Heh—I’m amazed at how many things in this post I alternately strongly agree or strongly disagree with.
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
OK… I honestly can’t comprehend how someone could simultaneously believe that “2” is just a symbol and that “two” is a blob of pure meaning. It suggests the inferential distances are actually pretty great here despite a lot of surface similarities between what Landsburg is saying and what I would say.
Instead, the point of the Peano axioms was to model what mathematicians do when they’re talking about numbers
I’m happy with this (although the next sentence suggests I may have been misinterpreting it slightly). Another way I would put it is that the Peano axioms are about making things precise and getting everyone to agree to the same rules so that they argue fairly.
Like all good models, the Peano axioms are a simplification that captures important aspects of reality without attempting to reproduce reality in detail.
I’d like an example here.
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
This is defining numbers in terms of sets, which he explicitly criticizes Yudkowsky for later. I’ll take the charitable interpretation though, which would be “Y thinks he can somehow avoid defining numbers in terms of sets… which it turns out he can’t… so if you’re going to do that anyway you may as well take the more straightforward Russell approach”
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
I really like this definition of logic! It doesn’t seem to be completely standard though, and Yudkowksy is using it in the more general sense of valid reasoning. So this point is basically about the semantics of the word “logic”.
But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.
I think this is a slight strawmanning of Yudkowsky. Here Yudkowsky is trying to define the meaning of one particular system—the natural numbers—not define the entire concept of meaning.
he’s effectively resorted to taking sets as primitive objects
So when studying logic, model theory etc. we have to make a distinction between the system of reasoning that we are studying and “meta-reasoning”. Meta-reasoning can be done in natural language or in formal mathematics—generally I prefer when it’s a bit more mathy because of the thing of precision/agreeing to play by the rules that I mentioned earlier. I don’t see math and natural language as operating at different levels of abstraction though—clearly Landsburg disagrees (with the whole 2/two thing) but it’s hard for me to understand why.
Anyway, if you’re doing model theory then your meta-reasoning involves sets. If you use natural language instead though, you’re probably still using sets at least by implication.
full-fledged Platonic acknowledgement
I think this is what using natural language as your meta-reasoning feels like from the inside. Landsburg would say that the natural numbers exist and that “two” refers to a particular one; in model theory this would be “there exists a model N with such-and-such properties, and we have a mapping from symbols “0”, “1“, “2” etc. to elements of N”.
It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”
So he’s saying that the existence of numbers implies the existence of the physical universe, and therefore that the existence of numbers is more likely?? Or is there some quality of “just out there”ness that’s distinct from existence?
Just to be clear, I assume we’re talking about the second order Peano axioms here?