I Am a Strange Loop (28 page)

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of mathematicians who had set their collective bead on the “big game” of Fermat’s Last Theorem (the notorious claim, originally made by Pierre de Fermat in the middle of the seventeenth century, that no positive integers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
a
ⁿ
+
b
ⁿ
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 in order to be sure that the rule will work exactly right for a manipulator (human or otherwise) who is
not
thinking about the symbols’ intended meanings.

To give a trivial example, suppose the symbol “∨” were intended to stand for the concept “or”. Then a possible rule of inference would be:

From any formula “P ∨ Q” one can derive the reversed formula “Q ∨ P”.

This rule of inference is reasonable because whenever an or-statement (such as “You’re crazy or I’m crazy”) is true, then so is the flipped-around or-statement (“I’m crazy or you’re crazy”).

This particular ∨-flipping rule happens not to be one of
PM
’s rules of inference, but it could have been one. The point is just that this rule shows how one can mechanically shunt symbols and ignore their meanings, and yet preserve truth while doing so. This rule is rather trivial, but there are subtler ones that do real work. That, indeed, is the whole idea of symbolic logic, first suggested by Aristotle and then developed piecemeal over many centuries by such thinkers as Blaise Pascal, Gottfried Wilhelm von Leibniz, George Boole, Augustus De Morgan, Gottlob Frege, Giuseppe Peano, David Hilbert, and many others. Russell and Whitehead were simply developing the ancient dream of totally mechanizing reasoning in a more ambitious fashion than any of their predecessors had.

Mechanizing the Mathematician’s Credo

If you apply
PM’
s rules of inference to its axioms (the seeds that constitute the “zeroth generation” of theorems), you will produce some “progeny” — theorems of the “first generation”. Then apply the rules once again to the first-generation theorems (as well as to the axioms) in all the different ways you can; you will thereby produce a new batch of theorems — the second generation. Then from that whole brew comes a third batch of theorems, and so on,
ad infinitum,
constantly snowballing. The infinite body of theorems of
PM
is fully determined by the initial seeds and by the typographical “growth rules” that allow one to make new theorems out of old ones.

Needless to say, the hope here is that all of these mechanically generated theorems of
PM
are true statements of number theory (
i.e.,
no false statement is ever generated), and conversely, it is hoped that all true statements of number theory are mechanically generated as theorems of
PM
(
i.e.,
no true statement is left ungenerated forever). The first of these hopes is called
consistency,
and the second one is called
completeness.

In a nutshell, we want the entire infinite body of theorems of
PM
to coincide exactly with the infinite body of true statements in number theory — we want perfect, flawless alignment. At least that’s what Russell and Whitehead wanted, and they believed that with
PM
they had attained this goal (after all, “s0 + s0 = ss0” was a theorem, wasn’t it?).

Let us recall the Mathematician’s Credo, which in some form or other had existed for many centuries before Russell and Whitehead came along:

X is true
because
there is a proof of X;
X is true
and so
there is a proof of X.

The first line expresses the first hope expressed above — consistency. The second line expresses the second hope expressed above — completeness. We thus see that the Mathematician’s Credo is very closely related to what Russell and Whitehead were aiming for. Their goal, however, was to set the Credo on a new and rigorous basis, with
PM
serving as its pedestal. In other words, where the Mathematician’s Credo merely speaks of “a proof ” without saying what is meant by the term, Russell and Whitehead wanted people to think of it as meaning
a proof within PM.

Gödel himself had great respect for the power of
PM,
as is shown by the opening sentences of his 1931 article:

The development of mathematics in the direction of greater exactness has — as is well known — led to large tracts of it becoming formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems yet set up are, on the one hand, the system of
Principia Mathematica
(
PM
) and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them,
i. e.,
reduced to a few axioms and rules of inference.

