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. — Srap Tasmaner

We don't see individual objects in isolation, but as embedded in and different from their surroundings, so difference if not a property of some putative completely isolated object, but a property it displays in its situatedness.

Do you mean numbers as abstracted from any particular instantiation if them? — Count Timothy von Icarus

Yes that sounds about right.

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.'" — Count Timothy von Icarus

I don't see any reason to think that we carve up the world arbitrarily, but rather I see many good reasons to think that we are constrained by its actual structures.

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. — Leontiskos

Well, I brought up the issue, so I'm bound to say there's something to this.

On the other hand, I'm hesitant to endorse what you say here because mathematics is special, and there's a sense in which mathematics is the goal of logic, the goal of thinking as such. (I think there are hints of the excitement of this discovery almost everywhere in Plato where he rattles off the list -- argument, mathematics, astronomy, and so on.) --- And that means "qua mathematics" is not generally a restriction of anything, a limiting of it to this one domain, but an idealization of it.

And it's historically backwards -- but maybe that was deliberate? Frege was trying to reduce mathematics to logic, not the other way around, and that turns out not quite to work, but in trying to do so, he came up with a formalization of logic which could be extremely useful to mathematics rather than providing its foundation. A sort of logic "adapted to" mathematics, or to the needs of mathematics, which is what I was suggesting --- although this time around I've already suggested this isn't necessarily a deformation of logic by focusing on a limited domain, so much as an idealization of logic by focusing on the domain that most cleanly, we might say, represents human thought. And as it happens, I think Frege thought so as well. I think he was mostly of the opinion that natural languages are too much of a mess to do sound work in.

Do all those steps amount to "logic qua mathematics"? Maybe kinda, in a dyer's hand sort of way. There's a lot that makes it look like a branch of mathematics, and the advanced stuff tends to be called "mathematical logic" and get taught in math departments. But that's a deeply tricky business because basic logic is the fundamental tool of everything done in mathematics, absolutely everything -- it's just taken as given at lower levels of learning, without any suggestion that you're actually borrowing from some rarefied advanced field of mathematics.

So I think advanced "mathematical logic" is something like "mathematized logic" -- that's qua-ish maybe in the sense you meant -- but what that means is applying the tools and techniques of mathematics to the given material that is logic, which mathematics can treat of, because mathematics is good at treating of anything. (That's the whole point.) And one of the techniques mathematics brings to bear in treating of logic is, well, logic, because mathematics was just borrowing it for free in the first place.

Still agree?

I actually think ↪fdrake's post may be most instructive and fruitful. — Leontiskos

Wouldn't be the first time, but he was addressing the topic, and I have yet to develop an interest in doing that.

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. — Leontiskos

Do as you like, I just don't see the point. We can talk about existence all we like without dragging quantifiers into it, and people -- they're always wandering around the forum -- who get worked up about the meaning of the "existential quantifier" are generally just confused by the name (a name I note Finn and Bueno would like to retire).

It's a funny thing. This is all Quine's fault, as I noted. "To be is to be the value of a bound variable" comes out as a deflationary slogan, but what we was really arguing for was a particular version of univocity: the idea was that if you quantify over it, you're committed to it existing, and he meant "existing" with the ordinary everyday meaning; what he was arguing against was giving some special twilight status to "theoretical entities". If your model quantifies over quarks, say, then your model says quarks are real things, and it's no good saying they're just artifacts of the model or something. --- The reason this is amusing is that all these decades later the consensus of neuroscientists and cognitive psychologists, so far as I can tell, is that absolutely everything we attribute existence to in the ordinary everyday sense -- medium-sized dry goods included -- is an "artifact of the model" or a "theoretical entity", so the threat to univocity Quine was addressing never actually existed, if only because the everyday meaning of "exist", the one Quine wanted to stick with, is in fact the "twilight" meaning he wanted to tamp down. And so it goes.

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.'" — Count Timothy von Icarus

I don't think we are any more justified in saying this than we are in saying the world is full of distinct objects. All we have is signal processing. Is the source one signal? Two? Two trillion? How can you tell when you're receiving and analysing them all at once? It makes a difference in your metaphysics, but in nothing else at all that I can see.

