Does infinity exist?

[wtf]

You know everything; therefore what I say is inconsequential.

On the contrary. I know nothing. You don't realize that about yourself; and thereby render your own opinions inconsequential.

I have said about all I can say here. @Serendipper, Thank you for an interesting and stimulating chat.

[barbarianhorde]

So infinity exists as a concept. Like we can all think the thought of infinite steps between 1 and 0, Serendipper showed he can do it too. But not as something you can hold in your hands.

I think Serendipper conceded infinity exists when he couldn't give the exact maximum number between 1 and 0 except as infinite.

That doesn't mean we can hold infinitely many apples in our hand.

But it does mean, infinity can be held in the mind.
Only to a mindless person that is nonexistent.

[Serendipper]

So infinity exists as a concept. Like we can all think the thought of infinite steps between 1 and 0, Serendipper showed he can do it too. But not as something you can hold in your hands.

I think Serendipper conceded infinity exists when he couldn't give the exact maximum number between 1 and 0 except as infinite.

That doesn't mean we can hold infinitely many apples in our hand.

But it does mean, infinity can be held in the mind.
Only to a mindless person that is nonexistent.

Well, like Gauss said, "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction."

Infinity can never be anything that can exist like forever can never arrive for when the boundless finally has bounds (infinity is completed, forever has arrived), then it's no longer boundless.

That is illustrated by the infinite series 1-1+1-1+1-1+1-1+1-1+.... which is either 1 or 0 if we stop to check, but continued forever and it's assumed to be 1/2 (equidistant between states).

The maximum number of conceptual steps between 1 and 0 is limited by the size of the universe or medium used for writing the number down (or remembering it). To determine a hard ceiling, find the volume of the observable universe in terms of the number of planck cubes. (Wildberger's point is that (((((((((10^10)^10)^10)^10)^10)^10)^10)^10)^10)+23 cannot be factored because the universe isn't big enough.)

The maximum number of divisions between 1 and 0 on a ruler would be the number of quarks contained, or if I'm wrong about that, then there will be some other point where locality loses meaning and there will be no way to distinguish one point from another because there will be no "thing" at that resolution to act as a reference.

Either way, conceptually or realistically, there is a limit which makes the divisions finite.

[barbarianhorde]

So infinity exists as a concept. Like we can all think the thought of infinite steps between 1 and 0, Serendipper showed he can do it too. But not as something you can hold in your hands.

I think Serendipper conceded infinity exists when he couldn't give the exact maximum number between 1 and 0 except as infinite.

That doesn't mean we can hold infinitely many apples in our hand.

But it does mean, infinity can be held in the mind.
Only to a mindless person that is nonexistent.

Well, like Gauss said, "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction."

Phaer enough. So lets put this in terms: there are infinite possible numbers between 1 and 0, but not actually an existing infinity of them.

Infinity can never be anything that can exist like forever can never arrive for when the boundless finally has bounds (infinity is completed, forever has arrived), then it's no longer boundless.

That is illustrated by the infinite series 1-1+1-1+1-1+1-1+1-1+.... which is either 1 or 0 if we stop to check, but continued forever and it's assumed to be 1/2 (equidistant between states).

The maximum number of conceptual steps between 1 and 0 is limited by the size of the universe or medium used for writing the number down (or remembering it). To determine a hard ceiling, find the volume of the observable universe in terms of the number of planck cubes. (Wildberger's point is that (((((((((10^10)^10)^10)^10)^10)^10)^10)^10)^10)+23 cannot be factored because the universe isn't big enough.)

I agree that if a number exists it should be possible to arrive at it.
But we can argue that to arrive at infinity you just have not divide by zero.
I know it is brutish but it kinda works.
Still I won't say this overthrows the point you're making, because 0 doesn't "exist" like 1 does of course.

The maximum number of divisions between 1 and 0 on a ruler would be the number of quarks contained, or if I'm wrong about that, then there will be some other point where locality loses meaning and there will be no way to distinguish one point from another because there will be no "thing" at that resolution to act as a reference.

