Just say that something has "a certain, je ne sais quoi. When you say it in French it becomes ineffable!

Speaking from a purely personal POV, I think that Natural numbers might objectively exist, and perhaps Real numbers as well. But when you get to stuff like the set of Complex numbers, things just don't make sense anymore. — Arcane Sandwich

Agree. But don't you think that the qualifier 'objective' might be inappropriate in the context? But then, what are the alternatives? The point being, 'objective' means 'inherent in the object/s'. But numbers are not objects per se, they're intellectual acts. We use mathematical techniques to determine what is objective. Not that they're subjective, either, but that their truth status is in some sense transcendental (but then, you can't use that, because 'transcendental numbers' are a special case in mathematics.)

Which is what leads me to speculate that the natural numbers are real but not existent. They are, in a sense other than the Kantian, 'noumenal' - objects of intellect (where 'object' is used metaphorically). But they are also indispensable to rational thought. That is part of the version of mathematical Platonism that makes sense to me.

I've seen this done in a few places actually. Normally the metaphor people use is one of a number line. You have 0 in the middle and positive and negative numbers extending in either direction, out to infinity. To get the imaginary numbers, imagine the real numbers as the x axis on a 2D graph. The imaginary numbers are the Y axis of the complex plain, they are pivoted orthogonal to the reals.

Hmmm... I guess my question is, if I agree or disagree with Mathematical Platonism, is it an all or nothing deal? I mean, does it make sense to say that Platonism might get it right for some mathematical objects (i.e., natural numbers) but not others (i.e., complex numbers)?

Is Platonism in mathematics, as you folks are discussing it, strictly restricted to a specific area, like Arithmetic? Or does it include all areas, like Geometry for example?

Does it make sense to agree with Platonism on some intellectual fronts but not on others?

Does it make sense to agree with Platonism on some intellectual fronts but not in others? — Arcane Sandwich

I'm sure it does. After all, just what Platonism is, over and above the actual dialogues, is always being refined and re-envisioned. My sole philosophical commitment is to what I consider an elementary philosophical fact: that number is real but not material in nature. Of course there are nowadays kinds of neo-pythagorean views, like Tegmark's, but that's not what I have in mind, as Tegmark is, perplexingly enough, still a pretty standard-issue scientific materialist in other ways.

But the point I've been pressing, pretty well ever since joining forums, is that some ideas are real in their own right, not reducible to neural activity or social convention or the musings of experts. We discussed Frege upthread, about whom I know not much, but he is nevertheless instinctively Platonist, in believing that numbers and basic arithmetical operations are metaphysically primitive, i.e. can't be reduced or explained in other terms. In other words, it's a defeater for materialism, and that is why it is so often rejected in the modern academy.

But there are many controversies. Take a look at What is Math? a Smithsonian Magazine article I cited earlier in this thread. I think it lays bare many of the contentious issues. (After reading that article, I purchased a copy of the book by the emeritus professor mentioned in it, James Robert Brown, Platonism, Naturalism, and Mathematical Knowledge, but alas, much of it was beyond my ken. Review here.)

My sole philosophical commitment is to what I consider an elementary philosophical fact: that number is real but not material in nature. — Wayfarer

I think I understand. It's like objective idealism, in some sense. But I'm just having a hard time trying to wrap my head around the underlying concept here. Something (i.e. a number) can be real without being material? How can that be? I'm admittedly a scientific materialist. The specific philosophy of mathematics that resonates the most with me is Mario Bunge's specific brand of mathematical fictionalism. He says that the number 3, for example, is just a brain process. And the same hold for every other abstract concept: from a humble number, to a tautology, to a scientific hypothesis, to a scientific theory, all of them are brain processes, but we feign that they exist as "autonomous ideas", as it were. It's like we're "fake Platonists", if only because all of our concepts are sort of like "useful fictions" in Nietzsche's sense of the term, if that makes any sense.

But I'm not so sure that this is true. Unlike Bunge and other mathematical fictionalists, I think that there's a very solid case that can be made for Platonism about, at the very least, the set of Natural numbers. Complex numbers are a more contentious issue, I think. Are they even "numbers", or are they just concepts? Is Infinity really a number, or is it "just as concept", as some folks say? It's a tough thing to argue, either way.

But I'll definitely read the references that you shared, thank you very much.

The specific philosophy of mathematics that resonates the most with me is Mario Bunge's specific brand of mathematical fictionalism. He says that the number 3, for example, is just a brain process. And the same hold for every other abstract concept: from a humble number, to a tautology, to a scientific hypothesis, to a scientific theory, all of them are brain processes, but we feign that they exist as "autonomous ideas", as it were. — Arcane Sandwich

What we could do is just use the concept of abstract objects as a placeholder. One day we might understand it better. Maybe it will turn out that Bunge is correct. Until we have a testable theory, all we have is biases.

