Does logical structure entail ontological commitments about things like grounding, “simples,” existence, Ǝx, and other tools of the trade? — J

Yes. (I actually think that the illusion that quantifiers are straightforwardly univocal is a deep problem in contemporary logic)

I was asked to comment on this discussion, which is getting away from the OP. As to the OP I would say that the problem of quantifier equivocation is significant but not insuperable. For those with a "low" epistemology ("ontology is beyond our pay grade!") it will appear insuperable, but for them it will always come down to a putative overreach of human inquiry. A form of positivism is also lurking here insofar as the presumption is that we can somehow scientifically or logically demonstrate the truth or falsity of QV, which seems to me a false assumption. Public demonstration is a limited epistemic tool which will not measure up to the task at hand.

About the only discussion I'm aware of that elucidates this distinction (albeit in relation to universals rather than number per se) is in Russell's Problems of Philosophy — Wayfarer

There is a longstanding dispute over the univocity of being (and predication) between the Thomists and the Scotists beginning in the Medieval period. The Scotists held to univocity (and Heidegger's first dissertation was on this topic, on a text then believed to be Duns Scotus').

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). — Srap Tasmaner

This is a good point, and points to the fact that "what we do" is presupposing ontological commitments, just as varieties of logic do.

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

Agreed. I think it reflects a "hermeneutic of despair," in the sense that the logical positivists and their progeny are saying something like, "It's not ideal, but it's the best we're capable of." Besides, no one disagrees that mathematics is something we do. That it is something we do does not answer the question which asks what is involved in mathematics.

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. — Wayfarer

I would want to slice the pie between epistemic optimists and epistemic pessimists, so to speak. The former believe that the human intellect has access to deeper levels of reality, whereas the latter do not. This is probably the biggest difference between you and Banno. The English-speaking tradition tends to fall in with the latter, especially in the secular sphere. Thinkers like Husserl and Heidegger are much more aligned with the former, at least in a relative contemporary sense. The differences are also strongly influenced by anthropology and, in due turn, experience. Plato, Aristotle, Aquinas, Heidegger, etc., are spiritual thinkers with a higher human anthropology. The difference between such thinkers and a Russell or even a Wittgenstein is that Russell hamstrings himself into a low anthropology, and this has the effect of limiting his epistemology and horizon. To be blunt, someone who lives their whole life with their head stuck in the sand will naturally come to the conclusion that only sand exists, and that Plato's divided line is a naive fiction. The difference is faith. One must have faith that something more than sand exists if they are ever to find anything other than sand. Without faith one hamstrings themselves and artificially truncates the horizon of knowledge and reality.

Intellectual naivete is, to my mind, a form of idolatry. Namely, it absolutizes the relative. The project of the logical positivists is a paradigm example of this idolatry. They absolutized one form of logic, assumed that it was associated with no controversial ontological commitments, and fell into all manner of folly. They made a nifty hammer and then assumed that everything was a nail. Students of philosophy should be wary of thinkers of this sort. They should begin with Plato and only descend to Russell if they feel the need. This is difficult because our inherited anthropology and epistemology is now very low, very technocratic. In any case, a general rule of thumb is that most intellectual perspectives or vantage points are not unconditioned. To take the example of the OP: quantifier meaning is not unconditioned by ontological commitments.

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. — Banno

I don't see any argument being presented for why this example must be a matter of domain and not quantification, and if this is right then you are begging the question. The example is intended to suggest the opposite conclusion, for the only linguistic difference pertains to quantificational terms. It should of course also be remembered that any quantification difference will also result in large or small domain differences (and as noted above, the meaning of quantification is conditioned by one's ontological domain, just as one's ontological domain is conditioned by quantification).

- I don't want to get embroiled in this thread, but a central question is to what extent quantifiers can be rigidly defined. The problem here is that quantification derives from the meaning of 'being' or 'exists', and this is one of the most elusive and foundational concepts, inextricably bound up with one's fundamental intellectual stance. "Being" is not like "apple" in that we can give a relatively straightforward definition and be done with it. Because of this adjudicating QV becomes increasingly difficult, and to stipulate a meaning for quantification is at the same time to make the dependent logic to that extent artificial. This is one of the places where the weaknesses of positivism begin to show.

I don't want to get embroiled in this thread — Leontiskos

Me neither. I've already spilled a lot of virtual ink on the forum about quantifiers.