We don't see individual objects in isolation, but as embedded in and different from their surroundings, so difference if not a property of some putative completely isolated object, but a property it displays in its situatedness. — Janus

And you don't see any circularity here?

Remember the issue was whether number could be a property of an object, and it just obviously can't unless sets count as objects. It's really straightforward and it pissed Quine off considerably.



But then you brought in this other stuff about "diversity, sameness, and difference being real" which just begs another pile of questions. I'm at a loss.

Thanks for your response. I have a full-on week of work starting tomorrow, so it may take me a while to respond further, but I'll just say one thing for now...each object is an instantiation of one, and groups of things obviously have different numbers of objects in them. You can see the difference between one object and a group of ten or any other small enough number.

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.'" — Count Timothy von Icarus

That it’s the kind of thing a Parmenides would say?

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

Not two kinds but two levels, phenomenal and noumenal - and the role of the mind in synthesizing them to produce a unity.

This is great, thanks! Didn't know the sample was available online.

each object is an instantiation of one — Janus

Is this a property it acquires naturally, along with its chemical composition, its mass, etc?

Or do we deem each object to be an instantiation of One?

@Count Timothy von Icarus @Wayfarer @Leontiskos et al.
Here's what I think, if you're interested.

Kant -- damn his eyes -- was right: we only understand of the world what we put into it.

We distinguish one bit from another, sort those bits and classify them, even paint them different colors to make it easier to keep track of them.

Mathematics is, first of all, our analysis of what we're doing when we do all that. More than that, it's a simplification and idealization of the process, to make it faster and more efficient.

It's all signal processing. The brain is not fundamentally interested in the world, but in the maintenance of the body it's responsible for, and the signals the brain deals with are about that body: they have an origin and and a type and a strength, and so on. Some of this is instrumented, so there's a reflective capacity to see how all these signals come together, and that's the beginning of mathematics.

Individual neurons themselves do this in microcosm, actively resisting firing until they absolutely have to, to sharpen and compress their signals from the analog toward the digital. And there's layer upon layer upon layer of this, simplifications of simplifications of simplifications. (The world itself is computationally very far away.)

Signals always have noise, and it's an efficient simplification not to pass through to the rest of the system the whole mess with a peak around 7 MHz and just say "7". We do this in well-known ways with phonemes, for example, counting a considerable range of sounds someone might make as an "r" or an "a".

Simplification and idealization makes it all possible, and that's mathematics. The world is in essence a mathematical construction of our brains, so of course it's a bit puzzling whether math is "in here" or "out there".

That's the gist, or part of a gist, of my view.

Just commenting on this to remember/"bookmark" it because I thought it was interesting.

Mathematics has this double role: it's the ideal we strive towards in our thinking, but it's what enables our thinking in the first place. Out brains have already been doing the sort of clarification and simplification we want when we model something mathematically -- so of course it feels like we're discovering that structure, not inventing it; we're just doing more of the same.

That's my working hypothesis anyway. Philosophy is almost entirely puzzling out the nature of idealization and its role in our thinking, and this approach makes some sense of that. To me at least.

Wouldn't be the first time, but he was addressing the topic, and I have yet to develop an interest in doing that. — Srap Tasmaner

I was addressing the topic as well, and so your attempt to address my post without addressing the topic would be difficult. If quantification and/or existence is straightforwardly univocal (as some here seem to hold), then it is hard to see how a theory like QV could even be entertained. @fdrake managed to "save the appearances" in both directions, so to speak.

Still agree? — Srap Tasmaner

Quickly, not quite. I do acknowledge that mathematics is a paradigm of logical thinking, and that Plato was heavily influenced by it, but I don't think logic is inherently mathematical, I don't think "mathematics is good at treating of [everything]," and I don't think mathematical logic is necessarily the epitome of logic. In fact at my university mathematical logic was very much acknowledged to be but one kind of logic, and I think this is correct. As someone who has formally studied computational logic, mathematical logic, and philosophical logic, I see no reason to universally privilege mathematical logic.