Either way, conceptually or realistically, there is a limit which makes the divisions finite.

Of physical matter surely you are 100% right. There are no physical infinitesimals, there is a minimal thinkable and thinkably also a minimal existable quantity of quality. But there may be infinite amounts of qualities. This is how you really arrive at infinity in an additive sense, if you look at the ways in which things are intertwined with each other, how many of them are there, because as soon as you understand something about them you intertwine with the web of intertwined ways of relating and then you see true infinity. But then you also see that infinity is what the world must become when it is held up by one perspective. So there can't be an infinite number, but there can be a number of infinities.

[MagsJ]

..though it can be imagined, infinity can never be reached/achieved, so it doesn't ever get to exist.. poor infinity, huh.

Yes that sums up the situation I think.

Perhaps the fact that it can be imagined is all that infinity needs to exist, or does a concept need to manifest into reality in order to.

Even the finite can be infinite..

[wtf]

Perhaps the fact that it can be imagined is all that infinity needs to exist ...

Question: Do purple flying elephants exist? I can imagin e them.

Bonus question: Is Ahab captain of the Pequod?

Second bonus question: If George Washington was the first president of the US, and Ahab was the captain of the Pequod, are those statements equally true? True in the same way? Both Washington and Ahab have an equal claim to existence? Along with the purple flying elephants?

I hope you can see that you need to greatly qualify your remark about imaginability being sufficient for existence.

[wtf]

Phaer enough. So lets put this in terms: there are infinite possible numbers between 1 and 0, but not actually an existing infinity of them.

Hard to know how you can allow one possibility without the other. First, of course we are not talking about the physical world. Numbers are abstract objects. So how can you allow a "potential" infinity of counting numbers 1, 2, 3, 4, ... but then deny the actual infinity of the set {1, 2, 3, 4, ...}?

In terms of set theory, we make the leap from 1, 2, 3, ... to {1, 2, 3, ...} via the axiom of infinity, which says that an infinite set exists. It's much more useful to allow infinite sets so we generally accept the axiom of infinity.

But if you deny the mathematical existence of the set of natural numbers, yet allow for infinitely many natural numbers, that seems like a philosophical quibble that you would need to justify.

And again, note that NONE of this has anything to do with the physical world. Numbers are abstract. By what philosophical principle can you say that infinitely many numbers 1, 2, 3, ... exist, but that I'm not allowed to put set brackets around the list?

[barbarianhorde]

Phaer enough. So lets put this in terms: there are infinite possible numbers between 1 and 0, but not actually an existing infinity of them.

Hard to know how you can allow one possibility without the other. First, of course we are not talking about the physical world. Numbers are abstract objects. So how can you allow a "potential" infinity of counting numbers 1, 2, 3, 4, ... but then deny the actual infinity of the set {1, 2, 3, 4, ...}?

It is not a mathematical quibble but a philosophical one which means it could end up telling mathematics what to do, after all philosophy still trumps mathematics by being the arbiter of what constitutes true correspondence. So what is actually in play here is what a "set" means, not what infinity means.

In terms of set theory, we make the leap from 1, 2, 3, ... to {1, 2, 3, ...} via the axiom of infinity, which says that an infinite set exists. It's much more useful to allow infinite sets so we generally accept the axiom of infinity.

I can now see how Serendipper considers this as strictly speaking an unwarranted shortcut. But I don't contest that it is useful. Shortcuts are very often useful, look at the Panama Canal.

But if you deny the mathematical existence of the set of natural numbers, yet allow for infinitely many natural numbers, that seems like a philosophical quibble that you would need to justify.

I personally don't deny the mathematical set. I just deny the philosophical set. I mean I deny that this set of infinitely many numbers has any meaning outside of how the set is being made useful.

And again, note that NONE of this has anything to do with the physical world. Numbers are abstract.

I know that. So the question is how far we want to allow mathematics to operate in defiance of physics.