The problem here is that quantification derives from the meaning of 'being' or 'exists', — Leontiskos

But this i disagree with, so here we are.

I don't think quantifiers have much of anything to do with existence or being or any of that. They're entirely about predication -- classification, categories, concepts. Quantifiers are about what things are, not that they are.

It's amusing that Quine is more or less directly responsible for the revival of metaphysics in English-speaking philosophy. By suggesting that there's not quite nothing to say about ontology, and that what little there is to say is covered by logic, he cracked the door open for everyone from Dummett to his own former students (Lewis and Kripke). He tried to build a dam to hold back modal speculation and caused a monumental flood of the stuff. And so it goes.

There is a longstanding dispute over the univocity of being (and predication) between the Thomists and the Scotists beginning in the Medieval period. The Scotists held to univocity (and Heidegger's first dissertation was on this topic, on a text then believed to be Duns Scotus'). — Leontiskos

I've learned that the dissident theological movement, Radical Orthodoxy, sees Duns Scotus' univocity (in combination with Ockham's nominalism) as the source of the decline of modern culture.

Like Macbeth, Western man made an evil decision, which has become the efficient and final cause of other evil decisions. Have we forgotten our encounter with the witches on the heath? It occurred in the late fourteenth century, and what the witches said to the protagonist of this drama was that man could realize himself more fully if he would only abandon his belief in the reality of transcendentals. The powers of darkness were working subtly, as always, and they couched this proposition in the seemingly innocent form of an attack upon universals. The defeat of logical realism in the great medieval debate was the crucial event in the history of Western culture; from this flowed those acts which issue now in modern decadence. — Richard Weaver, Ideas have Consequences

Many thanks for your perspective.

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. — Wayfarer

Can you state just why you think that incompatibility obtains? Many animals appear to have extraordinary capabilities, capabilities that we call instinctive without being able to explain them. How can we understand the idea that we and other animals have resources and capacities which would not be possible for the body/ brain alone when we really understand so little about these unimaginably complex organic beings.

It is questionable whether we have the intellectual capacity to comprehensively understand biological complexity.

Can you state just why you think that incompatibility obtains? — Janus

That's not my claim. It is in the article that I referred to, The Indispensability Argument in the Philosophy of Mathematics. That article says that mathematical knowledge and rationalist philosophy is incompatible with 'our best epistemic theories' which as @Banno pointed out is a reference to naturalism. Make of that what you will.

As for the unfathomable subtlety of living organisms, I'm all for it. I think many things we describe as 'instinct' are impossible to fathom, but that's a completely separate issue.

If it is not your claim, did you refer to the article in order to agree or disagree with it?

As for the unfathomable subtlety of living organisms, I'm all for it. I think many things we describe as 'instinct' are impossible to fathom, but that's a completely separate issue. — Wayfarer

It's not a separate issue if we include mathematical understanding as one of these unfathomable capacities of living organisms, a capacity much more developed in our own case, itself a fact which would seem to have much to do with our command of symbolic language.

They should begin with Plato and only descend to Russell if they feel the need. — Leontiskos

You mean I can't jump head into philosophers 3 thousand years deep into the dialogue? Jeez...

Can you state just why you think that incompatibility obtains? — Janus

He outlines his argument clearly in this post:

So instead of questioning why it is we can understand numbers, how about interrogating the claim that we are, in fact, 'physical creatures whose capacities for learning are exhausted by our physical bodies?' Or is that such an important principle in our 'best epistemic theories' that it has to be saved at all costs? That seems the point of the sophistry of the 'indispensability argument'. — Wayfarer

Putting into silly-willy terms:

We learn things through senses
Mathematical objects, if they exist, are abstract
Abstract objects can't interact with senses
So if mathematical objects exist we can't have knowledge of them

The nominalist will agree with the argument above. Wayfarer instead will deny that "we learn things [only] through senses".
I think it is clear as such, but it is his words so he can correct me anyway.

I don't see that quoted passage from @Wayfarer as an argument. I think he continues to ignore the fact that number is real; it is merely diversity, number is instantiated everywhere we look.

I have asked him to explain what could be meant by saying that numbers are real beyond our recognition of number in the world, and the formalization of the idea of number in the symbolic language of mathematics, but he does not seem able to proffer an answer.

:ok:


From an essay on the issue:

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 (i.e. number) 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?

