Over at Less Wrong, the estimable Eliezer Yudkowsky attempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
1. Yudkowsky poses, in essence, the following question:
Yudkowsky phrases the question a little differently. What he actually asks is:
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2″, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
2. Yudkowsky provisionally offers the following answer:
He then provisionally rejects this provisional answer on the grounds (with which I wholeheartedly agree) that “figuring out facts about the natural numbers doesn’t feel like the operation of making up assumptions and then deducing conclusions from them.” He goes on to say: “It feels like the numbers are just out there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them.”
He’s certainly right that it feels — to me, and, I am sure to almost everyone who has ever thought much about arithmetic — like the numbers are just “out there”. On the other hand, I’d quibble with Yudkowsky’s assessment of the point of Peano arithmetic. The point isn’t to “allow mathematicians to talk” about numbers; mathematicians from Pythagoras through Dedekind had absolutely no problem talking about numbers in the absence of the Peano axioms. Instead, the point of the Peano axioms was to model what mathematicians do when they’re talking about numbers. Like all good models, the Peano axioms are a simplification that captures important aspects of reality without attempting to reproduce reality in detail.
3. To reach a closer understanding of what numbers are, Yudkowsky imagines trying to explain them to a logician with a full grasp of logic but no grasp of numbers. Here I think Yudkowsky has fooled himself into imagining an impossibility. If you grasp logic, you grasp the idea of a proof. If you grasp the idea of a proof, you grasp the idea of a sequence of logical steps. If you grasp the idea of a sequence of logical steps, you grasp the idea of a sequence. If you grasp the idea of a sequence, you already know a lot about numbers. This is one reason why I believe that any attempt to account for numbers via logic must ultimately be circular.
4. Be that as it may, Yudkowsky goes on to try to explain to his fictional interlocutor what numbers are. He begins by essentially stating the first order Peano axioms: 0 is a number, every number has a successor, no two numbers have the same successor, and so forth. Eventually, he realizes that this approach isn’t taking him quite where he wants to go and makes a bit of a course correction (as we’ll see below). But I think more than a course correction is called for; he’s gone off in entirely the wrong direction. He’s listing the properties of numbers, but not even trying to explain what they are. If I were explaining numbers to a naif, I’d probably start with something like Bertrand Russell’s account of numbers: We say that two sets of objects are “equinumerous” if they can be placed in one-one correspondence with each other; a “number” is that which all sets equinumerous to a given set have in common. Whether or not that works in detail, it’s at least an attempt at a definition, as opposed to a mere list of properties.
5. Yudkowsky, in his fictional conversation with his fictional logician, eventually comes to realize that neither the first order Peano axioms nor any other first-order system can uniquely characterize the natural numbers. This is a consequence of Godel’s Incompleteness Theorem, or even more fundamentally of the Lowenheim-Skolem Theorem. What it means is that no matter what axioms you start with, there are going to be multiple systems that satisfy those axioms; the natural numbers are only one of those systems, so your axioms cannot collectively specify the natural numbers.
6. Yudkowsky solves his problem by passing to second order Peano arithmetic — “second order” meaning that, in addition to using variables to represent numbers, you can also use variables to to represent sets of numbers. He correctly notes that second order Peano arithmetic has a unique model. (I am using the word “model” here in the technical sense of logic, not in the informal social-sciencey sort of way that I used it in point 2 above.) This means that sure enough, there is one and only one system that satisfies all the axioms of second-order arithmetic. And he concludes that:
But this is disastrously wrong for at least two reasons, each of which deserves its own numbered point.
7. Yudkowsky leaps from “the natural numbers can be precisely specified by second order logic” to “the .. study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms”. This is wrong, wrong, wrong, because second order logic is not logic. Indeed, the whole point of logic is that it is a mechanical system for deriving inferences from assumptions, based on the forms of sentences without any reference to their meanings. (Thus if we assume that all bachelors are unmarried and that Walter is a bachelor, we can infer that Walter is unmarried, without having to know anything at all about who walter is, or what the words “bachelor” and ‘unmarried” mean.) That’s why you’re not allowed to set up an axiom system in which all the true theorems of arithmetic are taken as axioms — there is no mechanical procedure for determining whether a given statement is or is not a true theorem of arithmetic (see Tarski’s theorem on the undefinability of truth) and therefore no mechanical procedure for determining what is or is not an axiom in that system. In second-order Peano arithemetic, we have an analogous problem: The axioms can be identified mechanically, but the rules of inference can’t. A properly programmed computer can examine a first-order proof and tell you if it’s valid or not; that is, it can tell you whether each step does in fact follow logically from some of the previous steps. But no computer can do the same for second-order proofs.
So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the forms of sentences but their meanings. In other words, you have to understand meanings before you can carry out the operation of inference. But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.
8. Even putting all that aside, Yudkowsky is relying on a theorem when he says that second-order Peano arithmetic has a unique model. That theorem requires a substantial dose of set theory. So in order to avoid taking numbers as primitive objects, he’s effectively resorted to taking sets as primitive objects. But why is it any more satisfying to take set theory as “given” than to take numbers as “given”? Indeed, the formal study of numbers precedes the formal study of sets by millennia, which suggests that numbers are a more natural starting point than sets are. Whether or not you buy that argument, it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.
9. All of which leaves us with the problem of accounting for numbers, and for the meaning of statements like “two plus two equals four”. To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary. I’ve given an account in The Big Questions of how the existence of numbers can account for the existence of the physical universe; I think it would be very difficult to go in the opposite direction (though I’ve seen some pretty good attempts). Therefore, accepting numbers as primary and accounting for the universe as a necessary consequence of numbers seems to me to be the ontologically parsimonious thing to do, and I like parsimony.
10. Needless to say, point 9 is not a proof. But I know of no alternative story that strikes me as even remotely plausible. Moreover, the alternative stories all seem to go wrong in pretty much the same ways; for example, every single point above is one I’ve blogged about before in other contexts, but here they are, all being relevant again.