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Anatoly_Vorobey comments on [LINK] Steven Landsburg "Accounting for Numbers" - response to EY's "Logical Pinpointing" - Less Wrong Discussion

9 Post author: David_Gerard 14 November 2012 12:55PM

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Comment author: Anatoly_Vorobey 15 November 2012 06:53:49PM 2 points [-]

I disagree with your example:

The rest of this proof is really complicated, and uses things you might not think are valid, so you might not buy it. But second order arithmetic demands that if you believe the rest of the proof, you should also agree that the twin primes conjecture holds.

But set theory says the same thing. And set theory, unlike second-order arithmetic, is probably strong enough to formalize the large and complicated proof in the first place. Even if there are elements in the proof that go beyond ZFC (large cardinals etc.), mathematicians are likely to view them as additional assumptions on top of what they see as set theory.

Consider a non-logician mathematician to whom the induction principle is not primarily a formal statement to be analyzed, but just, well, induction, a basic working tool. Given a large proof as you describe, ending in an application of induction. What would be the benefit, to the mathematician, of viewing that application as happening in second-order logic, as opposed to first-order set theory? Why would they want to use second-order anything?

To phrase this differently, second order arithmetic gives us a universal way of picking the natural numbers out in any situation. Different models of set theory may end up disagreeing about what's true in the natural numbers, but second order arithmetic is a consistent way of identifying the notion in that model of set theory which lines up with our intuitions for the what the natural numbers are.

I don't see how that works, either.

Let G be the arithmetical statement expressing the consistency of ZFC. There are models of set theory in which G is true, and models in which G is false. Are you saying that second-order arithmetic gives us a better way, a less ambiguous way, to study the truth of G? How would that work in practice?

The way I see it, different models of set theory agree on what natural numbers are, but disagree on what subsets of natural numbers exist. This ambiguity is not resolved by second-order arithmetic; rather, it's swept under the carpet. The "unique" model "pinpointed" by it is utterly at the mercy of the same ambiguity of what the set of subsets of N is, and the ambiguity reasserts itself the moment you start studying the semantics of second-order arithmetic which you will do through model theory, expressed within set theory. So what is it that you have gained?

So apparently whatever I thought was the natural numbers was some other bizzarre object; what the second order induction axiom does for a Platonist is pin down which Platonic object we're actually talking about.

To a Platonist, what you used was not the second order induction axiom; it was just the familiar principle of induction.

Comment author: elseif 16 November 2012 02:09:54AM 2 points [-]

I think you're reading too much into what I'm saying. I'm not suggesting that second order arithmetic is useful as a mathematical framework to talk about reasoning, in the way that first-order logic can. I'm saying that second order arithmetic is a useful way to talk about what makes the natural numbers special.

I'm also not suggesting that second order arithmetic has anything deep to add relative to a naïve (but sufficiently abstract) understanding of induction, but given that many people don't have a sufficiently abstract understanding of induction, I think it usefully illustrates just how broadly induction has to be understood. (To be clear, I'm also not suggesting that second order arithmetic is the one true way to think about the natural numbers, only that it's a useful perspective.)

Consider a non-logician mathematician to whom the induction principle is not primarily a formal statement to be analyzed, but just, well, induction, a basic working tool. Given a large proof as you describe, ending in an application of induction. What would be the benefit, to the mathematician, of viewing that application as happening in second-order logic, as opposed to first-order set theory? Why would they want to use second-order anything?

I'm not claiming there is any such benefit. This is the wrong question to ask; neither Eliezer's original post nor my comment were written for an audience of mathematicians concerned with the optimal way to do mathematics in practice.

Let G be the arithmetical statement expressing the consistency of ZFC. There are models of set theory in which G is true, and models in which G is false. Are you saying that second-order arithmetic gives us a better way, a less ambiguous way, to study the truth of G?

No, I am not saying that.

The way I see it, different models of set theory agree on what natural numbers are, but disagree on what subsets of natural numbers exist.

If there are models where G is true and models where G is false then different models of set theory don't agree on what natural numbers are.

This ambiguity is not resolved by second-order arithmetic; rather, it's swept under the carpet. The "unique" model "pinpointed" by it is utterly at the mercy of the same ambiguity of what the set of subsets of N is, and the ambiguity reasserts itself the moment you start studying the semantics of second-order arithmetic which you will do through model theory, expressed within set theory. So what is it that you have gained?

No, the ambiguity is not resolved by second-order arithmetic. The ambiguity is not resolved by anything, so that doesn't seem like a very useful objection.

Also, note that I didn't refer to either a unique model nor to pinpointing. (I'm aware that Eliezer did, but you're replying to me.) What I said was: "[S]econd order arithmetic gives us a universal way of picking the natural numbers out in any situation. Different models of set theory may end up disagreeing about what's true in the natural numbers, but second order arithmetic is a consistent way of identifying the notion in that model of set theory which lines up with our intuitions for the what the natural numbers are"

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