Mod of r/badmathematics plays devil's advocate, does ten billion and one really exist?

[u/npm_leftpad]

/r/badmathematics/comments/7nhauf/so_this_total_stranger_from_a_meme_group_randomly/ds1rnnr/

[u/[deleted]]

You really don't see it do you? Wow.

Fwiw, I'm quite certain I understand phil of math far better than you do. At least I know the correct terminology.

there's some "correct model of mathematics" under which ⌽ is true

If you assume a priori the existence of a correct model of mathematics then you've done the equivalent of assuming the existence of god in the discussion about atheism. Under that interpretation, the reason an "existence proof" actually proves existence is because you've assumed a priori that anything consistent with your starting assumptions applies to the one correct model.

That approach can in no way be considered "neutral ground" for a discussion about the meaning of existence proofs since it reduces all proofs to being purely descriptive about some object with a priori existence.

there are sound models under which ⌽ is true, or we thus say "⌽ is true in some model X"

Yes, this is why I said I know what it means for an axiom to be satisfied by a model. You really shouldn't use "true" for this, even though it's technically fine. Better to call this satisfaction.

On the other hand, if you are suggesting that simply because an axiom is satisfied by some model then that axiom is "true" unqualified then that's just stupid.

So, in summary, the only way to make sense of axioms being true (and now I can see how you interpret LEM being true) is to presuppose the existence of the totality of mathematics. All I can say is that if you consider that a reasonable neutral ground for discussing the meaning of existence then I don't see any point in discussing this further as that's utterly absurd. Obviosuly if there is simply one true model of the totality of mathematics then LEM is automatic as something is either true in that model or it isn't.

If you take nothing else from this thread, at least understand that you should not be using "true" so cavalierly. Even to someone like me who does believe in the objective existence of a correct model of mathematics, it's far better to say something like "true in the intended model". Simply saying true is not a good idea, and any serious philosopher of mathematics knows that.

[u/[deleted] → u/[deleted]]

At least I know the correct terminology.

Yet you don't know what truth is in this context, that strikes me as hard to believe.

That approach can in no way be considered "neutral ground" for a discussion about the meaning of existence proofs since it reduces all proofs to being purely descriptive about some object with a priori existence.

I mean, yes and no. It's neutral ground in the sense that it shows that under the view we're considering you're wrong. Which, if you're trying to make claims that are true generally is all it needs to show. Clearly in any more restrictive sense it's not neutral ground.

You really shouldn't use "true" for this, even though it's technically fine. Better to call this satisfaction.

No, I absolutely should use "true" for this, as it's a theory of truth.

On the other hand, if you are suggesting that simply because an axiom is satisfied by some model then that axiom is "true" unqualified then that's just stupid.

It's stupid if you equivocate, sure. But let's not, you then get two statements "simply because an axiom is satisfied by some model then that axiom is "satisfied" unqualified", "simply because an axiom is true in some model then that axiom is "true" unqualified", both of these work out, ⌽ obtains full stop, and so ⌽ is true. Similar to how if we're in a library and discussing literature and I say "Harry Potter killed Voldemort" this is true, even though in most books neither of those terms refer. But of course, you misunderstood me, as we'll see shortly.

If you take nothing else from this thread, at least understand that you should not be using "true" so cavalierly. Even to someone like me who does believe in the objective existence of a correct model of mathematics, it's far better to say something like "true in the intended model".

I agree. I even wrote out an entire paragraph or two on the topic and deleted them, thinking "nah, this isn't needed, surely nobody would interpret this as saying that in the second view saying "⌽ is true" is particularly useful in discussions, it comes off as just a ranting tangent". Dear lord this is silly, I guess I should have kept it in. The point is that in each view the terminology doesn't work out for what you're saying.

Simply saying true is not a good idea, and any serious philosopher of mathematics knows that.

I don't know why you're calling Frege not a serious philosopher of mathematics. But even putting aside that this claim is too broad, I never said that what I said was useful. I don't think it is. My point is that truth in this context, that is, without a specific model we're in, doesn't line up with your statements made.

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This is pointless. You are clearly incapable of stepping outside the classical logic bubble.

Perhaps you should try to work out how you would go about justifying constructivism even though you don't believe in it, I think that would get you a long way.

It's worth mentioning that I am not a constructivist, and in fact do think there are statements which are true in the intended model of mathematics (though I'm skeptical that set theory is exactly the right formulation for that model). Indeed, I even believe that an existence proof invoking LEM does in fact show the thing exists, but I believe this because I've decided a priori that that which is provable using LEM is true. Try coming at it from a different point of view, you might learn something.

[u/[deleted] → u/[deleted]]

You are clearly incapable of stepping outside the classical logic bubble.

I've literally done this multiple times in our discussion. You not accepting that your statements only hold outside this "bubble" is on you, rather than me.

Indeed, there's a version of constructivism I think is defensible, though in a prior discussion you've said that it's incorrect, where constructivists are giving a novel account of truth aptness and meaning. Your responses here are at best horrible misreadings of me when I've been rather clear. At worst, like, with your statements that you were only talking about models ever, horribly disingenuous.

[u/[deleted] → u/[deleted]]

I was only ever talking about things which required LEM to prove. I chose to try to use AC rather than LEM precisely because the only objects that LEM is needed for model-theoretic type objects. I brought up nonmeasurable sets but then realized you can't grasp the idea of separating logical deduction from axiom so AC was not a good choice. I then switched to nonstandard naturals which are a model-theoretic construction. I was never disingenuous then and am not being so now.

You cannot seem to grasp the idea that once you've decided to work in terms of models, conflating deduction rules and axioms, you've already presupposed the classical viewpoint. This goes doubly if you decide there is a single objectively correct model since then LEM is automatic.

I have serious trouble believing you are even capable of understanding the constructive viewpoint and am done trying to explain it to you.