If we want to see this we need look no further than to one of Plato's direct successors, Aristotle. Aristotle is the father of formal logic, and his logic seems to have more to do with knowledge, biology, and classification ("substance") than with mathematics. In particular, as a scientist Aristotle would begin to develop systematic ways of thinking about non-necessary properties of real objects (proper accidents, accidents, etc.). Aristotle was more interested in representing the way the human mind draws conclusions than adhering to an a priori mathematical paradigm, and I think this makes for a much stronger logic. I think one could pick out mathematical logicians and philosophers throughout history (Plato, Descartes, Leibniz, right up to contemporaries like Frege), but to reduce logic to mathematics (or to privilege the mathematical paradigm as primary) would be to overlook lots of other, more natural-scientific thinkers along the lines of Aristotle. I don't think a priori mathematization is ever more plausible than Aristotle's a posteriori approach.

Do as you like, I just don't see the point. We can talk about existence all we like without dragging quantifiers into it, — Srap Tasmaner

The question here is different. It is the question of whether we can speak about quantifier variance without talking about notions of existence.

It's a funny thing. This is all Quine's fault, as I noted. "To be is to be the value of a bound variable" comes out as a deflationary slogan, but what we was really arguing for was a particular version of univocity: the idea was that if you quantify over it, you're committed to it existing, and he meant "existing" with the ordinary everyday meaning; what he was arguing against was giving some special twilight status to "theoretical entities". If your model quantifies over quarks, say, then your model says quarks are real things, and it's no good saying they're just artifacts of the model or something. --- The reason this is amusing is that all these decades later the consensus of neuroscientists and cognitive psychologists, so far as I can tell, is that absolutely everything we attribute existence to in the ordinary everyday sense -- medium-sized dry goods included -- is an "artifact of the model" or a "theoretical entity", so the threat to univocity Quine was addressing never actually existed, if only because the everyday meaning of "exist", the one Quine wanted to stick with, is in fact the "twilight" meaning he wanted to tamp down. And so it goes. — Srap Tasmaner

Interesting, but this seems to prove the point insofar as Quine's notion of existence (and quantification) differs from the approach of neuroscience. Here enters again the questions of the OP.

That's the gist, or part of a gist, of my view. — Srap Tasmaner

It is a common view these days. I will leave my objections for another day. :wink:

Thanks for your thoughts! Have to work, but I'll definitely get back to you after a bit.

It is a common view these days — Leontiskos

Glad to hear you say that. I'm not innovating here, I think, just trying to connect the dots.

I don't think logic is inherently mathematical, I don't think "mathematics is good at treating of [everything]," and I don't think mathematical logic is necessarily the epitome of logic. In fact at my university mathematical logic was very much acknowledged to be but one kind of logic, and I think this is correct. — Leontiskos

I get that. I'm using "mathematics" pretty broadly. What I have in mind is the mathematical impulse, the attempt to understand things by schematizing them, abstracting, simplifying, modeling. A musical scale is such an abstraction, for example, and "mathematical" in the sense I mean.

You're right, of course, that as commonly used the phrase "mathematical logic" is just a branch of mathematics, but to me logic is very much a product of the mathematical impulse, as when Aristotle abstracts away the content of arguments and looks only at their form -- and then follows up by classifying those forms! And we end up with the square of opposition, which is a blatantly mathematical structure. You see what I mean, I'm sure.

Aristotle was more interested in representing the way the human mind draws conclusions than adhering to an a priori mathematical paradigm — Leontiskos

As am I, in fact. I think the foundation of logic is the idea that one thought "follows from" another, and this in many more senses than are covered by material implication, for example. But I also think this is so because this is how our brains work, though we are not privy to the details. Hume noticed this, that the mind passes in some cases freely and in other cases with difficulty from one thought to another.

But I still say the foundation here is mathematical because with the brain we're really talking about prediction, and thus probability. The brain is a prediction engine that is constantly recalibrating. It instantiates a machine for calculating probabilities. The "following from" here is neural activity, which is messy and complicated, but has effects that are in principle measurable, and whose functioning itself is parametrized (concentration of ions and neurotransmitters, number of incoming connections and their level of excitation, distance to be covered by transmission, and so on).

this seems to prove the point insofar as Quine's notion of existence (and quantification) differs from the approach of neuroscience — Leontiskos

But his just thinking that doesn't get you there, to my mind. He was mistaken -- only because he was too early, really, and I think he'd be fine with how cognitive science has naturalized epistemology -- but does that eo ipso ground an alternative but legitimate meaning? Does QV amount to a claim that no one can be mistaken?

