Pretty good. So do we want to say that this example has met the challenge?:

OK, the challenge is to come up with something that is both a) inexpressible, and b) whose inexpressibility can be explained. It also ought to be something worth worrying about, — J

I'd be inclined to say yes. You? I suppose we could quibble about whether your account, above, really counts as an explanation, but I think it does. It's certainly an elucidation. Nor do I see us falling into the contradiction dilemma; we're not saying "ineffable" with one mouth while making it effable (is that a word?!) with the other.

Numbers exist. 2 is a number, therefore there are numbers. — Banno

What? Let's formalize the argument a bit:

First Premise: Numbers exist.
Second Premise: 2 is a number.
Conclusion: Therefore, there are numbers.

What? I'm not buying it. I'm not even sure that's valid, deductive reasoning. But let's suppose that your argument is somehow valid. Then I'll just deny the First Premise, mate. Technically speaking, numbers don't exit. You can pretend that they do, but they don't. If you say something insane and wild from an ontological POV, a statement like "Numbers exist", I'll tell you one of two things:

1) That no, numbers don't exist. Or:
2) That they exist in the same sense as Pegasus, or any other fictional entity. As in, they don't exist. So here's where Bunge would step in and he would say "Right, but people need to believe in something. So, let's talk about "conceptual existence" and "real existence". To exist conceptually is to be a part of a conceptual context, such as Greek mythology, or Euclidean geometry. And to exist really, is to be a mereological part of the world. This is where I disagree with Bunge. I believe that to exist is to have a spatiotemporal location. But I'll just quote you my favorite quote of his, to see what you make of it:

For example, the Pythagorean theorem exists in the sense that it belongs in Euclidean geometry. Surely it did not come into existence before someone in the Pythagorean school invented it. But it has been in conceptual existence, i.e. in geometry, ever since. Not that geometry has an autonomous existence, i.e. that it subsists independently of being thought about. It is just that we make the indispensable pretence that constructs exist provided they belong in some body of ideas—which is a roundabout fashion of saying that constructs exist as long as there are rational beings capable of thinking them up. Surely this mode of existence is neither ideal existence (or existence in the Realm of Ideas) nor real or physical existence. To invert Plato’s cave metaphor we may say that ideas are but the shadows of things—and shadows, as is well known, have no autonomous existence. — Mario Bunge

Well, there's a quibble here about what it is to express something. I don't think we've said something that is ineffable. We might have waved at something ineffable. That was the reservation I wanted to capture, when I said:

If something is inexpressible, then by that very fact one cannot say why... Doing so would be to give expression to the inexpressible. — Banno

In that spirit, we haven't explained its inexpressibility as much as exhibited it.

There will be plenty of folk who say Moore has proven that there is a hand, and others who say he has done no such thing, just as there are folk who see the duck but no the rabbit.

What? I'm not buying it. — Arcane Sandwich

Me either. It should be

2 is a number
Therefore there are numbers.
Hence numbers exist.

Which is an instance of f(a) ⊢ ∃x(f(x))

I thought we'd agreed on this.

Me either. It should be

2 is a number
Therefore there are numbers.
Hence numbers exist.

Which is an instance of f(a) ⊢ ∃x(f(x))

I thought we'd agreed on this. — Banno

But like, mate, you know what we're arguing right now? As in, right now at page 'effin 12 of this thing? Here's how I would describe it: we are debating the semantics of the rules. Like, I told you what I think, I gave you the example of Pegasus:

(1) ∀x(x = x) - Principle of Identity.
(2) p = p. From (1), by universal elimination.
(3) ∃x(x = p). From (2), by existential introduction.

Now, what does that mean? It means this:

(1) Everything is identical to itself.
(2) So, Pegasus is identical to Pegasus.
(3) So, Pegasus exists.

So, I ask you, have you seen a flying horse somewhere, mate? Of course you haven't. And that's my whole point. Like, here's a hand mate, here's a Moore-like argument that refutes solipsism. Now, that's not a particularly difficult thing to do, ey. To refute solipsism, that is. Again, here's a hand, mate. It's not a big deal. There are more important things to discuss. For example, do mathematical entities exist? Like, literally, outside of space and time themselves, structuring Reality itself? That's what a fan of Max Tegmark would say and I think he's wrong. Tegmark is wrong about that, and anyone who agrees with Tegmark about that is just plain wrong. And I have the right to say that. Nay, I have the epistemic right to say that, just as much as any professional physicist. That's how I would phrase it.

