(i) quantifiers cannot vary their meaning extensionally by changing the domain of quantification; (ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism; (iii) quantifier variance is not an ontological doctrine; (iv) quantifier variance is not compatible with charitable translation and as such is internally inconsistent.

In contemporary metaontological discussions, quantifier variance is the view according to which there is no unique best language to de- scribe the world. Two equivalent descriptions of the world may differ for a variety of pragmatic purposes, but none is privileged as providing the correct account of reality. — Finn and Bueno

I say “numbers exist”; you say “numbers do not exist”. — J

"in the same way", Frege says "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets." — Frege on Knowing the Third Realm, Tyler Burge

Consider such a proposition as 'Edinburgh is north of London'. Here we have a relation between two places, and it seems plain that the relation subsists independently of our knowledge of it. When we come to know that Edinburgh is north of London, we come to know something which has to do only with Edinburgh and London: we do not cause the truth of the proposition by coming to know it, on the contrary we merely apprehend a fact which was there before we knew it. The part of the earth's surface where Edinburgh stands would be north of the part where London stands, even if there were no human being to know about north and south, and even if there were no minds at all in the universe. ...We may therefore now assume it to be true that nothing mental is presupposed in the fact that Edinburgh is north of London. But this fact involves the relation 'north of', which is a universal; and it would be impossible for the whole fact to involve nothing mental if the relation 'north of', which is a constituent part of the fact, did involve anything mental. Hence we must admit that the relation, like the terms it relates, is not dependent upon thought, but belongs to the independent world which thought apprehends but does not create.

This conclusion, however, is met by the difficulty that the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. If we ask 'Where and when does this relation exist?' the answer must be 'Nowhere and nowhen'. There is no place or time where we can find the relation 'north of'. It does not exist in Edinburgh any more than in London, for it relates the two and is neutral as between them. Nor can we say that it exists at any particular time. Now everything that can be apprehended by the senses or by introspection exists at some particular time. Hence the relation 'north of' is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something.

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'....In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness. The connected ambiguity in the word 'idea', which we noted at the same time, also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.

We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless.

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

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

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

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

Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.

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

If we are even to recognise that there are two domains, we must thereby hold quantification constant. — Banno

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

XII. Y de esto se sigue, primer lugar, que aunque cada individuo sea la realidad formalmente uno, sin intervención de la consideración de la mente, sin embargo, muchos individuos de quienes afirmamos ser de la misma naturaleza, no son algo uno con verdadera unidad que exista en las cosas, a no ser sólo fundamentalmente o mediante el entendimiento. Y por eso siempre que Aristóteles dice que muchas cosas forman un uno en esencia o razón formal, explica dicha unidad en orden al entendimiento, concretamente, porque concebidos bajo una razón o definición, como se echa de ver en el lib. V de la Metafisica, c. 6, texto 11, y en el lib. X, al principio. Y Santo Tomás en el De ente et essentia, c. 4, dijo en este sentido que la naturaleza no tiene esencialmente unidad común, porque, de lo contrario, no podría convertirse en singular. Segundo, se deduce que una cosa es hablar de unidad formal y otra de la "comunidad" de dicha unidad; porque la unidad se da en las cosas, según se explicó; en cambio, la "comunidad" propia y estrictamente no se da en las cosas, porque ninguna unidad que exista en la realidad es común, según demostramos, sino que en las cosas singulares hay cierta semejanza en sus unidades formales, en la cual se funda la comunidad que el entendimiento puede atribuir a tal naturaleza en cuanto concebida por él, y esta semejanza no es propiamente unidad, porque no expresa la indivisión de las entidades en que se funda, sino solo la conveniencia o relación, o la coexistencia de ambas. — Disputaciones Metafísicas

12. And from this it follows, in the first place, that although each individual is in reality formally one, without the intervention of the mind's consideration, nevertheless, many individuals of whom we claim to be of the same nature are not one thing with true unity existing in things, unless only fundamentally or through the understanding. And therefore whenever Aristotle says that many things form a one in essence or formal reason, he explains this unity in order to the understanding, namely, because they are conceived under one reason or definition, as we see in the fifth book of the Metaphysics, c. 6, text 11, and in the tenth book, at the beginning. And St. Thomas in De ente et essentia, c. 4, said in this sense that nature has essentially no common unity, because otherwise it could not become singular. Secondly, it follows that it is one thing to speak of formal unity and another of the ‘community’ of that unity; for unity is given in things, as explained. On the other hand, ‘commonness’ properly and strictly speaking does not occur in things, because no unity existing in reality is common, as we have shown, but in singular things there is a certain similarity in their formal unities, on which is founded the commonness which the understanding can attribute to such a nature as is conceived by it, and this similarity is not properly unity, because it does not express the indivision of the entities on which it is founded, but only the convenience or relation, or the coexistence of the two. — DeepL translation, edited

All the problems come up when someone then asks you, Why make that choice? I don't mean just you, I mean anyone who wants to say something using words like "real" and "exist". What sort of case are philosophers supposed to make for their choices here? — J

Existence refers to what is finite and fallen and cut of from its true being. Within the finite realm issues of conflict between abound between autonomy and heteronomy. Resolution of these conflicts lies in the essential realm (the Ground of Meaning/the Ground of Being) which humans are both estranged and yet also dependent on. In existence man is that finite being who is aware both of his belonging to and separation from the infinite. Therefore existence is estrangement.

See Popper — J

...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

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

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

Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

"Platonism", as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)

If the truth of mathematical statements can be confirmed just by thinking about them

And by identifying the indispensable components invoked in the explanation of various phenomena, and noting that mathematical entities are among them, the platonist is then in a position to make sense of the success of applied mathematics.
[...]
to accommodate that success is often taken as a significant benefit of platonism. Less controversially, the platonist is certainly able to describe the way in which mathematical theories are actually used in scientific practice without having to rewrite them. This is, as will become clear below, a significant benefit of the view. — SEP's Nominalism in the Philosophy of Mathematics

Rather, they are something we do. — Banno

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

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. — Banno

In his seminal 1973 paper, “Mathematical Truth,” Paul Benacerraf presented a problem facing all accounts of mathematical truth and knowledge. Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to deny that knowledge of mathematical objects is possible.

Mathematical objects are in many ways unlike ordinary physical objects such as trees and cars. We learn about ordinary objects, at least in part, by using our senses. It is not obvious that we learn about mathematical objects this way. Indeed, it is difficult to see how we could use our senses to learn about mathematical objects. We do not see integers, or hold sets. Even geometric figures are not the kinds of things that we can sense. Consider any point in space; call it P. P is only a point, too small for us to see, or otherwise sense. Now imagine a precise fixed distance away from P, say an inch and a half. The collection of all points that are exactly an inch and a half away from P is a sphere. The points on the sphere are, like P, too small to sense. We have no sense experience of the geometric sphere.

(Rationalist) philosophers 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.

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

(ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism;