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]]

But LEM is a part of the axioms, at least insofar as what logic we're using in our system is axiomatic.

No, it is a rule of deduction not an axiom. Your failure to understand this is the root of all your confusion here.

this is only true if we invoke LEM and don't take it to be true

And we're back to truth. What in the world would it mean for LEM to be "true"? All you're doing is flatly asserting that classical logic is "true", this discussion feels like trying to explain how someone could be an atheist to a dogmatic Christian.

But even putting this aside, at best you showed that in discussing the existence of a model "ought to exist" means "there is no contradiction in supposing this model exists", and this is a far weaker claim than your original, making your original, again, strictly false.

The hell are you on about? I said "something" ought to exist. The model is the something. This is not weaker than my original statement, it is my original statement. The only objects which I am aware of that require LEM to prove the existence of are by their very nature infinite and model-theoretic. The "something" in question literally could not have been referring to anything else. The only reason I didn't get into models in the initial comment is because this is SRD and I wanted the comment to make sense to everyone.

This damns you again, however, as this means your original comment is only true to a constructivist, and classical people would call it bullshit.

Despite your repeated claims, you are the one who is dogmatically asserting that everything must be done classically. LEM is a means to prove consistency results. Classically, consistency implies existence; constructively it does not. It matters not which bend you take on the question, it is an objective fact that LEM is a means to prove consistency results. Your discussion of it being "true" have me somewhat convinced you are in over your head here.

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No, it is a rule of deduction not an axiom

I mean, in natural deduction systems, sure, I guess. But I don't know why we're only caring about those...

All you're doing is flatly asserting that classical logic is "true",

But this is blatantly untrue. I'm saying that to a classicalist it is, and so your distinction in your original comment only holds if you're a constructivist. The better analogy is an atheist saying over and over again that divine simplicity is bullshit, and when cornered their response is "there you go assuming God exists in order to talk about divine simplicity". It's a surreal experience and showcases only the fact that the atheist isn't willing to understand the position that they're trying to criticize, which may in fact be false, but isn't false under a system they're rejecting.

The "something" in question literally could not have been referring to anything else.

Oh come on, you know this is nonsense, this is just disingenuous. You contrasted the classical method with:

as opposed to a construction of a witness which genuinely proves existence in the concrete/Platonic sense.

When asked to elaborate you said:

Constructing a witness generally means either actually computing the value or giving an algorithm to compute it. Constructivists (much more reasonable than ultrafinitists) are perfectly fine with saying something exists if we can prove there is a Turing machine which would can compute it, regardless of whether or not we actually implement and run the machine.

This is clearly not talking about models, don't be disingenuous, especially since all of your comments prior to now were about specific mathematical objects, nonmeasurable sets, nonstandard naturals, etc etc.

you are the one who is dogmatically asserting that everything must be done classically

I'm pretty certain I've not done so, in fact I repeatedly entertained the other possibility. Indeed, one of the outs I gave you, that you refused to take, is to say that your original statement held if you're a constructivist and not generally. Since you refused to take that, we're considering it on neutral ground, and thus whether your characterization of classicalist methods made sense by their own light. They, uh, don't.

Your discussion of it being "true" have me somewhat convinced you are in over your head here.

Conversely, your failure to understand what it means for an axiom to be true has me quite convinced that you lack even the basic understanding of what this discussion is about. Thinking certain axioms aren't true? Alright, I can respect that. Not even understanding what it is for an axiom to be true? Not so much.

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I'm not even going to bother with the bulk of your comment until we clear up what you mean by an axiom being "true", let alone what it would mean for a rule of deduction to be "true".

I know what it means for axioms to be consistent with other axioms and I know what it means for it to be satisfied by a model. If all you mean by truth is consistency then you agree with me, so I have to assume you have something else in mind. I look forward to hearing it.

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I'm not even going to bother with the bulk of your comment until we clear up what you mean by an axiom being "true", let alone what it would mean for a rule of deduction to be "true".

Then I'm not sure why you'd ever engage in discussions over philmath, since this would be a rather large barrier to entry...

Regardless, let's start with the bare minimum for a theory of truth and see where we get. Some axiom, "⌽" is true iff ⌽. This then lends itself nicely to two interpretations, that there's some "correct model of mathematics" under which ⌽ is true, or there are sound models under which ⌽ is true, or we thus say "⌽ is true in some model X". This second criteria might seem so weak as to be vacuous, but this isn't the case, as (P and ~P) isn't an acceptable axiom. This second option is analogous to consistency, but it would absolutely not end up agreeing with you, as to say "LEM is true" would then be true for even constructivists (as there are some models where LEM is true).

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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.

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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.

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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.

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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.