Something (i.e. a number) can be real without being material? How can that be? I'm admittedly a scientific materialist. — Arcane Sandwich

Well, I think we have - no offense or anything - a flawed understanding of what is real. (After all, it's the business of philosophy to make such judgements.)

He says that the number 3, for example, is just a brain process. — Arcane Sandwich

This is 'brain-mind identity theory' which was prominent in the work of a couple of Australian philosophers, J J C Smart and D M Armstrong.

The fly in the ointment here is what exactly is meant by 'the same'. When you say that a brain process is 'the same as' a number then you're already well into the symbolic domain. There's no feasible way to demonstrate that a particular brain process - in fact, there are no particular brain processes, as brains are fiendishly irregular and unpredictable - 'means' or 'is' or 'equates to' anything like a number (or any other discrete idea.) And indeed an argument of this type has an ancient provenance. It appears in Plato's dialogue The Phaedo, in which Socrates makes a vital point about the implication of our ability to perceive the nature of 'equals'. When we see two things that are of equal dimensions, say, two stones or two pieces of wood, we are able to discern that they are equal - but only because we innately possess the idea of 'equals':

Socrates: "We say, I presume, that there is something equal, not of wood to wood, or stone to stone, or anything else of that sort, but the equal itself, something different besides all these. May we say that there is such a thing or not?"

Simmias: "Indeed, let us say most certainly that there is. It is amazing, by Zeus."

"And do we know what it is?"

"Certainly," he replied.

"From where did we obtain the knowledge of this? Isn't it as we just said? From seeing pieces of wood or stone or other equals, we have brought that equal to mind from these, and that (i.e. 'the idea of equals') is different from these (i.e. specific things that are equal)".

The Phaedo 74a ff

Another argument is a version of Putnam's multiple realisability - that a number (or any item of information) can be realised in any number of ways by different brains (or in different media or symbolic types, for that matter.) Neuroplasticity demonstrates the brains of injured subjects can be re-configured to grasp language or number with neural areas not usually associated with those functions. That's also a version of multiple realisability.

And in even thinking about these problems, you're all the time making judgements and reasoned inferences ('if this, then that', 'this must be the same as that' etc.) You can't even define what is physical without relying on those rational faculties, yet brain-mind identity claims that they are somehow the same.

So my view would be that, wherever rational sentient beings exist, there must be a core of real ideas that they are able to grasp, and these are discovered, not invented.

So my view would be that, wherever rational sentient beings exist, there must be a core of real ideas that they are able to grasp, and these are discovered, not invented. — Wayfarer

OK. I can understand that (I think?). It's something that I can agree with, if only for the sake of argument. Numbers are real, and they're not material. My follow-up question would be, are they physical? Like, are they somewhere, in spacetime? Are they in our head, in some sense? Not necessarily in the brain, but then where? In "the mind", assuming that "the mind" is something other than the brain? Are they outside the brain? Where are they? In the things, themselves? I think that might be true of Natural numbers, I can agree with Aristotle's notion that "quantity" is a real accident, a real property of "substances" themselves. Is that what you are saying? Or are you saying something different?

It's like objective idealism, in some sense. — Arcane Sandwich

In typical philosophical parlance, I find "objective idealism" pretty close to a contradiction in terms.

• Subjective   ≈   existentially mind-dependent
· Objective is not

• Idealism   ≈   mental monism
· Realism is not
· An analysis of the rationale leads toward solipsism

But, hey, in the rabbit hole of metaphysics, one can come up with whatever. :)
Never mind me, carry on.

Never mind me, carry on. — jorndoe

I'll mind, I'll take that bet. I think you have an excellent point when you say that "objective idealism" is a contradiction in terms. Plato would not be an objective idealist, then. He would be a "metaphysical realist". But that sounds somewhat "odd", at least to my ear.

, would "abstraction realist" work? Hm Also comes through odd (to me anyway).

Sure, one doesn't need to use imaginary numbers to count apples. Why should that make them more or less real than integers? Moreover, what does "real" do here.

But if something can't be said, it might be important to say why and surely philosophy has a role to play there. — Wayfarer

Yep.

If something is inexpressible, then by that very fact one cannot say why... Doing so would be to give expression to the inexpressible. — Banno

If someone believes something to be inexpressible, then they have a reason why. The ones who are willing to say why are the philosophers.

(And if the object of inexpressibility cannot be referenced in any way whatsoever, then there is nothing which is inexpressible in the first place.)