Because philosophy is about reliability and not about speed. The power gained from seeing sets as potentially having infinite size may come with a drawback of making it doubtful if sets can be trusted, if they can still logically correspond to another set.

I guess what I mean is all questions like, how does the set of integers correspond to the set of real numbers? Does the fact that the second is infinity squared make the former into the root of infinity? If this can't be addressed there is a logical problem with the infinite set, even if it can still do mathematical work, creating a subprime mathematics bubble.

By what philosophical principle can you say that infinitely many numbers 1, 2, 3, ... exist, but that I'm not allowed to put set brackets around the list?

You're allowed, but all you can do is use that for a specific mathematical operation. That you put brackets around it to mean it goes on infinitely does not mean it actually goes on infinitely. What do I mean by actually, if not physically?

I mean in the sense that it identifies infinity. Thats where philosophy begins to not be bored.

[wtf]

It is not a mathematical quibble but a philosophical one which means it could end up telling mathematics what to do, after all philosophy still trumps mathematics by being the arbiter of what constitutes true correspondence.

I don't necessarily agree, but I will stipulate for sake of discussion that philosophy trumps math. BUT philosophy has perfectly well accepted infinitary math. Wasn't Russell a philosopher? The philosopher of math Penelope Maddy has written Believing the Axioms (parts I and II. I linked to part I). She walks through each axiom of ZFC (Zermelo-Fraenkel set theory with Choice), places it in historical and philosophical context, and describes the principles by which we accept it today.

So it is NOT true that philosophy says one thing and math another. Some philosophers argue for finitary math, I suppose, just as some mathematicians study finitary math. It's interesting to study! But mainstream math accepts infinity and the mainstream philosophers of math do too.

So I don't agree with your conclusion even if I accept your premise. There's no dispute between philosophy and math when it comes to the mathematical infinite.

So what is actually in play here is what a "set" means, not what infinity means.

Yes that is very true and also insightful. In elementary education we tell people that a set is a "collection of objects." But of course that was Frege's idea, demolished by Russell. A set is, in fact, a highly technical gadget with no definition at all. A set, in mathematics, is any object that obeys the axioms of set theory. And what axioms are those? Any axioms you like, subject only to consistency and interestingness. And we don't even know for sure if our axiom systems ARE consistent.

If anyone wants to argue that mathematics is based on a pile of sand, you will get no argument from me. It's the job of philosophy to explain why all this obvious nonsense is so damn useful.

I can now see how Serendipper considers this as strictly speaking an unwarranted shortcut. But I don't contest that it is useful. Shortcuts are very often useful, look at the Panama Canal.

Great example! And I don't disagree with Serendipper on this point either. There is no logical or moral reason why we should prefer one assumption over another, when it comes to allowing the actual infinite into math. All we have is a centuries of experience that when it comes to understanding the physical world, mathematics is indispensable. SEP has an article on this indispensability argument.

I personally don't deny the mathematical set. I just deny the philosophical set.

Ok. But will you stipulate at least that most philosophers accept modern math? The axiom of infinity dates back to Frege and Russell and Zermelo and all those other ancients.

I mean I deny that this set of infinitely many numbers has any meaning outside of how the set is being made useful.

Ah. Ok. Meaning outside of utility. Well, tell me this. A Martian physicist comes to earth and sees a traffic light. She can tell red light from green light by the wavelength. But she cannot tell you which is stop and which is go. That's a socially constructed fact that has meaning only because it's useful. We could make green mean stop and red mean go, and that would be just as valid a choice. There's no inherent meaning in the colors.

So I would say that it's true there's no "meaning" to set theory outside of how we use it. But so what? Most of reality is that way. Civilization is one abstraction piled on another. None of it has any meaning outside of how we as humans use it. Your criticism of set theory is a criticism of the foundation of civilization: namely, the human power of abstraction. Our ability to make the abstract real.

I know that. So the question is how far we want to allow mathematics to operate in defiance of physics.

But it's not. A lot of math comes directly from physics. Physics finds modern infinitary math indispensable. Even though the universe might be discrete, the math used by the physicists is infinitary. That may be a puzzle; but it is also a FACT.

