~~

:rofl:

perhaps logic has advanced since then?

Now, if you want to say "numbers are a human invention," that seems like fair game. But there has to be some sort of explanation of their usefulness and development across disparate, isolated societies. — Count Timothy von Icarus

If you want to say "nouns are a human invention," that seems like fair game. But there has to be some sort of explanation of their usefulness and development across disparate, isolated societies.

Will we say that the world consists of objects, and we just give them names? Or will we say that the names are arbitrary, we just invent them?

Is the world already divided up, or do we divide it up arbitrarily? But that's a false dilemma. carcinization works.

Ho hum
.

Another point that seems to need reinforcing is the nature of quantification. If our domain is {a,b,c} then "U(x)fx" is just "fa & fb & fc"; and "∃(x)fx" is just "fa v fb v fc". If the domain changes to {a',b',c'} then "U(x)fx" is just "fa' & fb' & fc'"; and "∃(x)fx" is just "fa' v fb' v fc'". That is, the definition of each quantification doesn't change with the change in domain; but remains a conjunct or disjunct of every item in the domain. — Banno

Now to be sure there are issues when applying this to quantification in modal logic. But those issues are to do with the nature of the domain, not the nature of quantification. They concern whether a,b,c... are the unique to each possible world or alternately if say "a" can refer to a in any possible world in which a occurs, and so on.

There are different ways of applying quantification in modal logic. But each is a way of applying quantification, not a different way of quantifying. Which is "correct"? Well, asking that question that shows a fundamental misunderstanding of the nature of logic. Which is "correct", French or German? Better to ask which is more appropriate, or more useful in a given situation.

Let's add Gillian Russell to the mix: Logical Nihilism: could there be no logic?. Lemma incorporation is also preferable to monster-barring, in which Russell argues that ad hoc logical pluralism to be preferable to both arbitrary monster-baring and to nihilism.

Specifically,

...quantifier variance is not meant to entail a multiplicity of logical systems, each with its own quantifiers and conception of validity, but rather it requires that, within a single logic, there should be multiple (existential) quantifiers operating differently. And so, logical pluralism should not be equated with quantifier variance, as having a choice between logical systems is not the same as having a choice of quantifier meaning within a system of logic. — Quantifier Variance Dissolved

There remains a difference between quantification and ontological commitment that is not recognised by quantification variance. Quantification sits within a logical system, ontological commitment remains external to logical systems.

Logic gives us a variety of ways in which we might talk about how things are. It does not commit us to this or that ontology.

Oh, and this bit is salient:

What all of this illustrates, is that in tying quantification to existence, two distinct roles are ultimately conflated:
(a) The quantificational role specifies whether all objects in the domain of quantification are being quantified over or whether only some objects are.
(b) The ontological role specifies that the objects quantified over exist.
These are fundamentally different roles, which are best kept apart. By distinguishing them and letting quantifiers only implement the quantificational role, one obtains an ontologically neutral quantification. Ontological neutrality applies to both the universal and the particular quantifier (that is, the existential quantifier without any existential, ontological import). — Quantifier Variance Dissolved

And the conclusion to that section,

However, once again, no variance in any quantifier is involved.

, it seems is talking about some supposed ontological role, the E, not quantification, ∃.

Knowing what mathematics is seems like one of the biggest philosophical questions out there. Not to mention that a number of major breakthroughs in mathematics have been made while focusing on foundations, so it hardly seems like a useless question to answer either. — Count Timothy von Icarus

Sure. I don't see how what I have said counts against this. Maths as a language, a set of (or sets of) grammatical rules that set out what we might consistently say.

Why this huge difference? — Count Timothy von Icarus

Good questions. The property analogy will only go as far as "counts as..." or "as if...". And as I've said, we do treat numbers to quantification, equivalence and predication - all nice neat uses of "is". Numbers are in many ways not like property.

So where would causation fit here? I don't see that it does.

With all respect to Banno, the formula "Numbers are something we do" could use some clarification. — J

Yep An incipient notion. It probably relates to Austin's treatment of abstracts in Are There A Priori Concepts

Austin carefully dismantles this argument, and in the process other transcendental arguments. He points out first that universals are not "something we stumble across", and that they are defined by their relation to particulars. He continues by pointing out that, from the observation that we use "grey" and "circular" as if they were the names of things, it simply does not follow that there is something that is named. In the process he dismisses the notion that "words are essentially proper names", asking "...why, if 'one identical' word is used, must there be 'one identical object' present which it denotes". — Wiki article

Something I wrote quite a ways back. The salient line for this discussion is "from the observation that we use "grey" and "circular" as if they were the names of things, it simply does not follow that there is something that is named".