We went over that previously. Yes, Pegasus exists, in that Pegasus is the subject of a quantification. But nothing in that proof implies that Pegasus is physical.

If folk want to say that, in addition, Pegasus is in the stables down the road, it's up to them to present their case. If they want to say that primes have physicality, it's over to them to show how. If they claim that infinitesimals exist in the "Platonic"sense, then they can explain what they mean.

But we can all affirm and agree that there are numbers.

Exactly, which is why I think the original question of the OP has been successfully answered, and if someone wishes to make a case for some particular ontology (mythical realism, mythical physicalism, platonism, etc.), then they should make a case for it. Until then, we have nothing more to discuss.

If folk want to say that, in addition, Pegasus is in the stables down the road, it's up to them to present their case. — Banno

But if they want to point out that, literally and unequivocally, Pegasus and numbers don't exist, it's up to you to explain how this

Yes, Pegasus exists, in that Pegasus is the subject of a quantification. — Banno

isn't equivocal.

Are they "out there", like apples and trees are? That's the actual "existence debate", — Arcane Sandwich

"Apples and trees"—a number of apples and trees—how many? So of course, number is out there if apples and trees are. So, number is out there—are numbers out there? That's a different question, no? Is 5 out there over and above all the collections of five objects? Is 5 out there in the sense that digits and numerals are—as written or spken?

"Apples and trees"—a number of apples and trees—how many? So of course, number is out there if apples and trees are. So, number is out there—are numbers out there? That's a different question, no? — Janus

No, I would say that you have argued well for the existence of natural numbers, like 1, 2, 3, 4, etc.

But then we need to talk about fractions, and then, the number Pi, and then, the square root of minus one, etc. I would draw the line somewhere, but I don't know where. Maybe there is no line to draw, as in, maybe it's an "all or nothing" deal.

Ok.

Are we in agreement? Might be a first!

If there are no perfect circles in nature, then I guess Pi does not exist out there as an actual ratio—I mean it is an idealised ratio. Maybe the same for infinitesimals and the square roots of negative numbers, even negative numbers themselves as well as imaginary numbers.

I don't know—I could be talking shit.

Are we in agreement? — Banno

No.

Then I havn't understood your post. That's fairly normal.

What bit of your post is where you think we differ?

Then I haven't understood your post. — Banno

That was my impression.

That's fairly normal. — Banno

Oh, you!

What bit of your post is where you think we differ? — Banno

About you equivocating between fact and fiction. I suppose that does sound like us.

About you equivocating between fact and fiction. — bongo fury

Where?

As I specified, here:

Yes, Pegasus exists, in that Pegasus is the subject of a quantification. — Banno

So, is that the same as admitting that Pegasus is fiction, and doesn't literally exist? Or not?

I'll leave this chat there. I'm not following you at all, again.

No worries.

My take on some of this differs from what's been discussed here, especially since I'm an Idealist.

I'll take the liberty of repeating something I wrote previously:

No, my beef is with the term "existence", which I think we should retire from the field with all due honors. Same for "real". I believe we will learn a lot more about the concepts that those terms try to refer to, if we stop the endless, unresolvable bickering about them. — J

The posts on the last couple of pages make an excellent argument for my case. What would happen if we tried to reframe the "existence" question in terms of structure, grounding, and quantification, retaining full rights to claim metaphysical truth, but did so without once using the term "exist"?

What would happen if we tried to reframe the "existence" question in terms of structure, grounding, and quantification, retaining full rights to claim metaphysical truth, but did so without once using the term "exist"? — J

But, my perhaps ignorant question would be, why would you do that? The concept of existence should be retired from the field of mathematics and logic, I can agree with that (and I actually do). It does not follow, however, that it should be retired from metaphysics, to say nothing of ordinary language.

I'm fine with the concept of existence; you can even use it in logic as long as you confine it to existential predication. What I'm urging (tongue a bit in cheek, obviously, since it will never happen) is to retire the term "exist". I'm recommending a separation of term from concept and/or content.

