If mathematicians say something has mathematical existence, then it does. There is surely no objective standard. It's a mistake to believe that there is. — fishfry

From our past discussions about this, I understand the underlying sensibility here, but I think that it goes too far toward the subjective. Again, I endorse Charles Peirce's definition, which he adopted from his father Benjamin: "Mathematics is the study of what is true of hypothetical states of things." For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. Mathematicians may not yet recognize something as following necessarily from them, so it is not a matter of whether they do say that it has mathematical existence, but whether they would say that it has mathematical existence upon discovering a proof.

Certainly mathematics is a social enterprise. And what constitutes a proof is a kind of consensus among those who practice mathematics. However, when I discovered last night a fact about attracting fixed points in polynomials that minor discovery immediately assumed mathematical existence, regardless of whether it is publicized. And it is possible someone else had arrived at this trivial conclusion, so it might have had mathematical existence already. But, in the larger social scheme there is a kind of mathematical existence based upon an agreement that a revelation is important.

Have you considered irrational numbers, imaginary numbers, and infinite electrical resistance (as measured by ∞ on a simple ohm meter)?

Infinity is a principle that arises while reasoning from first principles, such as in mathematics, but not while experimental testing, such as in science. Furthermore, the models for number theory and set theory are never the physical universe. These models are collections of formal language strings. They are 100% abstract only.

Last but not least, you would not be able to prove anything about infinity in the physical universe, because you cannot prove anything at all about the physical universe. We do not have a copy of the Theory of Everything of which the physical universe is a model. Hence, there is no syntactic entailment ("proof") from theoretical axioms possible about the physical universe. — alcontali

Well put.

A thing has mathematical existence when a preponderance of working professional mathematicians say it does.
— fishfry

So are you saying that when Georg Cantor first defined infinite sets ca. 1871 and there was great resistance among the world's mathematicians, infinity didn't exist yet? — Daz

Yes that's what I'm saying. Because the opposite of that proposition is the claim that transfinite set theory was "out there" somewhere waiting to be discovered. But if that's true, where was it? When Ogg the caveperson first made marks in the dirt to count the mastodons killed by the tribe, did all of modern transfinite set theory already exist? Where, exactly? What else exists there? God? The Baby Jesus? The Flying Spaghetti monsters? Platonism is just as hard to defend.

Did Captain Ahab exist before Melville wrote him into existence? Perhaps you can answer me that.

Of course I understand your point, that mathematical objects often seem inevitable after we discover them. But math is a social process. When the preponderance or consensus of working mathematicians accepts a new idea, that social process is what brings that idea into existence. Or at best, when it's first published.

You could ask if the sculpture is there in the stone before the sculptor gets to work.

I understand that we could never get to the bottom of these questions. I'm trying to cut through the existence arguments that are being made here. I don't think you can point to mathematical objects and say that there is an ultimate or absolutely true answer to whether they exist. People argue about whether certain mathematical objects exist or not. Your question is a good one. Where were the transfinite numbers before Cantor discovered them? I have no idea. If you don't think Cantor brought them into existence in an act of human creation; then where exactly do you think they lived? Were they created as part of the Big Bang? That idea is not tenable.

Certainly mathematics is a social enterprise. And what constitutes a proof is a kind of consensus among those who practice mathematics. However, when I discovered last night a fact about attracting fixed points in polynomials that minor discovery immediately assumed mathematical existence, regardless of whether it is publicized. And it is possible someone else had arrived at this trivial conclusion, so it might have had mathematical existence already. But, in the larger social scheme there is a kind of mathematical existence based upon an agreement that a revelation is important. — jgill

Ok, good point. When someone first makes a discovery, that's when the existence happens. Social acceptance decides the importance.

But still, where was the discovery before it was discovered? As much as it's problematic to claim it had no existence beforehand, it's just as problematic to say it did. If it existed before you discovered it, did it exist at the moment of the Big Bang? How did that happen?

Platonism's harder to defend than fictionalism.