Your beef is with the physicists, not the mathematicians! The mathematicians invented this crazy non-Euclidean geometry, but it was the physicists who decided it was the best way to understand the world. I hope you see my point!

Because philosophy is about reliability and not about speed. The power gained from seeing sets as potentially having infinite size may come with a drawback of making it doubtful if sets can be trusted, if they can still logically correspond to another set.

Well maybe set theory can't be trusted. Make your case. What does that mean? What if it can't?

I guess what I mean is all questions like, how does the set of integers correspond to the set of real numbers?

The integers are a proper subset of the reals. Additionally, the reals can be set-theoretically constructed from the integers. That is, if all we had was the integers, we would first create the rationals as certain equivalence classes of integers; then we'd create the reals as certain subsets of the rationals.

Does the fact that the second is infinity squared make the former into the root of infinity?

I would not say the reals are infinity squared and in fact that's wrong. What is true is that the cardinality of the reals is the same as the cardinality of the set of subsets of the integers. Is that what you meant?

If this can't be addressed there is a logical problem with the infinite set, even if it can still do mathematical work, creating a subprime mathematics bubble.

I hope I addressed it. There is no sense in which "the reals are infinity squared" is meaningful. The reals are in fact essentially the same set as the collection of subsets of the integers. You can encode each as the other.

I mean in the sense that it identifies infinity. Thats where philosophy begins to not be bored.

All the philosophers of math I know accept mathematical infinity. I must be reading the wrong philosophers. What does "believe in" mean? Just that we accept it for being useful; and we rely on experience that the history of math is the history of weird stuff that someone realized was actually useful. Negative numbers, complex numbers, irrational numbers, non-Euclidean geometry. So ... what is the meaning of the mathematics of infinity? Perhaps we'll know in a hundred years.

[barbarianhorde]

I don't necessarily agree, but I will stipulate for sake of discussion that philosophy trumps math. BUT philosophy has perfectly well accepted infinitary math. Wasn't Russell a philosopher?

Yes but his paradox (of the set which includes every set which isn't included in itself) caused him to find set theory debunked and invent type theory, didn't it?
So philosophy through Russell doesn't redeem the infinite set. Im not aware of philosophers that do redeem it...and to be honest im not so sure physics rely on it except string theory?

I do understand of course that much if not most computer programming requires the tool of the infinite set.
But this doesn't mean lets say that we can hypothesize an infinitely large bus which could compute an actual numerical infinity. Type theory is a bit more of an engineers thing than set theory.

I need to verify some stuff to get back to you on some of the other points.

[wtf]

Yes but his paradox (of the set which includes every set which isn't included in itself) caused him to find set theory debunked and invent type theory, didn't it?
So philosophy through Russell doesn't redeem the infinite set.

I'm sure Russell believed in infinite sets but I am not a Russell scholar. Do you know which philosophers deny mathematical infinity? Is this point important?

Im not aware of philosophers that do redeem it...

Penelope Maddy would be one such. But I'm no expert on what philosophers think.

and to be honest im not so sure physics rely on it except string theory?

Quantum physics uses Hilbert space, a part of functional analysis, which is full of infinitary mathematics. Likewise relativity relying on differential geometry. You can't do any modern physics without infinitary reasoning. For that matter, calculus (eventually formalized via infinitary math) was invented by Newton to describe gravity.

I do understand of course that much if not most computer programming requires the tool of the infinite set.

Computers are finite and do not require infinitary mathematics.

But this doesn't mean lets say that we can hypothesize an infinitely large bus which could compute an actual numerical infinity. Type theory is a bit more of an engineers thing than set theory.

Not sure where you're going with this.

[barbarianhorde]

Did you look at type theory at all? Unlike set theory it integrates with Peano. So it doesn't ignore the machinations of the logic of integers. That's a first step to draw this anywhere near philosophy.

The infinity you mean is just 1/0.
That's not actually an infinite number of values but a functional limit, a horizon.
This is not a nominal infinity, it is just the reification of receding.