What can't be said can't be said, and it can't be whistled either. — Frank Ramsey as quoted in Nagel's The Last Word

Sure, one doesn't need to use imaginary numbers to count apples. Why should that make them more or less real than integers? — Banno

True, that's an excellent point. I agree.

Moreover, what does "real" do here. — Banno

I'm not sure. The adjective "real", as far as I'm concerned, has an external referent: it refers to the quality of being real. And what is that quality, exactly? It's hard to say, and it's a contentious issue in the literature. Should we define "real" as a concept, as that which exists outside the mind? Outside the brain? Is it instead that which belongs to a res, a thing? Would weight be a real property? Perhaps mass would be a better candidate. Or even energy. You could say that the difference between real things and mere concepts is that the former have energy while the latter don't. But how can they not? Can there exist real things without energy? Do numbers, as you understand them (as real but not material) entities, have energy in the physical sense of term? I prefer to define "real" as anything that has spatiotemporality (in other words, that it is somewhere in space and "somewhere" in time, even if such locations are not entirely clear-cut). But all of this is up for debate I think, at least inside the "Ontology Room".

My follow-up question would be, are they physical? Like, are they somewhere, in spacetime? Are they in our head, in some sense? Not necessarily in the brain, but then where? In "the mind", assuming that "the mind" is something other than the brain? Are they outside the brain? Where are they? In the things, themselves? — Arcane Sandwich

I think that is due to the cultural impact of empiricism. Because of this we are enculturated to believe that what is real can only be in located in space-time. Notice in that Smithsonian essay:

...scholars—especially those working in other branches of science—view Platonism with skepticism. 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.

But there's another approach - that of phenomenology. @Joshs is adept at explaining that (see this post.) My take is that numbers and logical principles are necessary structures of consciousness. That doesn't mean they're the product of the mind i.e. they're not neurobiological structures but intentional structures in Husserl's sense.

My take is that numbers and logical principles are necessary structures of consciousness. — Wayfarer

That may well be the case.

That doesn't mean they're the product of the mind i.e. they're not neurobiological structures but intentional structures in Husserl's sense. — Wayfarer

That may well be the case, as well.

But then I have another follow-up question. There are four apples on the table. I claim (I might be wrong, of course) that those four apples are still four apples even when no one is looking at them (i.e., "intending" them in any way, as in Husserl's concept of intentionality as a subject-object relation). I would say, the number "one" exists, like an "Aristotelian accident", in each of the four apples. And that "one-ness", if you want to call it that, doesn't somehow "dissipate", or "cease to be", when no one is contemplating the apples, or thinking about them in any sort of way. It's just a brute fact that there are four apples on the table instead of five or three.

Not sure if I'm being collaborative here, Philosophy of Math is quite arguably the toughest branch of philosophy.

it refers to the quality of being real. — Arcane Sandwich

What's that, then? See

@Banno I don't agree with Austin's diagnosis. I think it makes sense to distinguish "real x" from "non real x", it depends on what "x" is, in each specific circumstance. For example, is there a difference between real basilisks and non real basilisks? The question assumes that there are real basilisks to begin with, and it's asking how are they different from non real basilisks. But it's the assumption that's mistaken: there are no real basilisks to begin with, just as there is no x, such that x is (or is not) the current king of France, to phrase it in Russellian parlance.
However, it does make sense to trace a distinction between real fruit and non-real fruit, as in, plastic fruit, not actual parts that were collected from a living plant.
If Mathematical Platonism is right, numbers are more like real fruits than plastic fruit, if that makes any sense. If, on the other hand, Mathematical Fictionalism is right, numbers are more like fake plastic fruit instead of real fruit from an actual plant.

Does that make sense?

But then I have another follow-up question. There are four apples on the table. I claim (I might be wrong, of course) that those four apples are still four apples even when no one is looking at them (i.e., "intending" them in any way, as in Husserl's concept of intentionality as a subject-object relation). I would say, the number "one" exists, like an "Aristotelian accident", in each of the four apples. And that "one-ness", if you want to call it that, doesn't somehow "dissipate", or "cease to be", when no one is contemplating the apples, or thinking about them in any sort of way. It's just a brute fact that there are four apples on the table instead of five or three. — Arcane Sandwich

What you're referring to is 'brute fact' is actually just direct realism, the view that the world is perceived exactly as it is. But that fails to account for the role of the mind in shaping our perception of order and numerical concepts. It fails to grasp the fact that the order we perceive in the world, numerical and other, arises as a consequence of the interaction of mind and world. I acknowledge that in practical, everyday terms, it may seem straightforward to assert the existence of four apples, but this perception is itself mediated by observation and verification, verifying that they're real apples and not fakes or holographs, etc, of which requires observation. This is the subject of another OP The Mind-Created World, also discussed here, a defense of a form of phenomenological idealism.