Better to say it didn't exist before you thought of it, just as Captain Ahab didn't exist before Melville thought of him. If we can figure out where Captain Ahab lived before Melville created him, then we can talk about whether $\aleph_{47}$ existed before the Big Bang.

From our past discussions about this, I understand the underlying sensibility here, but I think that it goes too far toward the subjective. — aletheist

Not subjective, social. My own hallucinations don't exist. But if I can convince enough people to believe in them, they do. But as @jgill pointed out, a mathematical discovery comes into being at the moment of discovery.

But what if the moment of discovery turns out to be a mathematical error? Then the community corrects it. There's a huge brouhaha in the math world going on right now in the [url=https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/[/url] Mochizuki's claimed proof of the abc conjecture. One group of mathematicians firmly believes a certain result has been proven; others firmly disagree. We have to wait to see how this is ultimately decided, perhaps a long time.

Again, I endorse Charles Peirce's definition, which he adopted from his father Benjamin: "Mathematics is the study of what is true of hypothetical states of things." For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. — aletheist

I think that's limiting. It puts trivial conclusions derived from meaningless axioms on the same level as the deepest results. Math isn't just cranking out theorems from axioms. It's cranking out theorems about mathematical objects. No number theorist believes that Fermat's last theorem is merely a theorem that falls out of the axioms of set theory. Wiles proved that FLT is true. True in a way that transcends axioms. It's a truth about the natural numbers; not merely a truth about proofs in a formal system. That's the Platonist in me speaking.

Mathematicians may not yet recognize something as following necessarily from them, so it is not a matter of whether they do say that it has mathematical existence, but whether they would say that it has mathematical existence upon discovering a proof. — aletheist

Yes ok. Mathematicians tend to believe they're studying mathematical objects, though; and not just searching for proofs. I think that when you do math, you tend to be a Platonist; but when you try to defend the activity rationally, you have to fall back on being a fictionlist.

Sure, math is a social enterprise. But that's not all it is.

As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it.

There is a vast universe of mathematics that has existence as potential. All the logical derivations that lie in wait to be discovered, accompanied by acts of creativity yet to appear - like works of art. :cool:

For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. — aletheist

I think that's limiting. It puts trivial conclusions derived from meaningless axioms on the same level as the deepest results. — fishfry

Not really. Notice that my definition requires the set of definitions and axioms to be established, which could be interpreted as consistent with your requirement for intersubjective agreement among practicing mathematicians. The "deepest results" come about when someone works something out that follows from those definitions and axioms, but either has not been noticed or has not been demonstrated previously.

No number theorist believes that Fermat's last theorem is merely a theorem that falls out of the axioms of set theory. Wiles proved that FLT is true. True in a way that transcends axioms. It's a truth about the natural numbers; not merely a truth about proofs in a formal system. — fishfry

It's a truth about the natural numbers as established by a certain set of definitions and axioms. The latter are the only way we know what anyone means by "natural numbers."

I think that when you do math, you tend to be a Platonist; but when you try to defend the activity rationally, you have to fall back on being a fictionlist. — fishfry

As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it. — Daz

I am more in agreement with @Daz on this, but would substitute "is" for "exists" since the latter has ontological implications that I wish to avoid. Platonism holds that mathematical objects exist in some ideal realm, while fictionalism holds that all properties of mathematical objects are dependent on what someone thinks about them, just like characters in a novel. I am a mathematical realist, but not a platonist; I hold that mathematical objects are real by virtue of having certain properties regardless of what any individual mind or finite group of minds thinks about them, but they do not exist because they do not react with anything. Fermat's last theorem would be a truth about the natural numbers, as established by a certain set of definitions and axioms, even if Fermat never conceived it and Wiles never proved it.

For me, I believe mathematical objects, as well as mathematical truths, exist. So I'm a Platonist.

This is by analogy with the intuition that material objects exist. These exist in the sense that they can be imaged, measured and perceived by many people who will get the same measurements of the same things, at least within a fine tolerance that would improve as our measuring instruments improve.

Similarly, mathematical objects and truths can be observed and measured by many people, or even software, and multiple computer programs (assumed to be debugged) and mathematicians (assumed competent) will observe the same truths about these objects.