A straight line is infinite in length and in infinity it forms a circle. But to say that this circle exists is not philosophically valid.

[Serendipper]

if you look at the ways in which things are intertwined with each other, how many of them are there, because as soon as you understand something about them you intertwine with the web of intertwined ways of relating and then you see true infinity. But then you also see that infinity is what the world must become when it is held up by one perspective.

If you're saying what I think you're saying (bolded part), then you're hitting the nail on the head for the point I'm ultimately driving at which is there is no way to make an observation without affecting the thing being observed and, per Goethe, observation includes deduction. So, the fundamental of whatever we behold invariably will be perceived as infinite due to the infinite regression involved by affecting the thing being beheld, but that doesn't mean there is an existent infinity, but it simply means the subject and object are the same thing. So I would consider infinity to be proof of unity (the camera observing its own monitor) since the only alternative is to concede infinity exists as a completed incompletion, which is too nonsensical to get my head around and if we open the door to nonsense, how will we know where to draw the line.

When I try to picture an infinite plane in my mind, it can only curve back on itself because I'm trying to grasp the full extents of it as one thing and when I do that, it turns concave until it eventually joins with where my mind is calling the center. An infinite plane that extends forever without end isn't something I can imagine. I can fool myself into believing I can, but I'm lying if I claim such ability; the best I can do is make the edges fuzzy and call that infinite (that's cheating). But if I REALLY make a plane that doesn't end, then there is no other place to go than where it started. The only way to exist without also having beginning nor end is to be a loop (and why the wedding ring is a symbol of eternity and also a symbol of unity).

I think there is a way we can work with notions of infinity by working with the inevitable ramifications without actually conceptualizing infinity as a thing, but even that doesn't pan-out in practice. For instance the Thompson Lamp where the switch is turned on after 1 min and off after 1/2 min and on after 1/4 min and so on. Eventually the speed of the switch will exceed the speed of light, so whatever state the lamp was in before crossing the velocity threshold (probably on) will be the final state of the lamp since the switch would be moving too fast for electrons to react. So we can say in our heads that the final state of the lamp should be half-on and half-off, but it can't work in practice due to our finite universe. Even infinite velocity doesn't make sense since speed can't be faster than instant, which is c. Otherwise things could arrive before they left, and not only that, but arrive infinitely sooner than they left (whatever that means).

Conceptualizations of infinity are like that dream I had as a kid where a cat had its head in its own mouth (back in the days that I didn't realize bicycle spokes held the bike up and trees didn't make the wind blow); we probably can't admit to ourselves infinity is absurd because at such an old age, we shouldn't be that silly anymore, so it's denial. Compounding that, people really want infinity to exist since it's a good substitute for god and answers so many questions. The incentive to cheat (not be scientifically objective/unbiased) is high. And there is no proof or even good reason to believe an unbounded thing can be beheld by either our hands or our minds or even be said to exist, much like zero, which is also the bounded unbounded thing: limitless nothingness in a tidy package. Obviously we can think about "nothing" as the absence of something, but that's not nothing. I don't think anyone can truly think about nothing because there's nothing there to focus on, conceptualize, and observe.

[Serendipper]

All the philosophers of math I know accept mathematical infinity.

What do you consider to be a philosopher of math? Is it someone who first accepts infinity then is bestowed with the title of philosopher of math while anyone who rejects it is labeled a crank? I'd be willing to bet. Much like the ones accepting the hiv causing aids hypothesis get the funding while deniers are lost to obscurity; or the sat fat / cholesterol deniers get ostracized; or climate change deniers. Science is hardly more than fashion: you either support what they tell you to support or they'll pull your funding and call you names.

All The Great Geniuses in Art, Music, Philosophy, Science and Literature Believed in God https://www.amazon.com/Greatest-Minds-B ... B00K598JF4

Even modern-day geniuses believe in god. Chris Langan (IQ 200) is hard at work trying to prove it. Show me a famous genius who isn't a theist. There might be one.

