See Popper
— J
I don't recall this - where is it? — Banno

I had in mind his Three Worlds conception, where Russell's individual brain-events would be thoughts in the World 2 sense, whereas the universals or objects of thought would be World 3 items. I like Popper's discussion because he recognized that the World 2/World 3 distinction isn't just about universals, but concerns any "contents of thought" or propositional meaning. That said, I don't know how seriously we need to take the talk of "Worlds".

That's one way of using ∃ as a quantifier and as a predicate - in this case, ∃!, such that ∃!t=df∃x(x=t). — Banno

Yes, good, and that does help me recapture my puzzle about using ∃ that way. One picture: The existential quantifier is austere, a mere operator, and doesn't add anything to whatever terms it operates upon. This matches up with the traditional arguments for why existence can't be a predicate. Another picture: When we make a statement in Logicalese to the effect that ∃x(x=t), we are indeed providing new information; we are predicating existence of 't'. And in that case, if we go on to say ∃!t=df∃x(x=t), we're having it both ways -- quantifier and predicate. This looks right, but . . . what does that commit us to, in re quantifier variance? We'd been exploring the idea, above, that ∃ doesn't actually vary, but rather the sentences differ in what they pick out as existing, i.e., having the predicate 'existence'. We're supposed to be able to hold some sense of 'existence' steady, and my puzzle is, Which one? Existence as ∃, or existence as the predicate 'exists'? What's worse, the more I try to put this into words, the less certain I am that the question is even a good one. I may have merely muddled the terms.

Perhaps relatedly, the "two domains" question is still murky for me. I'm not a strong enough logician to have a worthwhile opinion.

I wonder what you mean when you say that numbers are real.
— Janus

That they have a common reference, that the value of a number is not a matter of opinion or choice. — Wayfarer

I'm sympathetic to that view, and offer a homely analogy. We can say true and false things about Sherlock Holmes. That he had lodgings in Baker Street is true. That he wore a long beard is false. Etc. Now we also want to say that, in some important sense, Holmes didn't exist at all. So how can we make T/F assertions about a nonexistent item? This is where "reality" becomes a tempting term to introduce. Holmes didn't and doesn't exist, but he is real if we let "reality" mean "capable of T/F predications".

The analogy with numbers breaks down, though, when we acknowledge that Holmes is without question nonexistent, whereas a mathematical Platonist (not @Wayfarer) would disagree with the way I'm divvying up the terms -- for her, numbers also exist, just not as empirical objects . . . and the dispute goes on. (Perhaps Holmes himself also exists, as a Form, on this view.)

This is where "reality" becomes a tempting term to introduce. — J

Phenomena - apparent, appearing
Noumenal - object of nous/intellect
Imaginary - fictional and literary

What say you? — Wayfarer

There are three clear ways of using "is". Quantification, "There is something that is green"; equivalence: "Superman is Clark Kent"; and predication: "Wayfarer is a human".

That numbers are a way of doing things does not mean that we cannot quantify over them, equate them or predicate to them.

What we have done here is to hypostatise the action of counting. This is not at all an unusual thing to do, we do this sort of thing with stuff all around us. Property, for instance, marks a difference between the actions you can perform and the actions your neighbour can perform. Money marks a difference between what a pauper can enact as opposed to what can be done by a comfortable middle-class retiree. Rank marks a difference in ability between an officer and a civilian.

But we do not spend time arguing over whether property, money or rank are "real" in the way trees and such are. Your article says "We learn about ordinary objects, at least in part, by using our senses." We do not learn who owns an object or what it is worth by simply examining it. Value and ownership are not physical attributes of an object.

We stipulate what counts as your property, what counts as five dollars, who counts as an admiral. And we stipulate what counts as two, three or four. That is we make it so by treating it as if it were so. See the various threads on Searle.

And it seems to me that this utterly undermines the misguided search for a platonic world of numbers.

I can't help but think that it's obvious that humans do indeed have a 'non-sensory capacity for understanding mathematical truths' — Wayfarer

That capacity, if it is anything, consists in the capacity to have something count as... An act of social intentionality of the sort that underpins much of our world.

I had in mind his Three Worlds conception, — J

Oh, Ok. "world three" corresponds, in broad terms, with the stuff invented by playing language games that I describe in the post above, to @Wayfarer. See

institutional facts — Banno

For me the problem here is the lack of a clear account of what quantifier variance is. Hence,

This raises the issue of how the meaning of a quantifier can differ, and what the other meanings could be. And it is this issue that we tackle, arguing that one cannot make sense of variation in quantificational apparatus in the way that the quantifier-variance theorist demands. — Quantifier Variance Dissolved

So i think we can pass the argument back to those who might support quantifier variance, and ask them to set out explicitly what it is they might mean.