And yet, despite his generous hat-tip to Russell and Whitehead’s opus, Gödel did not actually believe that a perfect alignment between truths and
PM
theorems had been attained, nor indeed that such a thing could
ever
be attained, and his deep skepticism came from having smelled an extremely strange loop lurking inside the labyrinthine palace of mindless, mechanical, symbol-churning, meaning-lacking mathematical reasoning.

Miraculous Lockstep Synchrony

The conceptual parallel between recursively defined sequences of integers and the leapfrogging set of theorems of
PM
(or, for that matter, of any formal system whatever, as long as it had axioms acting as seeds and rules of inference acting as growth mechanisms) suggested to Gödel that the typographical patterns of symbols on the pages of
Principia Mathematica
— that is, the rigorous logical derivations of new theorems from previous ones — could somehow be “mirrored” in an exact manner inside the world of numbers. An inner voice told him that this connection was not just a vague resemblance but could in all likelihood be turned into an absolutely precise correspondence.

More specifically, Gödel envisioned a set of whole numbers that would organically grow out of each other via arithmetical calculations much as Fibonacci’s
F
numbers did, but that would also correspond in an exact oneto-one way with the set of theorems of
PM.
For instance, if you made theorem
Z
out of theorems
X
and
Y
by using typographical rule
R5,
and if you made the number
z
out of numbers
x
and
y
using computational rule
r5,
then everything would match up. That is to say, if
x
were the number corresponding to theorem
X
and
y
were the number corresponding to theorem
Υ,
then
z
would “miraculously” turn out to be the number corresponding to theorem
Z.
There would be perfect synchrony; the two sides (typographical and numerical) would move together in lock-step. At first this vision of miraculous synchrony was just a little spark, but Gödel quickly realized that his inchoate dream might be made so precise that it could be spelled out to others, so he started pursuing it in a dogged fashion.

Flipping between Formulas and Very Big Integers

In order to convert his intuitive hunch into a serious, precise, and respectable idea, Gödel first had to figure out how any string of
PM
symbols (irrespective of whether it asserted a truth or a falsity, or even was just a random jumble of symbols haphazardly thrown together) could be systematically converted into a positive integer, and conversely, how such an integer could be “decoded” to give back the string from which it had come. This first stage of Gödel’s dream, a systematic mapping by which every formula would receive a numerical “name”, came about as follows.

The basic alphabet of
PM
consisted of only about a dozen symbols (other symbols were introduced later but they were all defined in terms of the original few, so they were not conceptually necessary), and to each of these symbols Gödel assigned a different small integer (these initial few choices were quite arbitrary — it really didn’t matter what number was associated with an isolated symbol).

For multi-symbol formulas (by the way, in this book the terms “string of symbols” — “string” for short — and “formula” are synonymous), the idea was to replace the symbols, one by one, moving left to right, by their code numbers, and then to combine all of those individual code numbers (by using them as exponents to which successive prime numbers are raised) into one unique big integer. Thus, once
isolated
symbols had been assigned numbers, the numbers assigned to
strings
of symbols were
not
arbitrary.

For instance, suppose that the (arbitrary) code number for the symbol “0” is 2, and the code number for the symbol “=” is 6. Then for the three symbols in the very simple formula “0=0”, the code numbers are 2, 6, 2, and these three numbers are used as
exponents
for the first three prime numbers (2, 3, and 5) as follows:

2
²
· 3
⁶
· 5
²
= 72900

So we see that 72900 is the single number that corresponds to the formula “0=0”. Of course this is a rather large integer for such a short formula, and you can easily imagine that the integer corresponding to a fifty-symbol formula is astronomical, since it involves putting the first fifty prime numbers to various powers and then multiplying all those big numbers together, to make a true colossus. But no matter — numbers are just numbers, no matter how big they are. (Luckily for Gödel, there are infinitely many primes, since if there had been merely, say, one billion of them, then his method would only have let him encode formulas made of a billion symbols or fewer. Now that would be a crying shame!)