So does that prove god exists?

[Serendipper]

Computers are finite and do not require infinitary mathematics.

They do to escape loops. Actually, that's the only practical use for set theory that I could find.

https://mathoverflow.net/questions/1033 ... heory-have

When such programs involve loops and recursive calls (self-reference), we need methods for showing that the loops and recursive calls terminate, i.e., that the program won't run forever. The usual induction principle for natural numbers suffices for showing that a single loop terminates, but we need double induction for double loops, triple induction for triple loops, etc. The whole business can get very complicated when the program is more than just a simple combination of loops. Set theory helps sort it all out with the principle of transfinite induction and the calculus of (infinite) ordinal numbers. Transfinite induction covers all possible ways in which one could show that a program terminates, while the ordinal numbers are used to express how complex the proof of termination is (the bigger the number, the more complicated it is to see that the program will actually terminate).

[wtf]

Computers are finite and do not require infinitary mathematics.

They do to escape loops. Actually, that's the only practical use for set theory that I could find.

Computers are not the same as computer science. Computers are finite and have no use for set theory or infinitary math. Computer science, on the other hand, does employ infinitary math, as for example the little-o and big-O asymptotic notation.

[Serendipper]

..though it can be imagined, infinity can never be reached/achieved, so it doesn't ever get to exist.. poor infinity, huh.

Yes that sums up the situation I think.

Perhaps the fact that it can be imagined is all that infinity needs to exist, or does a concept need to manifest into reality in order to.

Even the finite can be infinite..

I can't imagine the infinite other than being a loop, but if I could I'd say that it exists in my imagination.

[Serendipper]

Computers are finite and do not require infinitary mathematics.

They do to escape loops. Actually, that's the only practical use for set theory that I could find.

Computers are not the same as computer science. Computers are finite and have no use for set theory or infinitary math. Computer science, on the other hand, does employ infinitary math, as for example the little-o and big-O asymptotic notation.

True but computers couldn't compute without the computer science. I don't know much about it and I'm just appealing to the authority of stackexchange.

[wtf]

True but computers couldn't compute without the computer science. I don't know much about it and I'm just appealing to the authority of stackexchange.

Computers are physical machines, subject to the constraints of physics and electrical engineering. If you run an "infinite" loop on a real computer, the loop will eventually terminate. Given enough time the nearby power plant will stop producing the electricity that the computer runs on. The computer's chips and circuits will degrade and fail. Electrical connections will corrode. The sun will explode and destroy the solar system. The universe may collapse into a black hole or expand into heat death. It's certain that any physical system will eventually stop. No math is needed. Only physics and engineering.

Computer science, on the other hand, is the abstract study of computation. Computations in CS are not subject to limits of time, space, or energy. If you have a loop in a theoretical computation, you do need some higher abstract math to determine if it will go on forever or not. That's because it has all the time, space, and energy that it needs; far more of these things than our physical universe can contain.

Computers are physical machines, subject to physical constraints. Theoretical computations in CS are abstract, not subject to physical limitations.

Of course CS does help us analyze real-world programs. But computing and computer science are literally two separate disciplines. That's why CS majors can get a degree yet have no idea how to program. Programming is not part of CS. Programming is part of software engineering. Related but separate.

[Serendipper]

True but computers couldn't compute without the computer science. I don't know much about it and I'm just appealing to the authority of stackexchange.

Computers are physical machines, subject to the constraints of physics and electrical engineering. If you run an "infinite" loop on a real computer, the loop will eventually terminate. Given enough time the nearby power plant will stop producing the electricity that the computer runs on. The computer's chips and circuits will degrade and fail. Electrical connections will corrode. The sun will explode and destroy the solar system. The universe may collapse into a black hole or expand into heat death. It's certain that any physical system will eventually stop. No math is needed. Only physics and engineering.

Computer science, on the other hand, is the abstract study of computation. Computations in CS are not subject to limits of time, space, or energy. If you have a loop in a theoretical computation, you do need some higher abstract math to determine if it will go on forever or not. That's because it has all the time, space, and energy that it needs; far more of these things than our physical universe can contain.

