I Am a Strange Loop (31 page)

Gödel, analogously, created a “subjectless formula fragment” (by which I mean a
PM
formula that is not about any specific integer, but just about some unspecified variable number
x
). And then, making a move analogous to that of feeding Quine’s Quasi-Quip into itself (but in quotes), he took that formula fragment’s Gödel number
k
(which is a specific number, not a variable) and replaced the variable
x
by it, thus producing a formula (not just a fragment) that made a claim about a much larger integer,
g.
And
g
is the Gödel number of that very claim. And last but not least, the claim was not about whether the entity in question was a full sentence or not, but about whether the entity in question was a
provable formula
or not.

An Elephant in a Matchbox is Neither Fish Nor Fowl

I know this is a lot to swallow in one gulp, and so if it takes you several gulps (careful rereadings), please don’t feel discouraged. I’ve met quite a few sophisticated mathematicians who admit that they never understood this argument totally!

I think it would be helpful at this juncture to exhibit a kind of hybrid sentence that gets across the essential flavor of Gödel’s self-referential construction but that does so in Quinean terms — that is, using the ideas we’ve just been discussing. The hybrid sentence looks like this:

“when fed its own Gödel number yields a non-prim number”
when fed its own Gödel number yields a non-prim number.

The above sentence is neither fish nor fowl, for it is not a formula of
Principia Mathematica
but an English sentence, so of course it doesn’t have a Gödel number and it couldn’t possibly be a theorem (or a nontheorem) of
PM.
What a mixed metaphor!

And yet, mixed metaphor though it is, it still does a pretty decent job of getting across the flavor of the
PM
formula that Gödel actually concocted. You just have to keep in mind that using quote marks is a metaphor for taking Gödel numbers, so the upper line should be thought of as being a Gödel number (
k
) rather than as being a sentence fragment in quote marks. This means that metaphorically, the lower line (an English sentence fragment)
has
been fed its own Gödel number as its subject. Very cute!

I know that this is very tricky, so let me state it once again, slightly differently. Gödel asks you to imagine the formula that
k
stands for (that formula happens to contain the variable
x
), and then to feed
k
into it (this means to replace the single letter
x
by the extremely long numeral
k,
thus giving you a much bigger formula than you started with), and to take the Gödel number of the result. That will be the number
g,
huger far than
k
— and lastly, Gödel asserts that
this
walloping number is not a prim number. If you’ve followed my hand-waving argument, you will agree that the full formula’s Gödel number (
g
) is not found explicitly inside the formula, but instead is very subtly
described
by the formula. The elephant’s DNA has been used to get a description of the entire elephant into the matchbox.

Sluggo and the Morton Salt Girl

Well, I don’t want to stress the technical points here. The main thing to remember is that Gödel devised a very clever number-description trick — a recipe for making a very huge number
g
out of a less huge number
k
— in order to get a formula of
PM
to make a claim about its own Gödel number’s non-primness (which means that the formula is actually making a claim of its own nontheoremhood). And you might also try to remember that the “little” number
k
is the Gödel number of a “formula fragment” containing a variable
x,
analogous to a subjectless sentence fragment in quote marks, while the larger number
g
is the Gödel number of a
complete sentence in PM notation,
analogous to a complete sentence in English.

Popular culture is by no means immune to the delights of self-reference, and it happens that the two ideas we have been contrasting here — having a formula contain its own Gödel number
directly
(which would necessitate an infinite regress) and having a formula contain a
description
of its Gödel number (which beautifully bypasses the infinite regress) — are charmingly illustrated by two images with which readers may be familiar.

In this first image, Ernie Bushmiller’s character Sluggo (from his classic strip
Nancy
) is dreaming of himself dreaming of himself dreaming of himself, without end. It is clearly a case of self-reference, but it involves an infinite regress, analogous to a
PM
formula that contained its own Gödel number directly. Such a formula, unfortunately, would have to be infinitely long!

Our second image, in contrast, is the famous label of a Morton Salt box, which shows a girl holding a box of Morton Salt. You may think you smell infinite regress once again, but if so, you are fooling yourself! The girl’s arm is covering up the critical spot where the regress would occur. If you were to ask the girl to (please) hand you her salt box so that you could actually
see
the infinite regress on its label, you would wind up disappointed, for the label on
that
box would show her holding a yet smaller box with her arm once again blocking the regress.

And yet we still have a self-referential picture, because customers in the grocery store understand that the little box shown on the label is the same as the big box they are holding. How do they arrive at this conclusion? By using analogy. To be specific, not only do they have the large box in their own hands, but they can see the little box the girl is holding, and the two boxes have a lot in common (their cylindrical shape, their dark-blue color, their white caps at both ends); and in case that’s not enough, they can also see salt spilling out of the little one. These pieces of evidence suffice to convince everyone that the little box and the large box are identical, and there you have it: self-reference without infinite regress!

In closing this chapter, I wish to point out explicitly that the most concise English translations of Gödel’s formula and its cousins employ the word “I” (“I am not provable in
PM
”; “I am not a
PM
theorem”). This is not a coincidence. Indeed, this informal, almost sloppy-seeming use of the singular first-person pronoun affords us our first glimpse of the profound connection between Gödel’s austere mathematical strange loop and the very human notion of a conscious self.

CHAPTER 11

How Analogy Makes Meaning

The Double Aboutness of Formulas in PM

I
MAGINE the bewilderment of newly knighted Lord Russell when a young Austrian Turk named “Kurt” declared in print that
Principia Mathematica,
that formidable intellectual fortress so painstakingly erected as a bastion against the horrid scourge of self-referentiality, was in fact riddled through and through with formulas allegedly stating all sorts of absurd and incomprehensible things about themselves. How could such an outrage ever have been allowed to take place? How could vacuously twittering self-referential propositions have managed to sneak through the thick ramparts of the beautiful and timeless Theory of Ramified Types? This upstart Austrian sorcerer had surely cast some sort of evil spell, but by what means had he wrought his wretched deed?

The answer is that in his classic article — “On Formally Undecidable Propositions of
Principia Mathematica
and Related Systems (I)” — Gödel had re-analyzed the notion of
meaning
and had concluded that what a formula of
PM
meant was not so simple — not so unambiguous — as Russell had thought. To be fair, Russell himself had always insisted that
PM
’s strange-looking long formulas had
no
intrinsic meaning. Indeed, since the theorems of
PM
were churned out by formal rules that paid no attention to meaning, Russell often said the whole work was just an array of meaningless marks (and as you saw at the end of Chapter 9, the pages of
Principia Mathematica
often look more like some exotic artwork than like a work of math).

And yet Russell was also careful to point out that all these curious patterns of horseshoes, hooks, stars, and squiggles
could
be interpreted, if one wished, as being statements about numbers and their properties, because under duress, one could read the meaningless vertical egg ‘0’ as standing for the number zero, the equally meaningless cross ‘+’ as standing for addition, and so on, in which case all the theorems of
PM
came out as statements about numbers — but not just random blatherings about them. Just imagine how crushed Russell would have been if the squiggle pattern “ss0 + ss0 = sssss0” turned out to be a theorem of
PM
! To him, this would have been a disaster of the highest order. Thus he had to concede that there
was
meaning to be found in his murky-looking tomes (otherwise, why would he have spent long years of his life writing them, and why would he care which strings were theorems?) — but that meaning depended on using a
mapping
that linked shapes on paper to abstract magnitudes (
e.g.,
zero, one, two…), operations (
e.g.,
addition), relationships (
e.g.,
equality), concepts of logic (
e.g.,
“not”, “and”, “there exists”, “all”), and so forth.