This perception of material objects, or mathematical truths, doesn't need to occur in order for the things at issue to exist; it just needs to be in principle possible.

Let me add that I believe that if any hypothetical beings, or even high-powered computers, were acting like mathematicians in distant parts of the universe, they would eventually come upon the very same mathematical truths that mathematicians on Earth would, perhaps in a different sequence. Math is like a landscape waiting to be discovered, no matter where the discoverers happen to be.

Yet the number of all those possible photos is not infinite. — Zelebg

The fact that you say "every photo, every dream..." does not entail that there is a finite number of photos or dreams.

It seems you have assumed that there is only one thing called "the universe", and that it is finite in space and time.

How do you know this? What is the basis for this claim in your argument?

How do you know this? What is the basis for this claim in your argument? — Cabbage Farmer

Depends on what you mean by "the universe". If the nature of the universe is established via the scientific method, whatever is the result must be finite.

Depends on what you mean by "the universe". If the nature of the universe is established via the scientific method, whatever is the result must be finite. — Echarmion

It seems to me that it's only what's called "the known universe" that is "established by scientific method".

But there's an important conceptual difference between the world as it is, and the world as it is known by us.

I see no reason to suppose that our knowledge of the world at any given time in history would give us complete knowledge of the whole world.

Is there some reason to suppose that what we know about the universe at any given time, in keeping with scientific method, is all that we will ever come to know?

Is there some reason to suppose that the sum of everything we could ever possibly know about the universe, in keeping with scientific method, would provide a complete account of everything that is in fact the case, across all time and all space, or across whatever "dimensions" we should name alongside or instead of time and space, and across whatever universes and multiverses and iterations of generation and decay of universes or multiverses there may be....?

It seems to me that it's only what's called "the known universe" that is "established by scientific method". — Cabbage Farmer

The known and the empirically knowable, yes. But beyond that, the meaning of "the universe" gets rather vague and nebulous.

But there's an important conceptual difference between the world as it is, and the world as it is known by us.

I see no reason to suppose that our knowledge of the world at any given time in history would give us complete knowledge of the whole world.

Is there some reason to suppose that what we know about the universe at any given time, in keeping with scientific method, is all that we will ever come to know? — Cabbage Farmer

This topic tends to run into language limitations. So, I get what you're saying, but the problem is that concepts like "knowledge" break down when we go beyond whatever we can somehow experience. Everything we'd "know" about the "world as it is" can only be based on deductions from first principles, something entirely different in nature from knowledge about the physical world.

Based on that, we cannot ever "come to know" anything about the "world as it is". If that information is available to us, we already have it, we merely need to make the correct deductions.

Is there some reason to suppose that the sum of everything we could ever possibly know about the universe, in keeping with scientific method, would provide a complete account of everything that is in fact the case, across all time and all space, or across whatever "dimensions" we should name alongside or instead of time and space, and across whatever universes and multiverses and iterations of generation and decay of universes or multiverses there may be....? — Cabbage Farmer

Well, yes, because by definition "what is in fact the case" is established by the scientific method. You probably mean that there might be large parts of reality forever hidden from any human mind. And that could be the case. Or it could not. But for practical purposes, it seems irrelevant.

Infinity is something else. Somewhere, in the number pi, are all the phrases you have uttered during your life and, moreover, in the same order in which they were uttered. A little further on, there are all the books that disappeared because of the burning of the Library of Alexandria. In another place, there are all the speeches that Demosthenes gave and that he never wrote, but with the letters inverted, as in a mirror. Yes, the conception of what is infinite is too vast for me to grasp well in finite examples.

