM, Gödel nonetheless realized that PM, though terribly cumbersome, had enormous power to talk about whole numbers — in fact, to talk about arbitrarily subtle properties of whole numbers. (By the way, that little phrase “arbitrarily subtle properties” already gives the game away, though the hint is so veiled that almost no one is aware of how much the words imply. It took Gödel to fully see it.)

For instance, as soon as enough set-theoretical machinery had been introduced in Principia Mathematica to allow basic arithmetical notions like addition and multiplication to enter the picture, it became easy to define, within the PM formalism, more interesting notions such as “square” (i.e., the square of a whole number), “nonsquare”, “prime”, and “composite”.