Computers are physical machines, subject to physical constraints. Theoretical computations in CS are abstract, not subject to physical limitations.

Of course CS does help us analyze real-world programs. But computing and computer science are literally two separate disciplines. That's why CS majors can get a degree yet have no idea how to program. Programming is not part of CS. Programming is part of software engineering. Related but separate.

I agree. Well said! The only question I have is if the computer cannot complete an infinite loop, then how does the math predicated upon infinite sets help solve the problem?

[wtf]

The only question I have is if the computer cannot complete an infinite loop, then how does the math predicated upon infinite sets help solve the problem?

I'm not a computer scientist and I'm not familiar with those proofs. The point is that real computers have constrained resources and abstract computations have unlimited resources. So you need abstract mathematical tools to analyze theoretical computations; whereas any physical device must eventually run down and fail simply due to mechanical causes. I don't know the specifics.

[wtf]

Did you look at type theory at all?

Yes. It had some adherents in the 1930's then lost out on mindshare to set theory. It's making a comeback via approaches like homotopy type theory, and on the philosophical side, neo-intuitionism.

Yes I'm aware of type theory. Maybe I don't get where you're coming from. Of course there are alternatives to set theory. The fact that foundations are in a constant state of turmoil, and that there are interesting alternatives, doesn't invalidate set theory, nor disturb its place as the dominant paradigm up to the first couple of years of grad school. In areas like abstract algebra, differential geometry, and algebraic geometry, set theory has already been supplanted by category theory.

Alternative approaches are perfectly fine. I get the feeling that you think we are having a difference of opinion over whether type theory exists or whether set theory may or may not be supplanted by a better foundation. I agree to all of it.

Can you just explain to me where you're coming from? The point is that if you, or anyone, out of a sense of what, contrariness or whatever, wants to argue that set theory is some kind of evil plot designed to suppress the Truth, well I just can't hold up my end. Set theory is a tool. Math is a big toolbox with many tools.

Unlike set theory it integrates with Peano. So it doesn't ignore the machinations of the logic of integers. That's a first step to draw this anywhere near philosophy.

No I disagree with you. There is intuitionist philosophy that corresponds to type-theoretic math. And there's also philosophy that corresponds to full infinitary math. I do not believe you are correct that all the philosophers are intuitionists or finitists. That's simply not the case.

The infinity you mean is just 1/0.

That expression is undefined. When I speak of mathematical infinity I speak of no such thing. A set is infinite if it may be bijected with a proper subset of itself. That's the working definition.

That's not actually an infinite number of values but a functional limit, a horizon.
This is not a nominal infinity, it is just the reification of receding.

Nonsense. The reification of reciding? What the hell does that mean?

A straight line is infinite in length and in infinity it forms a circle. But to say that this circle exists is not philosophically valid.

I don't care if circles don't exist. Mathematical circles do, and they're interesting and useful. And by Putnam's indispensability argument (linked earlier), circles are entitled to abstract existence by virtue of the fact that they are indispensable in understanding the world.

Take it up with Putnam and Quine, not me.

ps Here is the SEP link. These are high fallutin' philosophers making the case that because abstract mathematics turns out to be indispensible -- as I've been putting it, interesting and useful, but indispensability seems to be a higher standard -- that abstract mathematical objects have a claim on existence.

https://plato.stanford.edu/entries/mathphil-indis/

[barbarianhorde]

if you look at the ways in which things are intertwined with each other, how many of them are there, because as soon as you understand something about them you intertwine with the web of intertwined ways of relating and then you see true infinity. But then you also see that infinity is what the world must become when it is held up by one perspective.