Not really. Notice that my definition requires the set of definitions and axioms to be established, which could be interpreted as consistent with your requirement for intersubjective agreement among practicing mathematicians. The "deepest results" come about when someone works something out that follows from those definitions and axioms, but either has not been noticed or has not been demonstrated previously. — aletheist

I should be careful. You described the point of view of Peirce, and I criticized that viewpoint. But I don't know enough about Peirce to really comment intelligently. I don't have certainty in my own positions. I'm a formalist when it suits me and a Platonist when it suits me. I don't want to let my rhetoric exceed my actual understanding; so I'll just say that Peirce probably knows a lot more about this than I do. And I haven't ever given any thought to mathematical existence. Mathematical existence is perfectly obvious to me from the perspective of math. It wasn't till @Metaphysician Undercover challenged me to define it that I ended up with my current position. But I'm not very wedded to it, nor is getting the right definition important to me. When I do math, mathematical existence is perfectly obvious. The question never comes up.

So rather than try to defend my position, I'll just say that what I've written so far represents the limited extent of my thinking; and that I'm not going to think much about this anymore. It's perfectly obvious to me that $\sqrt 2$ exists; and nothing I say will ever convince @Metaphysician Undercover. I should just quit while I'm behind.

It's a truth about the natural numbers as established by a certain set of definitions and axioms. The latter are the only way we know what anyone means by "natural numbers." — aletheist

Yes, on this you're just wrong. No mathematician thinks that way. Philosophy is not about standing outside a given discipline and telling them they're doing it wrong. Philosophy has to be about explaining what practitioners are actually doing. FLT is a statement about the natural numbers, and everybody knows exactly what they are. You do too; and no invocations of philosophy or nonstandard models will change the fact that you have an intuition of the natural numbers, and that I know you do.

One must account for that in one's philosophy. There, now I'm being a Platonist again.

As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it.
— Daz
I am more in agreement with Daz on this, but would substitute "is" for "exists" since the latter has ontological implications that I wish to avoid. Platonism holds that mathematical objects exist in some ideal realm, while fictionalism holds that all properties of mathematical objects are dependent on what someone thinks about them, just like characters in a novel. I am a mathematical realist, but not a platonist; I hold that mathematical objects are real by virtue of having certain properties regardless of what any individual mind or finite group of minds thinks about them, but they do not exist because they do not react with anything. Fermat's last theorem would be a truth about the natural numbers, as established by a certain set of definitions and axioms, even if Fermat never conceived it and Wiles never proved it. — aletheist

Where does mathematical truth exist? Did it exist before there were humans? Before the solar system? Before the Big Bang? What else lives there? You can't avoid the ontological implications. Mathematical truths are abstract things, so they have abstract existence. But they're based on things outside, like counting pebbles.

I agree with you that FLT was true before Fermat. Mathematical truths are very strange philosophical things. On the one hand they're nothing more than valid inferences from arbitrary axioms; and on the other hand they are obviously eternal truths. Quite the mystery. Beyond my pay grade I think.

I should be careful. — fishfry

No worries, I always appreciate your point of view on philosophical aspects of mathematical matters.

Philosophy is not about standing outside a given discipline and telling them they're doing it wrong. Philosophy has to be about explaining what practitioners are actually doing. — fishfry

I am not telling anyone that they are doing something wrong. I suggest that philosophy is (among other things) about explaining what practitioners are actually doing, regardless of whether they accurately recognize it themselves. I have written extensively about philosophy of engineering, my own discipline, and colleagues find it fascinating because they never otherwise think about it in the way that I explain it; they just do it. For better or worse, most practitioners are not reflective practitioners in that sense.

FLT is a statement about the natural numbers, and everybody knows exactly what they are. — fishfry

Yes, in accordance with a certain set of definitions and axioms. Since we cannot point at a natural number to indicate what it is, all we have is a hypothesis from which we can and do draw necessary conclusions (like FLT).

Mathematical truths are abstract things, so they have abstract existence. — fishfry

In Peircean terms, "abstract existence" is an oxymoron. Some abstractions are real, because they are as they are regardless of what any individual mind or finite group of minds thinks about them; but no abstractions exist, because they do not react with other things in the environment.

The known and the empirically knowable, yes. But beyond that, the meaning of "the universe" gets rather vague and nebulous. — Echarmion

Your claim "If the nature of the universe is established via the scientific method, whatever is the result must be finite", seems fair enough if it's a claim about the finitude of the current results of scientific method at any point in history, a claim about our knowledge.