Actually I think there’s a sensible answer to that question, which is that empiricism is tremendously effective at finding things out and getting things done. But ‘the nature of mathematical objects’ is not itself an empirical question. That’s the nub of the issue.

I have asked him to explain what could be meant by saying that numbers are real — Janus

I presented the argument here https://thephilosophyforum.com/discussion/comment/902998

which was somehow misconstrued as Cartesian dualism, although with an acknowledgment that I had at least distinguished real from existent.

the fact that number is real — Janus

Many people will ignore that too because they will say that numbers aren't real (Field, Azzouni).

by saying that numbers are real beyond our recognition of number in the world — Janus

The dualist will say that they are abstract objects (not spatial, not temporal, causally inefficacious).

fortunately Theodore Sider is trying to help us out . . . see his Writing the Book of the World. — J

A sample

I read it and agree with @Banno that you are saying there are two realities—the physical ("sensable") and the mental (abstract) which is basically dualism.

You still haven't given an answer as to why the reality of number (as "sensably") instantiated could not possibly explain our understanding of mathematics. Nor have you explained in what sense an abstract "object" could be considered to be real beyond its being thought by humans (or other suitably competent beings).

The dualist will say that they are abstract objects (not spatial, not temporal, causally inefficacious). — Lionino

Yes, and unfortunately, we have no idea what it could mean to be such an object, apart from, as I said above, it being thought by some mind.

I don't think quantifiers have much of anything to do with existence or being or any of that. They're entirely about predication -- classification, categories, concepts. Quantifiers are about what things are, not that they are. — Srap Tasmaner

I would say that the logic will inevitably be applied to real things, at which point the logical domain must grapple with mapping itself to an existent domain. I actually find it odd to hear you say that quantifiers do not implicate existence (real or imagined).

that you are saying there are two realities—the physical ("sensable") and the mental (abstract) which is basically dualism. — Janus

I would like to believe that this position is nearer to Kant’s transcendental idealism. There’s no way I posit anything like Descartes ‘res cogitans’ or the seperatness of mind and body.

Not quite sure what you're driving at, but logic was always a formalization of what is understood to be the necessary characteristics of real things.

You always argue from an unquestioned empiricism and can’t see how anything that challenges that can ‘make sense’ in your terms.

There are things you can't 'learn from experience'. All the math experts on this forum know things that I know I'll never understand, even if sat in the same room and looked at the same symbolic forms. They have an intellectual skill that I and others lack. Nothing to do with experience, although it can be shaped and augmented through experience. But the innate skill has to exist first. You'll never teach the concept of prime to a Caledonian crow ;-)

I actually find it odd to hear you say that quantifiers do not implicate existence (real or imagined). — Leontiskos

I mean, of course they implicate it, in the exact sense that they presuppose it -- but they don't have anything to say about it. Rather like the status that "truth" has in logic ... (Existence being not a real predicate, and in any given language neither is "... is true" -- need the metalanguage for that.)

What's asserted in an existentially quantified formula is not really, say, "Rabbits exist," but the more mundane "Some of the things (at least one) that exist are rabbits." Or "Not all of the things that exist aren't rabbits," etc.

And then there's all the complications that arise --- sortals and unrestricted quantification, vacuous singular terms, the elimination of singular terms, projectibility, the substitutional interpretation, et bloody cetera.

Also I always think it's worth rememembering that Frege's quantifiers, and the rest of classical logic so many of us know and love, was not designed as an all-purpose logic at all, but was what was needed to formalize mathematics. It's got some very rough edges when applied more broadly, about which there's endless debate, but it runs like a champ on its home turf.

You'll never teach the concept of prime to a Caledonian crow — Wayfarer

Hmmmm. You know that sounds a lot like one of those things people say because it's so obviously true, right up until it's proven false. (Heavier than air flight? Are you mad?)

There are simple algorithms for determining whether a number is prime; it's a mechanical process that doesn't require what you call "rational insight". Our intellectual superiority to the crow, in this case, is our greater capacity for purely mechanical, algorithmic thought-work. (In similar fashion, teenagers with essentially zero grasp of the niceties of algebraic geometry can solve quadratic equations for you all day long.)

Ah, but the concept, you'll say -- what extraordinary insight did it take to come up with the concept of primality? Eh. Primality is not subtle or complicated. If you do a lot of arithmetic, you're bound to notice that some numbers are a bit incorrigible in a similar way.