Your article says "We learn about ordinary objects, at least in part, by using our senses." — Banno

Fair enough - but it’s not ‘my article’, it’s an encyclopedia article on a genuine controversy. Why it’s a controversy, and what the implications are, are what I’m interested in. Quite agree that rational abilities underpin our world, but not that they can be reduced to ‘social intentionality’ (which sounds like the kind of thing a Josh’s would say :yikes:)

Well, then the problem is yours, and not mine. The account I gave has no need to give further account of the nature of numbers.

The difference form Joshs is that Searle gives at least an outline of how social intentionality works. It's not complete, but it is better than looking for platonic realms.

Those quantifiers are introduced differently, and as the paper "Quantifier Variance Dissolved" notes that provides a strong argument for a form quantifier variance without a reduction of quantifier meaning to underlying entity type it quantifies over, and without committing yourself to the claim that there's a whole bunch of equally correct logics for the purposes of ontology. — fdrake

I wasn't quite able to follow your point here. Are we in agreement that advocates of quantifier variance have failed to give an adequate account? That

a mere difference in the domain of quantification is not enough to deliver a difference in the meaning of the quantifiers, rather a difference in the rules that govern the quantifiers would be required.

and that this has not been provided?

It's not complete, but it is better than looking for platonic realms. — Banno

Do you agree or disagree that mathematical knowledge is incompatible with ‘our best epidemic theories?’ That is the point on which the argument hinges.

Odd. These "best epistemic theories" are, as is set out in the section on Quine, naturalism.

Quine’s belief that we should defer all questions about what exists to natural science is really an expression of what he calls, and has come to be known as, naturalism.

Seems an odd position for you to be defending.

See this comment I made earlier today:

Along the same line of thought, a number (and any other mathematical entity) is a set of neurons that form a specific structure in my brain.
— bioByron
There's a real problem with this view. If "seven" is a structure in your brain, then your "seven" is not the same as my "seven", which would be a distinct structure in my brain.

But when we each say seven is one more than six, we both mean the same thing.

Hence we must conclude that "seven" is not just a structure in your brain. Rather, it is in some way common to both you and I.

Plato answered this problem by positing a world of forms in which we both share. I think there are better answers, to do with how we use words. — Banno

Seems an odd position for you to be defending. — Banno

How am I defending it? I’ll come back to this later.

Quine’s belief that we should defer all questions about what exists to natural science is really an expression of what he calls, and has come to be known as, naturalism. — Banno

And Lloyd Gerson lays out the thesis, in Platonism and Naturalism: The Possibility of Philosophy, that (1) philosophy proper is Platonist, and (2) is incompatible with naturalism. This is what I believe is the underlying issue but I’ll fill that out later

Well, you want to deploy the indispensability argument, no? Which is that mathematical entities are indispensable for naturalist methodology, naturalist methodology only uses things that exist, hence mathematical entities must exist. Hence you seem to be using naturalism to argue that mathematical entities must exist.

SO I must be misunderstanding what you are saying.

I just think there is a category error in supposing that numbers must exist or not exist.

Rather, they are something we do. A way of talking about things. A grammar. — Banno

I think there are better answers, to do with how we use words. — Banno

Why words, though? I'm not googling, but isn't there somewhat robust evidence that some non-linguistic animals (crows, isn't it?) and infra-linguistic children have some rudimentary understanding of arithemetic? (With numbers befitting their size, of course.)

What's more, there are, or have been, human languages -- and thus functioning human communities to speak them -- that only have "1, 2, many". So language doesn't directly lead to mathematics more advanced than crows and infants possess, even if it enables it (as it does, you know, everything).

I think the gist of your approach is right -- that numbers are to do with us. I just wonder why you think it's to do with how we talk.

I don't see a problem. A crow that collects three sticks or whatever is acting, as is a child who cries on seeing it's sibling has "more". An understanding of numbers is shown in collecting sticks or matching items, not in a magical sense that peers into a platonic realm.

Language allows far more complexity. That's all.

SO I must be misunderstanding what you are saying. — Banno

Indeed you are. I will reply later, dealing with domestic duties today.

A crow that collects three sticks or whatever is acting, — Banno

Take it aside and explain to it the meaning of ‘prime number’.

you want to deploy the indispensability argument, no? — Banno

I’m not arguing in favor of it. I’m asking why it’s even necessary. I’m questioning the claim that ‘according to our best epistemic theories, mathematical knowledge ought not to be possible.’ It obviously is possible, so what does that say about the shortcomings of ‘our best epistemic theories’?

I quoted the paragraph which says 'rationalists claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.' And I think the 'rational insight arising from pure thought' is, in fact, reason. But according to this article, this appears to contradict naturalism. So I'm saying, so much the worse for naturalism.

@Srap Tasmaner may recall the debate over the 'argument from reason'.

Here it is again, but as I'm yelling into a gale, I'll desist.