If, by contrast, you mean that "the nature of the universe" is itself identical to the results of scientific method at any point in history, then your claim is indeed the product of vague and nebulous confusions. It seems likely to me that this claim is closer to what you intended, so I'll proceed here on that assumption:

Such a claim would resemble Zelegb's claim to have provided "proof that there is no infinity". Both claims purport to aim beyond what is empirically knowable. At most you can claim to show that our knowledge of the world is finite. But you cannot claim to show -- or how would you show? -- that our knowledge of the world gives us a perfectly complete account of the world as it is in fact.

By my account, those claims of yours and Zelegb's amount to speculation beyond the limits of empirical knowledge, and seem motivated by unwarranted conceptions of the relation of knowledge and reality.


By contrast, I have not claimed that the world is infinite. Rather, I say

(i) it seems we cannot know whether the world is finite or infinite in the relevant sense

(ii) surely the fact that our knowledge of the world is finite, or that "the world as we know it" is finite, is no proof that the world itself is finite

(iii) your claims seems to contradict both (i) and (ii).


You have conflated our knowledge of the world with the world itself, and thus engaged in a sort of metaphysical speculation. My claims are claims about the limits of knowledge. Your claims are claims to know the limits of the world.

This topic tends to run into language limitations. — Echarmion

Indeed. It seems to me these limitations are very much at issue here.

Well, yes, because by definition "what is in fact the case" is established by the scientific method. You probably mean that there might be large parts of reality forever hidden from any human mind. And that could be the case. Or it could not. But for practical purposes, it seems irrelevant. — Echarmion

Here again, it seems to me you've let your speech drag your claims and your beliefs beyond the bounds of evidence and reason.

Don't you agree that what is in the fact the case is in fact the case, whether or not we know it? Or do you suppose our knowledge creates reality in every regard?

Our knowledge of what is in fact the case is informed by experience and is made rigorous by scientific method. That does not entail that experience and scientific method establish what is the case and create or determine the whole world.

Again you're arguments seem to conflate the concept of our knowledge of reality with the concept of reality.

Your claim "If the nature of the universe is established via the scientific method, whatever is the result must be finite", seems fair enough if it's a claim about the finitude of the current results of scientific method at any point in history, a claim about our knowledge. — Cabbage Farmer

Yes, our knowledge, and therefore whatever model of reality is based on that knowledge, can only ever be finite. There might be unknowable aspects of reality, but given that they are unknowable, speculation on them is moot.

Such a claim would resemble Zelegb's claim to have provided "proof that there is no infinity". Both claims purport to aim beyond what is empirically knowable. At most you can claim to show that our knowledge of the world is finite. But you cannot claim to show -- or how would you show? -- that our knowledge of the world gives us a perfectly complete account of the world as it is in fact.

By my account, those claims of yours and Zelegb's amount to speculation beyond the limits of empirical knowledge, and seem motivated by unwarranted conceptions of the relation of knowledge and reality. — Cabbage Farmer

My point was exactly that anything that is empirically knowable must be finite. I further contend that "the universe" should refer to something empirical, as a matter of practicality.

By contrast, I have not claimed that the world is infinite. Rather, I say

(i) it seems we cannot know whether the world is finite or infinite in the relevant sense

(ii) surely the fact that our knowledge of the world is finite, or that "the world as we know it" is finite, is no proof that the world itself is finite

(iii) your claims seems to contradict both (i) and (ii). — Cabbage Farmer

I just don't see why whether the "world behind the world" is or is not finite is "relevant".

Don't you agree that what is in the fact the case is in fact the case, whether or not we know it? Or do you suppose our knowledge creates reality in every regard? — Cabbage Farmer

I do not agree with that. I am a constructivist, so yes I do claim that, in a way, our knowledge creates reality. Not necessarily "in every regard" though, since I am not sure what you wish to imply with that.

Our knowledge of what is in fact the case is informed by experience and is made rigorous by scientific method. That does not entail that experience and scientific method establish what is the case and create or determine the whole world. — Cabbage Farmer

They may not create the world, but they nevertheless populate it with all the content. All we can say about the world absent experience is that it exists.

