Douglas Hofstadter - I Am a Strange Loop

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).