So i think we can pass the argument back to those who might support quantifier variance, and ask them to set out explicitly what it is they might mean. — Banno

Yes, since the discussion needs a natural stopping place, we could certainly toss the ball back to the friends of QV. And yet . . . haven't they given us a pretty good explanation of what they mean? To re-quote:

Quantifier-Variance is the doctrine that there are alternative, equally legitimate meanings one can attach to the quantifiers – so that in one perfectly good meaning of ‛there exists’, I may say something true when I assert ‛there exists something which is a compound of this pencil and your left ear’, and in another, you may say something true when you assert ‛there is nothing which is composed of that pencil and my left ear’. — Bob Hale and Crispin Wright

I think a lot of our discussion on this thread has focused on whether these "equally legitimate meanings" are attached to the quantifiers at all. The best anti-QV position is that the existential quantifier always means the same thing, it's the predication of "existence" that changes. So let's say that's true. With a wave of the hand, we can eliminate quantifier variance strictly understood. But as I was saying in a previous post, I don't think we've laid to rest, or explained, the doubts that Hale and Wright express. They want to know whether the "perfectly good meanings" of "there exists" are equally legitimate, equally assertive of truths, equally undistinguished in terms of how well they correspond with or describe reality. They propose QV as a possible explanation of how this could be. Even if we reject that view, the real problem remains, which is, I suggest, ontological pluralism. But that can wait for a new OP.

"Numbers are something we do," suggests the question: "why are numbers something we (and animals) do?" All activities have causes, right?

IMO, attempting to answer that question is going to bring us back to questions about the nature of numbers, their ontic status, the "presence" of mathematics in nature, etc.

It's the same thing with words and meaning. We can say words and their meaning are part of social practices, but there remains the questions: "why are social practices what they are? why do they evolve the way they do? etc."

I don't see how an account that is social practice or activity "all the way down," is going to work.

I’m not arguing in favor of it. I’m asking why it’s even necessary. I’m questioning the claim that ‘according to our best epistemic theories, mathematical knowledge ought not to be possible.’ It obviously is possible, so what does that say about the shortcomings of ‘our best epistemic theories’? — Wayfarer

Well, I've long argued the incompleteness of naturalism. So I don't agree with the premise of the argument - that naturalism is our "best" epistemic theory.

'(R)ational insight arising from pure thought' is a bit of nonsense, so far as I can see. I've set out an outline of how our language permits the invocation of intentional facts- things that we bring into being by collective intentionality, such as money, property, and the prime numbers that crows find so difficult to follow.

My intuition about the matter is simply that numbers are real but that they don't exist. — Wayfarer

I've tried to have you fill this out explicitly. If what you say here were so we would have a neat case of quantification variance to work with - the difference between real and existent. But i do nto think you have been able to proved a coherent account.

We quantise over numbers, a clear sense in which they do exist.

I don't think we've laid to rest, or explained, the doubts that Hale and Wright express. — J

I'm thinking that in order to make explicit quantifier variance we would need a case in which it is clear that the difference between two languages was not found in the domain, but in their quantification.

Take the example:

I may say something true when I assert ‛there exists something which is a compound of this pencil and your left ear’, and in another, you may say something true when you assert ‛there is nothing which is composed of that pencil and my left ear’. — Bob Hale and Crispin Wright

This is pretty clearly a case in which one language has in its domain a thing which is a compound of this pencil and your left ear, and the other does not.

That's not a difference in the use (meaning) of quantification.

All activities have causes, right? — Count Timothy von Icarus

I'm not so enamoured with causes. Nor do I take evolutionary explanations as inherently fundamental.

But leaving that to one side, isn't it enough that we want to share the six fruit equally amongst the three of us, to explain the need for counting?

I've tried to have you fill this out explicitly. If what you say here were so we would have a neat case of quantification variance to work with - the difference between real and existent. But i do nto think you have been able to proved a coherent account. — Banno

Thanks for that, we have some agreement. Maybe to you the issue I'm commenting on 'goes without saying' but I think there's something that needs to be spelled out:

I don't agree with the premise of the argument - that naturalism is our "best" epistemic theory. — Banno

So you wouldn't endorse

Quine’s belief that we should defer all questions about what exists to natural science is really an expression of what he calls, and has come to be known as, naturalism

//

I will try and re-state what I see the difference between real and existent at a high level.

By 'existent' I refer to manifest or phenomenal existence. Broadly speaking, this refers to sensable objects (I prefer that spelling as it avoids the equivocation with the other meaning of 'sensible') - tables and chairs, stars and planets, oceans and continents. They're phenomenal in the sense of appearing to subjects as sensable objects or conglomerates.