We would never finish writing down the natural numbers. — TheMadFool

Wasn't this enough?

Oh right, it's a Philosophy forum. :wink:

Infinity is something else. Somewhere, in the number pi, are all the phrases you have uttered during your life and, moreover, in the same order in which they were uttered. A little further on, there are all the books that disappeared because of the burning of the Library of Alexandria. In another place, there are all the speeches that Demosthenes gave and that he never wrote, but with the letters inverted, as in a mirror. Yes, the conception of what is infinite is too vast for me to grasp well in finite examples. — Borraz

This property has been conjectured for pi and certain other constants, but it has not been proven. In any case, knowing that a certain sequence is buried somewhere in that infinite stream is not as helpful as it might seem, because on average, the index that points to the beginning of the sequence that you are looking for would be so large that it would contain more information than the sequence itself. Think Borges's The Library of Babel. Anyway, this is indeed fun to think about, and the above mentioned conjecture has kept number theorists busy.

It is not a conjecture. There is proof that between two real numbers there is always another real number. That is valid in base 10 and, in Spanish, in base 27 (just assign a number for each letter). In English, in base 26. But it is not a conjecture. About the theorem cited, see the following:
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

It is indeed true that between two real numbers there is always another real number. The same is true about rational numbers. This property is called dense ordering, and its proof is very simple - much simpler than Cantor's diagonal argument, which proves something else entirely.

However, the hypothesized property of pi to which you were referring - that it contains every finite sequence of digits - does not follow from this elementary property of real numbers. This would actually be a weaker version of absolute normality - the property of containing every finite sequence of digits in every base with "equal frequency" (scare quotes because this is more complicated than it sounds). While it is has been shown that "almost all" numbers are absolutely normal, it is surprisingly difficult to prove this property about a specific number. As far as I know, this has not been proven about any known number, including pi, although experimentally it has been confirmed for its calculated digits.

https://en.wikipedia.org/wiki/Transcendental_number

This would actually be a weaker version of absolute normality - the property of containing every finite sequence of digits in every base with "equal frequency" (scare quotes because this is more complicated than it sounds). — SophistiCat

Interesting. Thanks.

As far as I know, this has not been proven about any known number, — SophistiCat

I believe I've read that Chaitin's Omega is known to be normal. It's not actually a specific number, just a class of numbers; and you can't write down any particular one, since none of them are computable. So you may or may not take this as a counterexample to the statement that no particular real is known to be normal.

https://en.wikipedia.org/wiki/Chaitin%27s_constant

https://cs.stackexchange.com/questions/67695/chaitins-constant-is-normal

Right, I was being sloppy, I must have had in mind computable numbers. Thanks.

↪fishfry Right, I was being sloppy, I must have had in mind computable numbers. Thanks. — SophistiCat

Not at all. You could still be right. At best the Chaitin number shows that there's a normal number among a class of numbers we can define in first-order logic. But we already knew that normal numbers exist. So it would still be fair to say that we don't know a particular one.

Yes, our knowledge, and therefore whatever model of reality is based on that knowledge, can only ever be finite. There might be unknowable aspects of reality, but given that they are unknowable, speculation on them is moot. — Echarmion

Specific speculative claims about what is unknowable are unwarranted conjectures.

It seems to me that claims to know the actual limits of the world -- as distinct from the limits of the "known world" -- are instances of such speculative claims. In my view you have signed on to such a claim, seemingly without realizing that this is what you are doing. I agree such claims are unwarranted, which is why I've been objecting to them here.

By contrast, the claim that it seems we cannot know whether some facts or features of the world are unknowable in principle for creatures like us is arguably not speculative at all. It's not an empirical or metaphysical claim about what the world is like. It's an epistemological claim that seems to follow from any reasonable conception we might assign to terms like "know" and "world".

My point was exactly that anything that is empirically knowable must be finite. — Echarmion

I'm not sure I would agree with this.

The fact that we "know something" or "know about something" -- a dog or a table, for instance -- does not entail that we know everything about it. It's not clear that we ever have "complete knowledge" of a thing we know or know about, or what it might mean to say that we do have complete knowledge of a thing.

