I Am a Strange Loop - Douglas R. Hofstadter [81]
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of mathematicians who had set their collective bead on the “big game” of Fermat’s Last Theorem (the notorious claim, originally made by Pierre de Fermat in the middle of the seventeenth century, that no positive integers a, b, c exist such that an + bn equals c n, with the exponent n being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 and square 144 will never have any perfect-power mates in Leonardo di Pisa’s recursive sequence (except for 1). Though extremely recondite, the reason behind the infinite mutual-avoidance dance had been found. This is just one more triumph of the Mathematician’s Credo — one more reason to buy a lot of stock in the idea that in mathematics, where there’s a pattern, there’s a reason.
A Tiny Spark in Gödel’s Brain
We now return to the story of Kurt Gödel and his encounter with the powerful idea that all sorts of infinite classes of numbers can be defined through various kinds of recursive rules. The image of the organic growth of an infinite structure or pattern, all springing out of a finite set of initial seeds, struck Gödel as much more than a mere curiosity; in fact, it reminded him of the fact that theorems in PM (like theorems in Euclid’s Elements) always spring (by formal rules of inference) from earlier theorems in PM, with the exception of the first few theorems, which are declared by fiat to be theorems, and thus are called “axioms” (analogues to the seeds).
In other words, in the careful analogy sparked in Gödel’s mind by this initially vague connection, the axioms of PM would play the role of Fibonacci’s seeds 1 and 2, and the rules of inference of PM would play the role of adding the two most recent numbers. The main difference is that in PM there are several rules of inference, not just one, so at any stage you have a choice of what to do, and moreover, you don’t have to apply your chosen rule to the most recently generated theorem(s), so that gives you even more choice. But aside from these extra degrees of freedom, Gödel’s analogy was very tight, and it turned out to be immensely fruitful.
Clever Rules Imbue Inert Symbols with Meaning
I must stress here that each rule of inference in a formal system like PM not only leads from one or more input formulas to an output formula, but it does so by purely typographical means — that is, via purely mechanical symbol-shunting that doesn’t require any thought about the meanings of symbols. From the viewpoint of a person (or machine) following the rules to produce theorems, the symbols might as well be totally devoid of meaning.
On the other hand, each rule has to be very carefully designed so that, given input formulas that express truths, the output formula will also express a truth. The rule’s designer (Russell and Whitehead, in this case) therefore has to think about the symbols’ intended meanings