The words "exist" and "existence" cause nothing but trouble, because they call like Sirens to philosophers, inviting us to argue about which use of the word is correct. "My use is correct!" says one group, "because when I use it, I mean concept A." "No, my use is correct!" says another group, "because when I use it, I mean concept B." "Well, Plato used it for concept A." "Well, Kant used it for concept B."

Oh dear, which concept is the right one to be called "existence"? The answer is, Neither, none, because the word doesn't matter. What matters -- and it matters a great deal, if you believe metaphysics is worthwhile -- is getting straight on the conceptual territory, on concepts A, B . . . n. But you can do that with any vocabulary you please. So pick one that doesn't bring 2,000 years of ambiguity and dispute along with it.

The words "exist" and "existence" cause nothing but trouble, because they call like Sirens to philosophers — J

But, you see, this is my argument. Ordinary people philosophize from time to time. How could they not? Everyone does. But we, as philosophers, have a responsibility towards them, because, like it or not, they are indeed our colleagues when they philosophize. We have the moral responsibility to vindicate their use of the very "word" existence like they mean it in ordinary language, as something that rocks have and that winged horses from Greek mythology don't. That's on us, philosophers. We have to explain why appealing to a rock is not a fallacy, why it's not fallacious to rely on good common sense in all matters, not just the ones involved in ordinary life.

That's one of the better defenses of continuing "existence"-talk that I've heard. I woudn't want to cut all ties with one common usage (and I'll come back that) if it's genuinely helpful to folks. The ethical obligation to remain in communication with as many interested parties as possible is a real one.

That said . . . is it genuinely helpful? A great deal of damaging nonsense on this subject is spoken in the name of "good common sense," which too often means "ways of thinking that are common to me and the people who share my views." People who think only physical stuff exists -- materialists, in other words -- are the same people who often want to say that "rights" and "truth" and "justice" also don't exist. What they believe exists may influence them on issues from abortion to contract law. I know, it looks very unproblematic to point to Pegasus vs. rocks, and if that's all one ever needs the concept for, I guess no sweat. But if we really have an obligation to help clarify thought when it gets difficult, then we can't stop there.

My other response also refers back to what's "good common sense." I dunno, is it really your experience that the average non-philosopher you know is quite settled in the opinion that rocks exist but numbers don't? I get into a fair number of semi-philosophical discussions with friends and acquaintances, for obvious reasons, and when they're not ordering me a cup of hemlock, they seem to be very alive to why this question of what exists is not cut and dried. They also seem to move quite quickly to noticing that it looks like a terminological dispute. So again, I think we should be really wary of invoking a notion of common sense that may not stand up under inspection.

People who think only physical stuff exists -- materialists, in other words -- are the same people who often want to say that "rights" and "truth" and "justice" also don't exist. — J

So wouldn't what you say provide reason for going in the other direction - for showing that rights and truth and justice do exist?

So wouldn't what you say provide reason for going in the other direction - for showing that rights and truth and justice do exist? — Banno

He's wrong about that, in my humble opinion, Banno. He's right when he says that most materialists are like that, but not all of us (materialists, that is) deny emergence in an ontological sense.

People who think only physical stuff exists -- materialists, in other words -- are the same people who often want to say that "rights" and "truth" and "justice" also don't exist. — J

I'm a proponent of Emergent Materialism, more or less how Bunge has articulated it throughout his publications. And, as a materialist, I can confidently say the existence of rights, truth and justice is not incompatible with the materialist premises and conclusions of my philosophy.

EDIT:

So wouldn't what you say provide reason for going in the other direction - for showing that rights and truth and justice do exist? — Banno

My thoughts exactly.

Yes, if I thought there was a hope of ever settling it. But using the "existence" terminology to do so just doesn't seem to get anywhere. Instead, let's talk about the ways that rocks show up in our lives, and what we can say about them -- also the ways that justice shows up in our lives, and what we can say about that -- and whether there might be various grounding relations obtaining between physical things and values -- but do it all without trying to award the Grand Prize of Existence to anything.

As a materialist, I can confidently say the existence of rights, truth and justice is not incompatible with the materialist premises and conclusions of my philosophy. — Arcane Sandwich

Good! But that must mean that "existence" is being given a much broader interpretation than "made of material stuff." So here we go again . . .

I won't disagree with that.