Accordingly, I see no reason to object to the claim that partial knowledge of an infinite thing would count as knowledge. So, if the world is infinite in some respect, say in space or time, and our knowledge of it is finite, this would not entail that we don't know the world, but only that our knowledge of the world is partial and incomplete. But in this respect it would resemble our knowledge of many "finite things", like dogs and tables.

I further contend that "the universe" should refer to something empirical, as a matter of practicality. — Echarmion

The claim we began by addressing is a claim to have proved that "there is no infinity". I take it you and I are still considering that claim when we use words like "universe" and "world" in this conversation.

What is the practical value of a conversation about whether "there is infinity"?

On my view, at least part of the practical value is that it directs our thoughts to consider the limits of our knowledge of the world.

Accordingly, I reject your ad hoc definition on practical as well as theoretical grounds.

I just don't see why whether the "world behind the world" is or is not finite is "relevant". — Echarmion

I do not claim there is a "world behind the world". I say, by definition, there is one world; and it seems that world is knowable at least in part, on the basis of appearances.

My reply to your remark about practicality should suffice to indicate my position on the matter of relevance.

I'll add this: If it's true, then it's relevant. It's my aim to practice clear, coherent, and honest speech in philosophical conversation, and to offset our tendency to error, confusion, and insincerity.

It seems to me that philosophical confusion, even in small and abstract matters, may have far-reaching personal and political implications.

I would characterize philosophical discourse as pursuit of a sort of personal and political harmony.

I do not agree with that. I am a constructivist, so yes I do claim that, in a way, our knowledge creates reality. Not necessarily "in every regard" though, since I am not sure what you wish to imply with that. — Echarmion

In what way do you say our knowledge creates reality?

I agree we have a peculiar way of participating in the world as sentient animals and as cultural animals with powerful conceptual capacities. I suppose we can say each sentient animal "creates" its way of participating in the world just by living, and this participation includes a way of perceiving the world and a way of acting in the world.

I see no reason to say that to perceive a world is to "create a world", nor that to perceive a dog is to create a dog. Nor that to understand a state of affairs is to create that state of affairs, nor that to run into a wall is to create that wall. And so on. So far as I can tell, that would be getting carried away with talk of our "creativity".

They may not create the world, but they nevertheless populate it with all the content. All we can say about the world absent experience is that it exists. — Echarmion

Do you mean to say that experience and scientific method "populate the world with all the content" of the world? What does it mean to say this?

Does it mean that when we perceive a dog, our minds somehow "populate the world" with a dog that in fact does not otherwise exist in the world, or with a dog that in fact is not otherwise "contained" in the world?

Here again it seems you may be conflating a conception of the world as it is in fact with a conception of our knowledge of the world.

Do you suppose the dog is not "in the world" and "does not exist", unless we know it?

Specific speculative claims about what is unknowable are unwarranted conjectures.

It seems to me that claims to know the actual limits of the world -- as distinct from the limits of the "known world" -- are instances of such speculative claims. In my view you have signed on to such a claim, seemingly without realizing that this is what you are doing. I agree such claims are unwarranted, which is why I've been objecting to them here

By contrast, the claim that it seems we cannot know whether some facts or features of the world are unknowable in principle for creatures like us is arguably not speculative at all. It's not an empirical or metaphysical claim about what the world is like. It's an epistemological claim that seems to follow from any reasonable conception we might assign to terms like "know" and "world". — Cabbage Farmer

I think I agree with this. I just don't think claiming that infinity cannot exist in the "world" is a specific claim in that sense. It's a general statement about the concept of infinity.

I'm not sure I would agree with this.

The fact that we "know something" or "know about something" -- a dog or a table, for instance -- does not entail that we know everything about it. It's not clear that we ever have "complete knowledge" of a thing we know or know about, or what it might mean to say that we do have complete knowledge of a thing.