I don't say that a crow would notice. I'm just pointing out that, as with everything, it's practice first then theory, if ever, and that what gets noticed is something about the experience of doing arithemetic -- no portal opens to reveal the crystalline realm of mathematics, with an altar to primality at the center.

Neither am I denying that the noticing is where the action is, and we're damn good noticers. I would just want to be clear about what the noticing is and how it occurs before drawing any conclusions. --- And none of this says anything about whether numbers "exist" or whatever. That's the tail wagging the dog.

There are simple algorithms for determining whether a number is prime; it's a mechanical process that doesn't require what you call "rational insight — Srap Tasmaner

Machines are artefacts, are they not?

I'd be interested in your take on this paper I often cite, Frege on Knowing the Third Realm, Tyler Burge - about Frege's implicit Platonism concerning number.

Frege accepted the traditional rationalist account of knowledge of the relevant primitive truths, truths of logic. This account, which he associated with the Euclidean tradition, maintained that basic truths of geometry and logic are self-evident. Frege says on several occasions that such primitive truths - as well as basic rules of inference and certain relevant definitions- are self-evident. He did not develop these remarks because he thought they admitted little development. The interesting problems for him were finding and understanding the primitive truths, and showing how they, together with infer- ence rules and definitions, could be used to derive the truths of arithmetic.

It's about the extent of my knowledge of Frege, but I've always found it an interesting paper.

Well yeah, Frege was a platonist. He was a pretty good logician, but he wasn't a god, and platonism is an inevitable and understandable mistake. :smiley-face:

There are simple algorithms for determining whether a number is prime; it's a mechanical process that doesn't require what you call "rational insight — Srap Tasmaner

Machines are artefacts, are they not? — Wayfarer

What of it? Natural selection, for instance, is a mechanical, algorithmic process. Nature is full of them, without the need of a mind to have conceived them. That recognition is why Dewey thought Darwin would finally put paid to platonism in its many guises. That was over a hundred years ago, I believe, and people have yet to get the message. And so it goes.

I would like to believe that this position is nearer to Kant’s transcendental idealism. There’s no way I posit anything like Descartes ‘res cogitans’ or the seperatness of mind and body. — Wayfarer

Are you not arguing for two kinds of reality—the reality of the body and the different reality of the mind?

↪Janus You always argue from an unquestioned empiricism and can’t see how anything that challenges that can ‘make sense’ in your terms. — Wayfarer

This just seems to be an ad hominem deflection. I have asked for an account of the reality you want to claim for abstracta, which is alternative to the account that says they are real insofar as they are thought, and you have not been able to give me any account to comment on.

So, it's a bit rich for you to be claiming that my position is "unquestioned" and that I am incapable of seeing anything that challenges it, It is my long and deeply considered view that the only kinds of intersubjective justification that are possible for beliefs are the empirical and the logical/ mathematical, but I am open to having my mind changed if someone presents an alternative to those that is convincing. I hold that view because I am yet to see such an account.

And I shouldn't have to remind you that it is my position that people can be justified in believing things which cannot be empirically or logically supported on the basis of their own intuitions or experience, but I maintain that that cannot be justification for anyone else believing those things (unless of course they share the same intuitions and/or experience).

So, my position is not positivism, despite your repeated attempts, despite repeated corrections, to paint it as such. And in any case even if it were pure positivism, that does not let you off the hook from being called upon to actually give an account instead of the constant hand-waving and claims to be misunderstood you do generally present.

:up:

I think if you're going to argue for something more along the lines of "mathematics is invented/arbitrary," a compelling argument at least needs a good explanation of why such a practice arises, is so incredibly useful, and seems so certain. By way of analogy, swimming is also something we "do," but any decent explanation of how swimming works is going to involve, at the very least, mentioning water. Certainly, you don't need an in-depth explanation of "how swimming works," to swim, but swimming itself, or the fact that it is an activity, is not an explanation of swimming.

But since mathematics underpins all of science, it's obviously going to be an area of intense curiosity, which is why...

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 would say no. 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.