I'm extending Austin's point, made about universals, to individuals. We do, after all, use names for things that don't exist. Frodo, Sherlock Holmes, and so on. That we talk as if Frodo walked into Mordor does not imply that you could walk in to Mordor, nor that you might met Frodo on the road.

We can quantise over these non-existent things. That Frodo is a hobbit implies that at least one thing is a hobbit.

The point is here applied to numbers. From the observation that we use "7" and "One Million" as if they were the names of things, it simply does not follow that there is something that is named. And we can quantise over numbers. That seven is a prime implies that at least one thing is a prime.

So we have an apparent contradiction; as if we are to say that there is a hobbit who does not exist, or there is a prime number that does not exist. Hence the temptation to treat these as cases of different uses of "exits", and the view that fictional characters and numbers exist in a way that is different to you and I.

One might supose that talk of numbers is different to talk of fictional characters not because they are quantified in a different way, but because the domain of quantification differs. Fictional characters and not numbers. But we do want to be able to talk about seven dwarves, for example. Hence we obliged to include "seven" in the domain of fictional characters.

All this by way of repeating the fairly obvious point that numbers are not like other individuals.

Another point that seems to need reinforcing is the nature of quantification. If our domain is {a,b,c} then "U(x)fx" is just "fa & fb & fc"; and "∃(x)fx" is just "fa v fb v fc". If the domain changes to {a',b',c'} then "U(x)fx" is just "fa' & fb' & fc'"; and "∃(x)fx" is just "fa' v fb' v fc'". That is, the definition of each quantification doesn't change with the change in domain; but remains a conjunct or disjunct of every item in the domain.

So on to ontological pluralism? — J

I'm going to maintain that the domain, and hence the ontology, one way or another, is stipulated. And see where that leads.

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.

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.

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 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.

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 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.

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.

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

Do I get a prize? :halo: — J

Respect.

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.

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?

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.

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.

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.

Your immortal soul is indentured to @Jamal for eternity on posting.

okay. For the sake of addressing the OP, it is worth pointing out that we do indeed quantify over numbers. There is an X such that X is greater than seven.

So whatever you mean when you say numbers don't exist, it can't be that.

Meh. I could say that that's a cop out. You are just excusing yourself from answering my critique. But that doesn't progress the discussion.

I've made an attempt to tighten up your claim by pointing out its relation to free logic and modality. You have not addressed this. Presumably, if I have not understood your argument, you can point out how what you claim differed from what I offered.

So here, we are in basic agreement:

But while the symbolic form exists, what it symbolises, a number, is an act, namely, the act of counting, which is grasped by the mind — Wayfarer

And presumably we agree there is some reification, where the act of counting is treated as if we were dealing with a series of individuals - 1,2,3...

But whereas you seem to be saying that these individuals are "real", I'm pointing out that they remain shorthand for an activity we can perform.

And sure, we can quantify them as needed.

So what is the argument I don't understand - is it the same one you could make a case for, but is too long? Then you might forgive my not understanding it until you present it...

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. I've filled this out elsewhere and in previous conversations with you.

But this is not the topic of this thread.

I can make the case for it, but it would be a very long one. — Wayfarer

Go on - you've nearly caught me, in terms of post count! :wink:

See Popper — J

I don't recall this - where is it?

Nice use of Russell. It looks to be a precursor to discussions of private language.

seems to want two sorts of quantifiers, real and exist. He's immediately committed at least to some sort of free logic. He is giving us permission to talk of things that do not exist, but are real - like numbers.

That's one way of using ∃ as a quantifier and as a predicate - in this case, ∃!, such that ∃!t=df∃x(x=t). But this is just to create a short form, and permit empty singular terms.

Are there two domains? Well, here I am not on secure ground, but I think there are good arguments for making use of multiple domains. For example, if there is only one domain then every individual exists in every possible world... Uy☐∃x(x=y). But I do wish to be able to say that it is possible that some things might not have existed. See Quantifiers in Modal Logic.

is content with mysticism; but I'm not. I'd prefer to remain silent than to lurch into inconsistency.

So I'll go back to the point made elsewhere, that it seems to me that domains are stipulated, not discovered. This by way of agreeing that

...it doesn't start by sending a team of metaphysicians to beat the bushes and bring back an actual sample of "existence" or "reality". — J

But, specifically, what about natural laws? Maybe they can be derived from some ethical consideration of the good... — Shawn

There's a logical gap between the ought of ethics and the is of natural laws.

Ok. See also 's other thread. And had one, too.

Amused to see that Hirsch's latest publication is "On ontology by stipulation".