You seem to be dragging me into the actual topic, but alas my lunch break is over.

Is this a property it acquires naturally, along with its chemical composition, its mass, etc?

Or do we deem each object to be an instantiation of One? — Srap Tasmaner

If an object has a unique identifying chemical composition and mass "naturally" would this qualify it as being one thing naturally? If we go down the road of thinking that some properties are merely "ascriptions" where would we draw the line?

And you don't see any circularity here?

Remember the issue was whether number could be a property of an object, and it just obviously can't unless sets count as objects. It's really straightforward and it pissed Quine off considerably. — Srap Tasmaner

When I think about the visual field, comprising many things, it has the property of number. We can think of it as one or many. Do you think our seeing it as comprised of many is constrained by actual structure, actual diversity, difference and configuration, or does the brain make it all up from scratch?

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

Not two kinds but two levels, phenomenal and noumenal - and the role of the mind in synthesizing them to produce a unity. — Wayfarer

Below is quoted from you on this;

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. — Wayfarer

Kant's phenomenal/ noumenal distinction as I understand it is not between sense objects and abstracta, but between what we can know and what we cannot.

You seem to be claiming there are two kinds of objects: the physical (phenomena) and the mental (abstracta)_and claiming that at least abstracta are real independently of the mind. If you claim both phenomena and abstracta are mind independently real, then that would be dualism. I guess if you claim that only abstracta are real then that would be idealism, but certainly not of a Kantian kind.

Just what your position consists in remains unclear to me.

Glad to hear you say that. I'm not innovating here, I think, just trying to connect the dots. — Srap Tasmaner

Sure. :up:

I get that. I'm using "mathematics" pretty broadly. What I have in mind is the mathematical impulse, the attempt to understand things by schematizing them, abstracting, simplifying, modeling. A musical scale is such an abstraction, for example, and "mathematical" in the sense I mean.

You're right, of course, that as commonly used the phrase "mathematical logic" is just a branch of mathematics, but to me logic is very much a product of the mathematical impulse, as when Aristotle abstracts away the content of arguments and looks only at their form -- and then follows up by classifying those forms! And we end up with the square of opposition, which is a blatantly mathematical structure. You see what I mean, I'm sure. — Srap Tasmaner

Okay, I understand. I tend to follow Aristotle and Aquinas, and for them the intellect or the reason deals with form (as opposed to matter); thus any kind of intellectual operation will deal with "abstracted" forms (e.g. shape, color, number, etc.). This is all the more true when it comes to discursive reason (and logic). Mathematics deals with one kind of form, namely number or quantity, and this is a very stark and useful form as far as pedagogy is concerned, but the intellect is able to handle all sorts of non-mathematical forms as well. So I don't know that we disagree very much, but I would want to say that mathematics is logical or rational rather than saying that logic or reason are mathematical.

Beyond that, I am wary of the mathematization of reason insofar as our technological age has a heavy predilection for mathematics. In other words, I think we have a bias in favor of math, given our modern Baconian desire to shape and control nature.

But I still say the foundation here is mathematical because with the brain we're really talking about prediction, and thus probability. The brain is a prediction engine that is constantly recalibrating. It instantiates a machine for calculating probabilities. The "following from" here is neural activity, which is messy and complicated, but has effects that are in principle measurable, and whose functioning itself is parametrized (concentration of ions and neurotransmitters, number of incoming connections and their level of excitation, distance to be covered by transmission, and so on). — Srap Tasmaner

Okay. I am not a physicalist and you will never hear me talk about the brain in these ways, but I don't really want to get into those things. I think that if one were to grant the premise that the human being is basically a kind of survival-oriented prediction machine, then a kind of Kantianism and pragmatism does follow, and human reason (and logic) will then be understood in this same light. Yet what I said earlier about faith is also relevant here, for I think that the reduction of the human mind or soul to logical-mathematical functions of this kind is "pidgeonholing" or "hamstringing." At the very least I think one should be open to the possibility that the human mind is able to engage in other, less pragmatic activities. (I get the sense that your understanding of mathematics is pragmatic, and that you would not be inclined to simply contemplate mathematical Forms with Plato.)

