I Am a Strange Loop

Many philosophers and mathematicians contributed to this noble goal, and around the turn of the twentieth century it appeared that the goal was coming into sight. Mathematical reasoning seemed to have been precisely characterized as the repeated use of certain basic rules of logic, dubbed rules of inference, such as modus ponens: If you have proven a result X and you have also proven X ⇒ Y (where the arrow represents the concept of implication, so that the line means “If X is true, then Y is also true”), then you can toss Y into the bin of proven results. There were a few other fundamental rules of inference, but it was agreed that not very many were needed. About a decade into the twentieth century, Bertrand Russell and Alfred North Whitehead codified these rules in a uniform if rather prickly notation (see facing page), thus apparently allowing all the different branches of mathematics to be folded in with logic, making a seamless, perfect unity.

Thanks to Russell and Whitehead’s grand work, Principia Mathematica, people no longer needed to fear falling into hidden crevasses of false reasoning. Theorems were now understood as simply being the bottom lines of sequences of symbol-manipulations whose top lines were either axioms or earlier theorems. Mathematical truth was all coming together so elegantly. And as this Holy Grail was emerging into clear view, a young boy was growing up in the town of Brünn, Austro-Hungary.

CHAPTER 10

Gödel’s Quintessential Strange Loop

Gödel Encounters Fibonacci

BY HIS early twenties, the boy from Brünn was already a superb mathematician and, like all mathematicians, he knew whole numbers come in limitless varieties. Aside from squares, cubes, primes, powers of ten, sums of two squares, and all the other usual suspects, he was familiar with many other types of integers. Most crucially for his future, young Kurt knew, thanks to Leonardo di Pisa (more often known as “Fibonacci”), that one could define classes of integers recursively.

In the 1300’s, Fibonacci had concocted and explored what are now known as the “Fibonacci numbers”:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, . . .

In this rapidly growing infinite sequence, whose members I will henceforth refer to as the F numbers, each new element is created by summing the two previous ones (except for the first pair, 1 and 2, which we simply declare by fiat to be F numbers).

This almost-but-not-quite-circular fashion of defining a sequence of numbers in terms of itself is called a “recursive definition”. This means there is some kind of precise calculational rule for making new elements out of previous ones. The rule might involve adding, multiplying, dividing, whatever — as long as it’s well-defined. The opening gambit of a recursive sequence (in this case, the numbers 1 and 2) can be thought of as a packet of seeds from which a gigantic plant — all of its branches and leaves, infinite in number — grows in a predetermined manner, based on the fixed rule.

The Caspian Gemstones: An Allegory

Leonardo di Pisa’s sequence is brimming with amazing patterns, but unfortunately going into that would throw us far off course. Still, I cannot resist mentioning that 144 jumps out in this list of the first few F numbers because it is a salient perfect square. Aside from 8, which is a cube, and 1, which is a rather degenerate case, no other perfect square, cube, or any other exact power appears in the first few hundred terms of the F sequence.

Several decades ago, people started wondering if the presence of 8 and 144 in the F sequence was due to a reason, or if it was just a “random accident”. Therefore, as computational tools started becoming more and more powerful, they undertook searches. Curiously enough, even with the advent of supercomputers, allowing millions and even billions of F numbers to be churned out, no one ever came across any other perfect powers in Fibonacci’s sequence. The chance of a power turning up very soon in the F sequence was looking slim, but why would a perfect mutual avoidance occur? What do nth powers for arbitrary n have to do with adding up pairs of numbers in Fibonacci’s peculiar recursive fashion? Couldn’t 8 and 144 just be little random glitches? Why couldn’t other little glitches take place?