In previous work the author suggested that many ontological disputes can be viewed as merely verbal, in that each side can be charitably interpreted as speaking the truth in its own language. Critics have objected that it is more plausible to view the disputants as speaking the same language, perhaps even a special philosophy-room language, sometimes called Ontologese. This chapter suggests a different kind of deflationary move, in a way more extreme (possibly more Carnapian) than the author’s previous suggestion. The chapter supposes we encounter an ontological dispute between two sides, the A-side and the B-side, and we assume that they are speaking the same language so that (at least) one of them is mistaken (perhaps the common language is Ontologese). The author’s suggestion is that we can introduce by stipulation two languages, one for each side, such that in speaking the A-side stipulated language we capture whatever facts might be expressed in the A-side’s position, and in speaking the B-side stipulated language we capture whatever facts might be expressed in the B-side’s position. In this way we get whatever facts there might be in this ontological area without risking falsehood. A further part of the argument consists in explaining why the stipulation maneuver applies to questions of ontology but not to questions of mathematics (such as the Goldbach conjecture). One basic point is that mathematics has application to contingencies in a way that ontology doesn’t. — Eli Hirsch

Might be interesting.

Davidson is just the ubiquitous On the very idea of a conceptual scheme. The argument presented there is that languages are translatable, an argument against relativism. Plenty of threads on that topic in the forum.

Midgley has it that there is a difference between various ways we talk about the world, especially between scientific and moral or intentional language. In various of her later books.

There's a prima facie disagreement here, but I think it is on the surface only, that Midgley is espousing something not too dissimilar to Davidson's anomalism of the mental.

This is mostly a problem for consistency in my own accounts, not something of direct relevance here.

But see the thread mentioned here:

I started a thread here a while back that might be of interest. — J

Ontology concerns bigger questions — Wayfarer

So you have said. But what they might be, apart from hand waving, remains obscure. And not so germane to this conversation.

I prefer to think of it more as an ontological question. — Wayfarer

Indeed, a distinction that I can't make sense of. Ontology is choosing between languages. It consist in no more than stipulating the domain, the nouns of the language.

An example might be helpful. I say “numbers exist”; you say “numbers do not exist”. Each of us would have to use Ǝ to formulate our position in Logicalese. What I’m arguing is that we’re each going to use Ǝ the same way, as we state our respective contradictory positions. The difference in our statements is not at the subsentential, quantifier level. We have no quarrel about "variation in quantificational apparatus." We differ on what exists, not on the use of the quantifier. — J

This looks agreeable.

To summarize: Is it the quantifier whose meaning changes, or the sentences in which the (unchanged) quantifier occurs? And if the latter, is it still QV? — J

Isn't there variation in the domain, in what we are talking about, while quantification remains constant?

That is, we can bring in Davidson's argument against relativism. If we are even to recognise that there are two domains, we must thereby hold quantification constant.

(This is too brief - just me trying to recall the line of thought I was following.)

Oh, that thread dropped of my list. I didn't see your last reply. Still the most anoying question on the forums.

Yes, that's the issue on Tarskian's thread.

It brings out the conflict in my own arguments, between Midgley and Davidson, and provides something of a logical frame for that discussion. No small topic. So I'm not surprised to see it here as well.

But for here, it seems that has come around to an antirealist position, after 's "the structures themselves are nothing more than formal systems". A more direct rout than I was taking. Nice work.

's is not just a "terminological question". It's (potentially) a choice between grammars, between languages. Which implies quantifier variance. Which I think we (you and I) are inclined here to deny.

Yet, take the example of good being defined, not by an individual; but, by the very values people or groups enshrine into laws. — Shawn

Ok. That's right, in so far as what is enshrined in law is what we enact. But of course there is no equivalence between the law and the good. There are bad laws.

My question was, "How could one decide if a proffered definition were correct, apart from comparing it to experience?" Along with Moore, I doubt that it is possible to give a satisfactory analysis of "the good"; and along with Wittgenstein I take it that one recognises it when one sees it.

But as well, we are talking here about our interactions with others. Ethics begins as one takes other folk into account.

I didn't say it did. My point was more that you might be accused of accept realism in your premise, in supposing that ℕ exists. But the point is now moot. Cheers.

PA does not create N. — Tarskian

Odd choice of phrasing. It might be thought of as defining ℕ.

N exists independently from PA. — Tarskian

I don't have a clear idea of what you mean by "exists" here. Same for "preexisting" in the next paragraph.

Model theory makes mathematics decisively correspondentist — Tarskian

Why?

Do you agree with this, namely that the notion of good in inherent in the primacy of experience, and not something that can be learned by simply looking up a definition and analyzing it? — Shawn

How could one decide if a proffered definition were correct, apart from comparing it to experience?

Yes, the meaning of "good" is shown, not said; found in use, not in analysis.