But his just thinking that doesn't get you there, to my mind. He was mistaken -- only because he was too early, really... — Srap Tasmaner

Perhaps, but I don't know that he has to be. I'd say that Hume's constant conjunction and the probability theories that tread similar ground are intellectually problematic insofar as they pre-pave a meta-rut for cognitive bias. For instance, we are now prone to mistake anthropological habits for natural probabilities. For example, one could look at our contemporary world and conclude that the human mind is inherently (and reductively) "mathematical" and technological, but I would contend that the evidence at hand is not a result of natural probability, and is instead a result of choices we have made, individually and collectively. Similarly, one might have grown up in Bavaria and have drawn the conclusion that all humans do, and always have, preferred Weizenbier. A wider scope would demonstrate that the preference for Weizenbier is conditioned, and is a human habit flowing from free choice and circumstance. When reason is reduced to constant conjunction or probability repeated decisions become self-justifying, and the distinction between knowledge and opinion dissolves.

You seem to be dragging me into the actual topic... — Srap Tasmaner

Hopefully. But I've said too much, and you are returning just as I am leaving, as I am planning to take a hiatus from TPF. Hopefully this doesn't draw me in too far. :grimace:

Frege was trying to reduce mathematics to logic . . . this isn't necessarily a deformation of logic by focusing on a limited domain, so much as an idealization of logic by focusing on the domain that most cleanly, we might say, represents human thought. And as it happens, I think Frege thought so as well. I think he was mostly of the opinion that natural languages are too much of a mess to do sound work in. — Srap Tasmaner

This thread has developed far and wide, with the discussion quality very high, and I don’t mind at all the divagations from my OP. Unsurprisingly, thinking about quantifier variance opens the door to basic questions about existence and the nature of philosophical thinking itself.

So, just a few responses: The above statements about Frege are right, I believe. Responding to the messiness of natural language, he/we’ve gone on to develop the quantificational apparatus and the ability to speak Logicalese, which really does clear up some of the mess, quite often. But it leaves us with puzzles too, like this one about whether quantifier variance is a coherent idea. The underlying problem doesn’t go away just because we declare (as I think we should) that Ǝ never actually changes its meaning, or, better, its use. If I say, in English translation, “Numbers exist,” and you say, “Numbers do not exist,” we’re disputing what it means to exist, not how to use the quantifier correctly. This is even clearer with a mereological example such as “There exists an item composed of my left nostril and the planet Venus”. We very much want to deny that such an item exists, but we can’t do it by claiming it’s impervious to quantification. To argue for a common-sense meaning (or any other) for “exist” is done in a natural language, not Logicalese.

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

This makes the same point well. To say that "Some of the things that exist are rabbits" doesn't tell you a thing about what "exist" means.

To take the example of the OP: quantifier meaning is not unconditioned by ontological commitments — Leontiskos

This would be the pro-QV position, but suppose we said instead, “The meaning of ‛existence’ is not unconditioned by ontological commitments.” This seems unproblematically true – in fact, if we’re not careful, it becomes redundant. But I’m recommending it because it rids us of the assumption that quantifier meaning is about ontological commitments – the very point that needs to be demonstrated. (And no, I don’t think simply saying “To be is to be the value of a bound variable” demonstrates it, catchy though that is.)

Does QV amount to a claim that no one can be mistaken? — Srap Tasmaner

It had better not. But it’s not the only position in the neighborhood that threatens that consequence, so denying QV is only a beginning. As I was saying to Banno previously, the real question is ontological pluralism, which at the very least seems to imply that, if you are mistaken (about basic ontological questions), you’d never know it.

basic logic is the fundamental tool of everything done in mathematics, absolutely everything -- it's just taken as given at lower levels of learning, without any suggestion that you're actually borrowing from some rarefied advanced field of mathematics. — Srap Tasmaner

One of my friends is a distinguished physicist who also knows a lot of philosophy. He is adamant that logic precedes math, in just the way you suggest. (I’m trying to get him to opine about Reality-with-a-Capital-R but he’s being coy. Claims he doesn't understand the question . . . what a cop-out.)

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.

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.