I don't agree with Austin's diagnosis. — Arcane Sandwich

Ok. I'll leave you to that.

"a certain, je ne sais quoi — Count Timothy von Icarus

Nice. That catches something of the drift.

@Banno I just think that mathematical fictionalism, as articulated by Bunge, can withstand Austin's philosophical and linguistic analysis of the word "real". And I mean no offense by that. Is Bunge right about everything involved in this issue? I don't think so. I think natural numbers could indeed be real in the sense that, if the human species suddenly became extinct for some reason, the four apples on my table would still be four apples (and not, five or three). I'm aware that I'm the habit of explaining things from the point of view of common sense. That is not to say that common sense is applicable to every problem: it isn't. But I, personally, as a creature, am inclined to approach common sense as a moth is inclined to approach a flame. And yes, the irony of that metaphor is not lost on me.

Ok.

I can't see that saying π is a fiction is any better than saying it is the subject of a quantification. Indeed, that π is the ratio of the diameter of a circle to it's circumference isn't in any useful way like saying Frodo walked into Mordor.

I can imagine Frodo walking into Mordor. I can imagine the number Pi (up to a certain point).

Is there a small, barefooted humanoid in the world, walking into a territory somewhere in Europe or some other place on planet Earth, that is a scorched landscape with a tower that has a supernatural Eye at the top (the Eye of Sauron)? I would say no. Is there a thing or property in the world, that "answers" in such a way to the number Pi that I can imagine (up to a certain point)? I think there might be. That would be the difference between Pi and Frodo. Unlike Bunge, who actually compares linear equations to Donald Duck as far as their ontology goes, I believe that natural numbers might indeed have a one-to-one correspondence with the "one-ness" of each ordinary object. Natural numbers could well exist really, outside the mind, in the things themselves. Our intellect merely "reflects" them or "abstracts" them or "represents" them in some way. We pick up on them, we become aware that they are there, just as we become aware that these four apples still exist -as four apples, not merely as a non-numerical bunch of fruit-, when no one is in the house.

I just don't see any of that as helpful. π in't imagined, it's r/c. To say that π exists is just to say it is the value of a bound variable.

∃(x)(x=r/c)

@Banno That's what Quine thought, that "to be, is to be the value of a bound variable". But physical objects existed before logic (propositional, first order, second order, etc.) was invented. Unless you are also Platonic about logic. In that case, I would say that the existential quantifier, symbolized by ∃, should be distinguished from a first-order existence predicate. I can provide examples to how that would work, in the context of first-order symbolic logic. I can also explain why interpreting the existential quantifier as if it had ontological import necessarily leads to a contradiction.

But physical objects existed before logic (propositional, first order, second order, etc.) was invented. — Arcane Sandwich

:meh:
How is that salient?

I would say that the existential quantifier, symbolized by ∃, should be distinguished from a first-order existence predicate. — Arcane Sandwich

I agree. Free logic. Not used here.

I can also explain why interpreting the existential quantifier as if it had ontological import necessarily leads to a contradiction. — Arcane Sandwich

Sure. Quine's point being that treating that π exists is just to say it is the value of a bound variable has no ontological import. That was kinda the joke.

Later.

@Banno Free Logic is not the only option. You can keep classical logic while tracing a distinction (as Bunge does) between real existence and conceptual existence. You can then say that God doesn't exist really, but He, or She, or They, exist conceptually, in the same sense that a Beethoven sonata exists conceptually. What are musicians doing, when performing a Beethoven sonata? They are performing. They are performing the physical act of emitting sounds, in a way that sounds pleasing to the human ear. What are religious leaders doing when they speak to their "human flock"? They are performing. They are performing the physical act of emitting sounds, in a way that sounds pleasing to the human ear (but for a different reason than Beethoven). What are mathematicians and philosophers doing when they speak of numbers? They are performing. They are performing a physical act that sounds pleasing "to the human ear", in a somewhat literal, somewhat poetic (non-literal) sense (but different from the musician, and different from the religious preacher).

Hmmm...

They are performing the physical act of emitting sounds, in a way that sounds pleasing to the human ear. — Arcane Sandwich

Note to Self: Well the irony here, of course, is that such way of speaking sounds "non-human", if that's even a thing. But if it is, then it would have a de-personalizing effect on the listener. Is that true? Do I even agree with this idea myself? But how couldn't I? I'm the one that has thought it. However, it is a thought experiment. Anyone can perform it, at least potentially. Thought experiments, that is. They are objective, though cerebral, and hence, physical.