Accordingly, I see no reason to object to the claim that partial knowledge of an infinite thing would count as knowledge. So, if the world is infinite in some respect, say in space or time, and our knowledge of it is finite, this would not entail that we don't know the world, but only that our knowledge of the world is partial and incomplete. But in this respect it would resemble our knowledge of many "finite things", like dogs and tables. — Cabbage Farmer

But even if "the world" is infinite in some respect, it is impossible to ever observe it in that infinity, since observing the infinite is, by definition, never complete. Even if we observed a seemingly endless succession, all we can ever say based on that is that the succession is indefinite.

So if we extend the meaning of "the world" to things that are, in principle, impossible to observe, we are doing unwarranted metaphysical speculation.

The claim we began by addressing is a claim to have proved that "there is no infinity". I take it you and I are still considering that claim when we use words like "universe" and "world" in this conversation. — Cabbage Farmer

The problem is that the claim "there is no infinity" isn't specific enough. Normally, when we talk about what is and isn't we're talking about physical reality, not theoretically possible worlds. We can discuss possible worlds, but I am not sure the conclusions would be very interesting.

What is the practical value of a conversation about whether "there is infinity"?

On my view, at least part of the practical value is that it directs our thoughts to consider the limits of our knowledge of the world.

Accordingly, I reject your ad hoc definition on practical as well as theoretical grounds. — Cabbage Farmer

Right. But you do see that my argument is essentially an epistemological one, so I does fit your definition of practical value.

I do not claim there is a "world behind the world". I say, by definition, there is one world; and it seems that world is knowable at least in part, on the basis of appearances.

My reply to your remark about practicality should suffice to indicate my position on the matter of relevance. — Cabbage Farmer

So do you agree infinity belongs to the part of the world that's not knowable?

I'll add this: If it's true, then it's relevant. It's my aim to practice clear, coherent, and honest speech in philosophical conversation, and to offset our tendency to error, confusion, and insincerity.

It seems to me that philosophical confusion, even in small and abstract matters, may have far-reaching personal and political implications.

I would characterize philosophical discourse as pursuit of a sort of personal and political harmony. — Cabbage Farmer

An interesting perspective. Of course, what "true" even means is itself a matter of philosophical discourse.

In what way do you say our knowledge creates reality?

I agree we have a peculiar way of participating in the world as sentient animals and as cultural animals with powerful conceptual capacities. I suppose we can say each sentient animal "creates" its way of participating in the world just by living, and this participation includes a way of perceiving the world and a way of acting in the world.

I see no reason to say that to perceive a world is to "create a world", nor that to perceive a dog is to create a dog. Nor that to understand a state of affairs is to create that state of affairs, nor that to run into a wall is to create that wall. And so on. So far as I can tell, that would be getting carried away with talk of our "creativity". — Cabbage Farmer

Well I think that there are neither dogs nor walls outside our heads. Those are categories put together by human minds on the basis of certain patterns in the information that these minds receive. There is some source of that information, but it's the mind that constructs the patterns. The patterns are not the information though, like the pattern on a sweater is not the fabric.

Do you mean to say that experience and scientific method "populate the world with all the content" of the world? What does it mean to say this?

Does it mean that when we perceive a dog, our minds somehow "populate the world" with a dog that in fact does not otherwise exist in the world, or with a dog that in fact is not otherwise "contained" in the world?

Here again it seems you may be conflating a conception of the world as it is in fact with a conception of our knowledge of the world.

Do you suppose the dog is not "in the world" and "does not exist", unless we know it? — Cabbage Farmer

If we abstract from all experience, talk of "the world" is meaningless. Fundamentally, the world is the collection of our knowledge about the world. We can reason, based on experience, that there are things we don't know about the world, and from that we get the idea of a larger world of which we know only some part. So the "conception of the world" is actually secondary and based on experience.

So if you lived on some undiscovered island and had never seen or heard of a dog, and neither did anyone you had ever seen or heard of, you might still know, based on experience, that you sometimes discover new animals. And hence you might say that "unknown animals exist". But would you say that "dogs exist"? You wouldn't, because the term "dog" is meaningless to you. Someone would have to explain a dog based on things you have already experienced for the claim to make any sense.