Kant's phenomenal/ noumenal distinction as I understand it is not between sense objects and abstracta, but between what we can know and what we cannot. — Janus

As I also noted somewhere in this thread, I am using the term slightly differently to Kant. Some points from the wiki article on noumenon:

The Greek word νοούμενoν, nooúmenon (plural νοούμενα, nooúmena) is the neuter middle-passive present participle of νοεῖν, noeîn, 'to think, to mean', which in turn originates from the word νοῦς, noûs, an Attic contracted form of νόος, nóos, 'perception, understanding, mind'. A rough equivalent in English would be "that which is thought", or "the object of an act of thought".

However, the article also notes that noumenon is customarily taken to denote 'an object that exists independently of human sense.' Elsewhere I quoted Russell saying that 'universals are not thoughts, but when known they appear as thoughts.' And this causes confusion, because we confuse them with 'the act of thinking' even though (and here's the clincher) they're independent of any particular act of thought. As Frege says (previously cited):

"in the same way", Frege says "that a pencil exists independently of grasping it. Thought contents (e.g. numerical value) 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

So, here's the intriguing thing. Empirical objects *cannot* be truly 'mind-independent' because information about them is received by the senses, which is invariably interpreted by the mind (through apperception). But as far as universals and other abstract objects are concerned, the mind must conform to them. I think this is the sense in which empiricist naturalism gets it backwards when it come to the metaphysics of cognition (putting descartes before dehorse, as Hofstadter said.)

That same Wikipedia entry also observes in respect of noumenon:

Vedānta (specifically Advaita)... talks of the ātman (self) in similar terms as the noumenon.

and

Regarding the equivalent concepts in Plato, Ted Honderich writes: "Platonic Ideas and Forms are noumena, and phenomena are things displaying themselves to the senses... This dichotomy is the most characteristic feature of Plato's dualism; that noumena and the noumenal world are objects of the highest knowledge, truths, and values is Plato's principal legacy to philosophy."

What I'm trying to understand and articulate is along these lines - more Platonist than Cartesian, but also drawing on non-dualism.

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Kant -- damn his eyes -- was right: we only understand of the world what we put into it.

We distinguish one bit from another, sort those bits and classify them, even paint them different colors to make it easier to keep track of them.

Mathematics is, first of all, our analysis of what we're doing when we do all that. More than that, it's a simplification and idealization of the process, to make it faster and more efficient.

It's all signal processing. The brain is not fundamentally interested in the world, but in the maintenance of the body it's responsible for, and the signals the brain deals with are about that body: they have an origin and and a type and a strength, and so on. Some of this is instrumented, so there's a reflective capacity to see how all these signals come together, and that's the beginning of mathematics. — Srap Tasmaner

I agree with you about Kant, but the later analysis is reductionist. I think it's a mistake to try and explain mathematics in terms of signal processing. Why? Because to explain it reductively requires that we are able to stand outside, apart from or above it - to treat it objectively. But, from Thomas Nagel's recent book (and in comments that are also germane to the overall subject):

In ...Engagement and Metaphysical dissatisfaction, Barry Stroud argues that the project (of metaphysics) cannot be carried out, because we are too immersed in the system of concepts that we hope to subject to metaphysical assessment. This "prevents us from finding enought distance between our conception of the world and the world it is meant to be a conception of to allow for an appropriately impartial metaphysical verdict on the relation between the two."

Stroud believes that we cannot succeed in reaching either a positive (often called realist), or a negative (anti-realist) metaphysical verdict about a number of basic conceptions – that we cannot show either that they succeed in describing the way the world is independent of our responses, or that they fail to do so. He argues for this claim in detail with respect to three of the most fundamental and philosophically contested concepts: causality, essentially, and value. The argument has a general and powerful form. Stroud contends that the use of the very concepts being assessed, and judgements of the very kind being questioned play an indispensable part in the metaphysical reasoning that is supposed to lead to our conclusions about these concepts and beliefs. — Analytical Philosophy and Human Life, Thomas Nagel, p 218

He's saying, in effect, that such constructs are 'too near for us to grasp'. And any account of signal processing, indeed neurological and evolutionary accounts of cognition, like all science, already assume the efficacy of numerical and logical analysis. We can't 'stand outside' those elements of our own cognition and observe how they arise from primitive constituents, as we must already be utilising these very elements to detemine what those constituents are.