I am differentiating this from what used to be called 'intelligible objects' - logical principles, numbers, conventions, qualifiers and so on. For example, if I were to say to you, 'show me the law of the excluded middle', you would have to explain it to me. It's not really an 'object' at all in the same sense as the proverbial chair or apple. You might point to a glossary entry, but that too comprises the explanation of a concept. The same with all kinds of arithmetical proofs and principles. Even natural laws - the laws of motion, for example. All of these can only be grasped by a rational intelligence. I could not demonstrate or explain them to a cow or a dog. They are what could be described as 'noumenal' in the general (not Kantian) sense, being 'objects of intellect' (nous) - only graspable by a rational mind. (Significant that the Collins Dictionary definition of ‘noumenal’ is ‘real’ as distinct from ‘phenomenal appearance’, echoing scholastic realism.)

As I said at the outset, in regular speech it is quite clear to say 'the number 7 exists'. But when you ask what it is, then you are not pointing to a sensable object - that is the symbol - but a rational act. (That's the sense in which I mean that 'counting is an act', but it doesn't mean that the demonstrations of rudimentary reasoning in higher animals amounts to reason per se.)

In Plato these levels or kinds of knowledge were distinguished per the Analogy of the Divided Line . Those distinctions are what have been forgotten, abandoned or lost in the intervening millenia due to the dominance of nominalism and empiricism. But In reality, thought itself, the rational mind, operates through a process of synthesis which blends and binds the phenomenal and noumenal into synthetic judgements (per Kant).

That is the back-story of why the need is felt for 'the indispensability argument for mathematics', and the difficulties of accomodating mathematical knowledge into the procrustean bed of empiricist naturalism.

Ok.

No surprise there. You've differentiated between things that exist and things that are real, and while there are issues here that at least makes some sense. You've just re-plastered Descartes mind-body dualism by calling it "manifest" and "ineligible". But the problem with any dualism is explaining how the two interact.

Nothing whatever to do with Cartesian dualism, which never made any such distinction.

//but thank you all the same//

Nothing whatever to do with Cartesian dualism — Wayfarer

You still want mind on one side and matter on the other. It's inveterate in your posts.

Only in your misreading of them. A perceptive reader would notice a much closer resemblance to form-matter dualism which is quite a different thing to Decartes’.

//there’s nothing in the post you’re responding to which suggests mind-matter duality. The contrast, rather than the dualism, was between sensable and intelligible. But in reality, mind synthesises both elements in arriving at judgement.//

"why are social practices what they are? why do they evolve the way they do? etc." — Count Timothy von Icarus

Because we have a brain with trillions of parameters capable of extremely complicated abstraction and inference tasks!

"Numbers are something we do," suggests the question: "why are numbers something we (and animals) do?" All activities have causes, right? — Count Timothy von Icarus

With all respect to @Banno, the formula "Numbers are something we do" could use some clarification. For one thing, it lends itself to the interpretation you're querying here -- that "doing numbers" is just a practice, something we might have chosen to do differently, or not do at all. I don't think this is right, and I don't think Banno needs to hold this position in order to make his point -- though he can tell us if that is so.

I read his position as saying that we wouldn't have numbers if we didn't have mathematics as a whole; that is, numbers "come into existence" as they assume their place within mathematical practice, which is a doing, an activity. You can't "find" 3 but overlook or do without 4, to put it crudely. With numbers, it's all or nothing.

But that understanding, if expressed as "Numbers are something we do," doesn't distinguish between two sets of alternatives, two different questions. The first set of alternatives is: Numbers are either a) found or b) invented. (Let's not worry about getting this more precise, for the moment.) The second set is: Numbers are either a) reflective of the basic structure of reality, or b) arbitrary/pragmatic. (Same caveat here.)

Now if we say "Numbers are something we do," this could mean that we perform them -- or, to put it in more ordinary talk, do mathematics -- as a kind of invention, rather as we might dance or sing. This would be option B of the first set of alternatives. Then again, "Numbers are something we do" might mean "Numbers are [just] something WE do" -- they are indeed arbitrary choices that might have been made differently, had we practiced a different mathematics. This would be option B of the second set of alternatives.

I want to hold out for option B in the first set, and option A in the second set, and I don't know whether "Numbers are something we do" represents a disagreement with me. It needn't, as I'm trying to show. My assertion could mean that numbers as such -- numerals, individual items with names like '7' -- aren't "out there," they aren't found, but nevertheless our choices within mathematics are far from arbitrary or free. This latter clause could be extended to the point of claiming that math (or logic) is perhaps the most basic structure there is, absolutely ontologically fundamental.

The challenge to that position is, How could something so basic not be "out there"? What do I mean by "structure" and "fundamental"? Yeah, that's worth a tome or two, and fortunately Theodore Sider is trying to help us out . . . see his Writing the Book of the World.

The more I think about this, the more I'm persuaded that this is the right line to take. It makes the most sense of some difficult concepts. So on to ontological pluralism? Would you agree that we can have that (or the threat of it) regardless of whether QV is workable?