Douglas Hofstadter - I Am a Strange Loop

Just as some people’s curiosity was fired by the fact of seeing a square in Fibonacci’s recursively defined sequence, so some people might become interested in the question as to whether there are any squares (or cubes, etc.) in the recursively defined sequence of wff numbers. They could spend a lot of time investigating such purely number-theoretical questions, never thinking at all about the corresponding formulas of Principia Mathematica.

One could be completely ignorant of the fact that Gödel’s wff numbers had their origin in Russell and Whitehead’s rules defining well-formedness in Principia Mathematica, just as one can study the laws of probability without ever suspecting that this deep branch of mathematics was originally developed to analyze gambling. What long ago inspired someone to dream up a particular recursive definition obviously doesn’t affect the numbers it defines; all that matters is that there should be a purely computational way of making any member of the club grow out of the initial seeds by applying the rules some finite number of times.

Now wff numbers are, as it happens, relatively easy to define in a recursive fashion, and for that reason wff-ness (exactly like F- ness) is just the kind of mathematical notion that Principia Mathematica was designed to study. To be sure, Whitehead and Russell had never dreamed that their mechanical reasoning system might be put to such a curious use, in which its own properties as a machine were essentially placed under observation by itself, rather like using a microscope to examine some of its own lenses for possible defects. But then, inventions often do surprise their inventors.

Prim Numbers

Having realized that some hypothetical volume of the series by Whitehead and Russell could define and systematically explore the various numerical properties of wff numbers, Gödel pushed his analogy further and showed, with a good deal of fancy machinery but actually not very much conceptual difficulty, that there was an infinitely more interesting recursively defined class of whole numbers, which I shall here call prim numbers (whimsically saluting the title of the famous three tomes), and which are the numbers belonging to provable formulas of PM ( i.e., theorems).

A PM proof, of course, is a series of formulas leading from the axioms of PM all the way to the formula in question, each step being allowed by some rule of reasoning, which in PM became a formal typographical rule of inference. To every typographical rule of inference acting on strings of PM, Gödel exhibited a perfectly matching computational rule that acted on numbers. Numerical computation was effectively thumbing its nose at typographical manipulation, sassily saying, “Anything you can do, I can do better!” Well, not really better — but the key point, as Gödel showed beyond any doubt, was that a computational rule would always be able to mimic perfectly — to keep in perfect synchrony with — any formal typographical rule, and so numerical rules were just as good.

The upshot was that to every provable string of Russell and Whitehead’s formal system, there was a counterpart prim number. Any integer that was prim could be decoded into symbols, and the string you got would be a provable-in- PM formula. Likewise, any provable-in- PM formula could be encoded as one whopping huge integer, and by God, with enough calculation, you could show that that number was a prim number. A simple example of a prim number is, once again, our friend 72900, since the formula “0=0”, over and above being a well-formed formula, is also, and not too surprisingly, derivable in PM. (Indeed, if it weren’t, PM would be absolutely pathetic as a mechanical model of mathematical reasoning!)

There is a crucial difference between wff numbers and prim numbers, which comes from the fact that the rules of inference of PM sometimes produce output strings that are shorter than their input strings. This means that the corresponding arithmetical rules defining prim numbers will sometimes take large prim numbers as input and make from them a smaller prim number as output. Therefore, stretches of the number line that have been visited once can always be revisited later, and this fact makes it much, much harder to determine about a given integer whether it is prim or not. This is a central and very deep fact about prim numbers.

Just as with squares, primes, F numbers, or wff numbers, there could once again be a hypothetical volume of the series of tomes by Whitehead and Russell in which prim numbers were defined and their mathematical properties studied. For example, such a volume might contain a proof of the formula of PM that (when examined carefully) asserts “72900 is a prim number”, and it might also discuss another formula that could be seen to assert the opposite (“72900 is not a prim number”), and so on. This latter statement is false, of course, while the former one is true. And even more complex number-theoretical ideas could be expressed using the PM notation and discussed in the hypothetical volume, such as “There are infinitely many prim numbers” — which would be tantamount to asserting (via a code), “There are infinitely many formulas that are provable in PM ” .

Although it might seem an odd thing to do, one could certainly pose eighteenth-century–style number-theory questions such as, “Which integers are expressible as the sum of two prim numbers, and which integers are not?” Probably nobody would ever seriously ask such an oddball question, but the point is that the property of being a prim number, although it’s a rather arcane “modern” property, is no more and no less a genuinely number-theoretical property of an integer than is a “classical” property, such as being square or being prime or being a Fibonacci number.

The Uncanny Power of Prim Numbers

Suppose someone told you that they had built a machine — I’ll dub it “Guru” — that would always correctly answer any question of the form “Is n a prime number?”, with n being any integer that you wish. When asked, “Is 641 prime?”, Guru would spin its wheels for a bit and then say “yes”. As for 642, Guru would “think” a little while and then say “no”. I suppose you would not be terribly surprised by such a machine. That such a machine can be realized, either in silicon circuitry on in domino-chain technology, is not anything to boggle anyone’s mind in this day and age.

But suppose someone told you that they had built an analogous machine — I’ll dub it “Göru” — that would always correctly answer any question of the form “Is n a prim number?” Would this claim — strictly analogous to the previous one — strike you as equally ho-hum? If so, then I respectfully submit that you’ve got another think coming.

The reason is this. If you believed Göru to be reliable and you also believed in the Mathematician’s Credo ( Principia Mathematica version), then you could conclude that your little Göru, working all by itself, could answer any number-theoretical question that you were interested in, just like a genie conjured from a magic lamp. How so? What makes Göru a magic genie?

Well, suppose you wanted to know if statement X is true or false (for instance, the famous claim “Every even number greater than 2 is the sum of two primes” — which, as I stated above, remains unsettled even today, after nearly three centuries of work). You would just write X down in the formal notation of PM, then convert that formula mechanically into its Gödel number x, and feed that number into Göru (thus asking if x is prim or not). Of course x will be a huge integer, so it would probably take Göru a good while to give you an answer, but (assuming that Göru is not a hoax) sooner or later it would spit out either a “yes” or a “no”. In case Göru said “yes”, you would know that x is a prim number, which tells you that the formula it encodes is a provable formula, which means that statement X is true. Conversely, were Göru to tell you “no”, then you would know that the statement X is not provable, and so, believing in the Mathematician’s Credo ( Principia Mathematica version), you would conclude it is false.