(This is something well known to non-dualist philosophies mentioned above. There is a well-known and often-cited passage from the Upaniṣad, 'the eye can see another, but cannot see itself, the hand can grasp another but not itself' (source)).

This video review is also worth the time. Neuroscience, it seems, is coming to terms with the way in which the mind 'constructs reality'. Names mentioned in these discussions include Beau Lotto, Donald Hoffman, Anil Seth, Bernardo Kastrup, David Chalmers, and Christoff Koch among others. There's a plethora of video presentations and panel discussions about it on social media. There are, of course, a huge range of views about what it all means (you'll notice a rather panicked cameo from Richard Dawkins at the end lamenting the 'whispering campaign' against objectivity :yikes: )

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, ∃.

I am planning to take a hiatus from TPF. — Leontiskos

Good call. Think I'll scarper as well. Cheers, everyone.

Wayfarer is talking about some supposed ontological role, the E, not quantification — Banno

I am interested in discussing ontology. By the way I checked in with ChatGPT about the relevance of quantifier variability, which produced some useful summaries and sources which can be reviewed here.

"in the same way", Frege says "that a pencil exists independently of grasping it. Thought contents (e.g. numerical value) 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

In the same way the rules of chess, or the value of money could be said to exist and be mind-independent. Likewise for the perfect form of the turd or the pile of vomit. Do you want to claim noumenal (in the platonic sense) status for those?

Do you believe logic existed begore it was formulated by humans? Frege in that passage says that planets and their interactions with other planets existed before they were known—I bet you don't agree with that.

The real problem I see with saying that universals are mind-independently existent or real is that no one has the foggiest notion of what kind of reality or existence they could enjoy.

Likewise for the perfect form of the turd or the pile of vomit — Janus

Unfortunately I don't have the rhetorical skills to fend of such exalted polemics. And, as always, you declare what you yourself don't understand as the limits to what anyone else might consider.

Unfortunately I don't have the rhetorical skills to fend of such exalted polemics. And, as always, you declare what you yourself don't understand as the limits to what anyone else would consider. — Wayfarer

I've noticed that if anyone disagrees with you or questions your ideas you fall back on the claim that they don't understand. I think that if you understood what you meant by saying that universals are real, you would be able to explain it. But no such explanation is ever forthcoming, which leads me to conclude that you don't understand it yourself.

I have no problem with you believing that universals are real entities in some way on the basis that it "feels right intuitively" or whatever, but when you enter a forum like this and want to argue for your belief then you'd better have a strong case to support it, otherwise discussions will devolve into "yes, it is", "no, it isn't".

I'm actually not saying that universals are not real, just as I don't positively claim that God doesn't exist, but I freely admit that I cannot positively imagine any such reality or existence or see any evidence to lead me to conclude that I would have rational justification for believing that there is such an existence.

Doffing my rational hat, I do tend to be intuitively drawn to such ideas, and I allow myself to entertain them in my feelings, and in my poetry and art practice, but I don't claim to have any rational arguments to support my doing that.

Responding to the messiness of natural language, he/we’ve gone on to develop the quantificational apparatus and the ability to speak Logicalese, which really does clear up some of the mess, quite often. But it leaves us with puzzles too, like this one about whether quantifier variance is a coherent idea. — J

I would say that natural language always takes precedence over artificial languages which derive from natural language, and that trying to grant an artificial language autonomy seems to go hand in hand with positivism. Whenever we move from natural language to an artificial language we must be mindful of what is happening, along with the limitations inherent in any artificial language.

This would be the pro-QV position... — J

No, I actually don't think so. As is happening at various points in this thread, you are jumping to an extreme. I think you are under the spell of a pseudo-exhaustive dichotomy (false dilemma), "Either quantification is perfectly univocal, or else QV holds." This is the standard dichotomy of univocity positions, but if we spin things around then it seems obvious that both options are false, and therefore there must be a third option. The key is to discover this third option, this tertium quid. For Aristotle the mean between univocal and equivocal predication is called "pros hen" predication; and by the time of Aquinas it was developed and called "analogical" predication. Cf. SEP.

fdrake's post deftly exorcises both sides of the false dilemma:

That is what you see in practice though. There are no modal operators in propositional logic. But both modal and propositional logic are great. Their semantics also differ considerably. When you write the possibility and necessity symbols in a modal logic, you quantify over possible worlds. When you write them in a quantified modal logic, you quantify over worlds, and there's also quantification within worlds in the usual logic way.