As for causes, in this case I don't think you can do without them. If you want to say that "three" is something "stipulated" in the way that the concept of private property is, the obvious next question is: "ok, conceptions of property vary radically across time and space. Numbers do not, and they are stipulated the same way across cultures, including those that have been isolated from one another. Moreover, we have a very easy time imagining worlds were private property, marriage, etc. do not exist, but people have long thought it not incoherent to be asked to imagine a world where basic arithmetic works differently, where two and two make five. Why this huge difference?"

Many people will ignore that too because they will say that numbers aren't real (Field, Azzouni).
I, personally, think mathematics is an empirical endeavor. — Lionino

I think mathematics is an empirical (as well as logical) endeavor also, so we agree on that. But note I said number is real, not numbers.

The dualist will say that they are abstract objects (not spatial, not temporal, causally inefficacious).
— Lionino

Yes, and unfortunately, we have no idea what it could mean to be such an object, apart from, as I said above, it being thought by some mind. — Janus

If number is real in the sense I say it is, that is in the sense that there are numbers of objects, then number would be a real attribute of objects, and the objects would be real, but the numbers themselves would only be real as ways of thinking and dealing with objects, and also as elements in formalized systems of rules elaborated upon that basis.

number would be a real attribute of objects — Janus

That's a tough sell, though. It was one of Frege's brilliant examples, that the logical form of "The king's carriage was pulled by black horses" is different from the logical form of "The king's carriage was pulled by three horses." This is the guy who (independently of Peirce, I believe) is going to invent our modern regime of quantifiers, because he noticed things like this.

I don't see why it should be a "tough sell". If diversity, sameness and difference are acknowledged as being real, then number would seem to naturally follow. It is easy enough to physically demonstrate all the basic operations of arithmetic with actual objects; you start with, say sixty objects and then group them to show how they can be variously added, subtracted, multiplied and divided.

If diversity, sameness and difference are acknowledged as being real — Janus

But what does that mean? Is "different" a property an object can have?

Yes, I'm being a little cagey, but you can do better than a shrug.

(And that's all for me tonight.)

number would be a real attribute of objects

How does this square with this?

but the numbers themselves would only be real as ways of thinking and dealing with objects, and also as elements in formalized systems of rules elaborated upon that basis.

Do you mean numbers as abstracted from any particular instantiation if them?

What do you think of the claim that discrete entities only exist as a product of minds? That is, "physics shows us a world that is just a single continuous process, with no truly isolated systems, where everything interacts with everything else, and so discrete things like apples, cars, etc. would exist solely as 'products of the mind/social practices.'"

- Okay good, it sounds like we are on the same page, or at least the same chapter.

I mean, of course they implicate it, in the exact sense that they presuppose it -- but they don't have anything to say about it. Rather like the status that "truth" has in logic ... (Existence being not a real predicate, and in any given language neither is "... is true" -- need the metalanguage for that.)

What's asserted in an existentially quantified formula is not really, say, "Rabbits exist," but the more mundane "Some of the things (at least one) that exist are rabbits." Or "Not all of the things that exist aren't rabbits," etc. — Srap Tasmaner

I may be reading between the lines of the OP, but here is what I see. I see a question about intractable philosophical disagreements and the possible answer of “quantifier variance.” That at least in some cases the culprit is a notion of quantification that is not shared between the two parties. Now if quantifier variance is occurring—superable or insuperable—then the existential quantifier is doing more than presupposing a univocal notion of existence. Or, if you like, the two secretly competing meanings of existential quantification are each “presupposing” a different notion of existence, and this is the cause of the disagreement. Thus arises the very difficult question of how to adjudicate two different notions of existence, and this is the point of mine to which you initially objected. ...Regarding metalanguage, my earlier contention was that language shapes metalanguage, and does not merely presuppose it. There are no metalanguage-neutral languages, and logicians are prone to miss this.

I actually think 's post may be most instructive and fruitful.

Also I always think it's worth rememembering that Frege's quantifiers, and the rest of classical logic so many of us know and love, was not designed as an all-purpose logic at all, but was what was needed to formalize mathematics. It's got some very rough edges when applied more broadly, about which there's endless debate, but it runs like a champ on its home turf. — Srap Tasmaner

Yes, and this is an important way that the logic reflects the commitments or intentions of its creators. It is not logic qua logic; it is logic qua mathematics.

---

- :up: I should say that while debates about universals—mathematical or otherwise—are interesting, I don’t want to enter that fray given my time constraints. It’s also one of those mountains that requires preparation and gumption—not something I would want to do impromptu. :smile: