I Am a Strange Loop

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 Brunn, Austro-Hungary.

CHAPTER 10

Godel’s Quintessential Strange Loop

Godel Encounters Fibonacci

BY HIS early twenties, the boy from Brunn 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?

To cast allegorical light on this, imagine someone chanced one day to fish up a giant diamond, a magnificent ruby, and a tiny pearl at the bottom of the great green Caspian Sea in central Asia, and other seekers of fortune, spurred on by these stunning finds, then started madly dredging the bottom of the world’s largest lake to seek more diamonds, rubies, pearls, emeralds, topazes, etc., but none was found, no matter how much dredging was done. One would naturally wonder if more gems might be hidden down there, but how could one ever know? (Caveat: my allegory is slightly flawed, because we can imagine, at least in principle, a richly financed scientific team someday dredging the lake’s bottom completely, since, though huge, it is finite. For my analogy to be “perfect”, we would have to conceive of the Caspian Sea as infinite. Just stretch your imagination a bit, reader!)

Now the twist. Suppose some mathematically-minded geologist set out to prove that the two exquisite Caspian gems, plus the tiny round pearl, were sui generis — in other words, that there was a precise reason that no other gemstone or pearl of any type or size would ever again, or could ever again, be found in the Caspian Sea. Does seeking such a proof make any sense? How could there be a watertight scientific reason absolutely forbidding any gems — except for one pearl, one ruby, and one diamond — from ever being found on the floor of the Caspian Sea? It sounds absurd.

This is typical of how we think about the physical world — we think of it as being filled with contingent events, facts that could be otherwise, situations that have no fundamental reason for their being as they are. But let me remind you that mathematicians see their pristine, abstract world as the antithesis to the random, accident- filled physical world we all inhabit. Things that happen in the mathematical world strike mathematicians as happening, without any exceptions, for statable, understandable reasons.

This — the Mathematician’s Credo — is the mindset that you have to adopt and embrace if you wish to understand how mathematicians think. And in this particular case, the mystery of the lack of Fibonacci powers, although just a tiny one in most mathematicians’ eyes, was a particularly baffling one, because it seemed to offer no natural route of access. The two phenomena involved — integer powers with arbitrarily large exponents, on the one hand, and Fibonacci numbers on the other — simply seemed (like gemstones and the Caspian Sea) to be too conceptually remote from each other to have any deep, systematic, inevitable interrelationship.

And then along came a vast team of mathematicians who had set their collective bead on the “big game” of Fermat’s Last Theorem (the notorious claim, originally made by Pierre de Fermat in the middle of the seventeenth century, that no positive integers a, b, c exist such that aⁿ + bⁿ equals c ⁿ, with the exponent n being an integer greater than 2). This great international relay team, whose final victorious lap was magnificently sprinted by Andrew Wiles (his sprint took him about eight years), was at last able to prove Fermat’s centuries-old claim by using amazing techniques that combined ideas from all over the vast map of contemporary mathematics.

In the wake of this team’s revolutionary work, new paths were opened up that seemed to leave cracks in many famous old doors, including the tightly-closed door of the small but alluring Fibonacci power mystery. And indeed, roughly ten years after the proof of Fermat’s Last Theorem, a trio of mathematicians, exploiting the techniques of Wiles and others, were able to pinpoint the exact reason for which cubic 8