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

Your claim that "Ǝ never actually changes its meaning" is refuted by the simple fact that there are different forms of quantification available in different kinds of logic; thus falls the first, univocal horn of the dilemma. We are well aware of the problems with the second, equivocal horn of the dilemma (unrestricted QV). He points to the paper for evidence that an unproblematic variety of "quantifier variance" is possible ("variance" is to my mind a poor choice of word for the theory, for what is truly at stake is incommensurable variance or equivocation). In Medieval terms the unproblematic variety would be an instance of analogy, where the semantic relation between various forms of quantification is one of analogy. Wittgenstein's notion of "family resemblance" is perhaps not too far off the mark, although I cannot speak to that notion with any expertise.

Edit: For a quick lesson in how quickly logicians become confused when they try to talk about natural language, see section "4. Ontological Pluralism," of Quantifier Variance Dissolved. There is some frightful confusion occurring there. What is happening? Logicians are treating natural language as if it were logical. Note, too, that one of the primary formal differences between natural language and logical languages is that the former includes analogical predication whereas the latter does not. Because of this the authors are not able to truly entertain the view they pretend to be considering, namely the view of ontological pluralism whereby there are "different ways of existing." Such an analogical claim is not representable in logic. This failure plagues their examples and argumentation. Wherever there can be found a border between natural and logical language these incommensurability problems arise, such as the border separating existence and quantification. This is the Achilles heel of analytic philosophy writ large. Logic is a two-edged sword, a tool like an exceedingly fine pencil that can be used to do marvelous, detailed work, but is incompetent in other, more broad-ranging matters. This is why some are apt to criticize analytic philosophy for being skilled at saying relatively unimportant things with exceptional precision. Aristotle was aware of all of this, along with the fact that substantial inferences are usually not perfectly sound, and that perfectly sound inferences are usually insubstantial. A work which combines enough strict demonstration to arrive at substantial conclusions without boring or losing the reader is very rare.

I've noticed that if anyone disagrees with you or questions your ideas you fall back on the claim that they don't understand. — Janus

I have provided references to many other sources, including, in this instance, Frege, Russell, Nagel, and Advaita Vendanta. I believe that I make a coherent philosophical case, but that you haven't demonstrated a grasp of what that is. I'm not saying that to 'anyone', I deal with every interaction on its merits, or lack thereof. I'm saying it to you.

Reading those you have cited, I would say they make the same mistake you do: the mistake of thinking that there is a rational case to be made in the absence of either empirical evidence or logical necessity, or else some other kind of evidence. As I see it, the fact that others make the same mistake as you do does not make it any the less a mistake.

What is it precisely you think I don't understand about your position? You should be able to pinpoint that and you should be able to lay out your case clearly if you have a cogent one, and I haven't seen you do that. When, or if, you do then I will respond.

I deal with every interaction on its merits, or lack thereof. — Wayfarer

Why is it that you cannot tolerate disagreement? Surely you know that when it comes to philosophical questions there never has been consensus, or any way to prove the truth or falsity of positions. I'm not demanding any kind of proof from you; there is no proof even when it comes to scientific theories. But you have stated many times that your views are in the minority on this forum, and there is nothing intrinsically wrong with that, but it should give you pause when you want to level accusations of "misunderstanding" to your interlocutors. It just makes you look defensive.

What is it precisely you think I don't understand about your position? — Janus

The fact that you think all the sources I cite are mistaken, would be a major one.

I can tolerate disagreement, but not pointless arguments, of which this is one.

Just tell me concisely, and without bringing in other authorities, what evidence or rational support you think there is for the reality of universals.

It wouldn't be a pointless argument if you could actually make an argument; then we might actually get somewhere.

If you don't want to try, then I'll conclude that you don't have such an argument.

If you don't want to try, then I'll conclude that you don't have such an argument. — Janus

Tell me, then, exactly where this goes wrong:

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.

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). — Wayfarer