If you're saying what I think you're saying (bolded part), then you're hitting the nail on the head for the point I'm ultimately driving at which is there is no way to make an observation without affecting the thing being observed and, per Goethe, observation includes deduction. So, the fundamental of whatever we behold invariably will be perceived as infinite due to the infinite regression involved by affecting the thing being beheld, but that doesn't mean there is an existent infinity, but it simply means the subject and object are the same thing. So I would consider infinity to be proof of unity (the camera observing its own monitor) since the only alternative is to concede infinity exists as a completed incompletion, which is too nonsensical to get my head around and if we open the door to nonsense, how will we know where to draw the line.

Yes, thats what I meant but damn thats nicely phrased. So an object is infinitely grounded in itself. A set that describes a function such as a rational number sequence ids infinite in its potential reflection of itself on itself, where each reflection produces another integer, but there is not any infinitude of integers given unless that set is taken as the vessel. So the infinity opt the set is always infinity+1, the infinity of the number brings along the notion of the set. Which already shows its is not really infinite in capacity.

Damn, I dont now everyone can follow my thought here. Capacity as different from potential... well, like the capacity of a hose tied to an opened hydrant, and the potential of the closed hydrant and the rolled up hose.

Yes, a number only acquires a capacity to mean anything under certain circumstances, such as existence.
lol.

Pi is one of these numbers.
An irrational number. That speaks volumes. Its infinity is not a neat row or axial system, but more like snow on a tv. It is a better infinity if you want to come close to existence.

When I try to picture an infinite plane in my mind, it can only curve back on itself because I'm trying to grasp the full extents of it as one thing and when I do that, it turns concave until it eventually joins with where my mind is calling the center. An infinite plane that extends forever without end isn't something I can imagine. I can fool myself into believing I can, but I'm lying if I claim such ability; the best I can do is make the edges fuzzy and call that infinite (that's cheating). But if I REALLY make a plane that doesn't end, then there is no other place to go than where it started. The only way to exist without also having beginning nor end is to be a loop (and why the wedding ring is a symbol of eternity and also a symbol of unity).

I think there is a way we can work with notions of infinity by working with the inevitable ramifications without actually conceptualizing infinity as a thing, but even that doesn't pan-out in practice. For instance the Thompson Lamp where the switch is turned on after 1 min and off after 1/2 min and on after 1/4 min and so on. Eventually the speed of the switch will exceed the speed of light, so whatever state the lamp was in before crossing the velocity threshold (probably on) will be the final state of the lamp since the switch would be moving too fast for electrons to react. So we can say in our heads that the final state of the lamp should be half-on and half-off, but it can't work in practice due to our finite universe. Even infinite velocity doesn't make sense since speed can't be faster than instant, which is c. Otherwise things could arrive before they left, and not only that, but arrive infinitely sooner than they left (whatever that means).

I think this is attained in the mirror loop metaphor for the set, where the set is a thing which is infinite inside but has no infinite capacity to change things, which would be infinite existence, which would mean infinite divisibility of meaning.

Conceptualizations of infinity are like that dream I had as a kid where a cat had its head in its own mouth (back in the days that I didn't realize bicycle spokes held the bike up and trees didn't make the wind blow); we probably can't admit to ourselves infinity is absurd because at such an old age, we shouldn't be that silly anymore, so it's denial. Compounding that, people really want infinity to exist since it's a good substitute for god and answers so many questions. The incentive to cheat (not be scientifically objective/unbiased) is high. And there is no proof or even good reason to believe an unbounded thing can be beheld by either our hands or our minds or even be said to exist, much like zero, which is also the bounded unbounded thing: limitless nothingness in a tidy package. Obviously we can think about "nothing" as the absence of something, but that's not nothing. I don't think anyone can truly think about nothing because there's nothing there to focus on, conceptualize, and observe.

Im compelled by this image of the trees that cause the wind I must say. Thats pretty damn cool. Yes, as kids we clearly have a lot more touch with the contradictions that are thrown at us, the way things are set against each other.

Out of the blue, it reminds me of my first solar eclipse when I was just a 4 year old kid walking home with my friend from getting some candy, I don't know why I was allowed, it was the 80s, and it suddenly got dark. It was a partial eclipse and no one was paying attention (it was the 80s) but for a moment I had the distinct sensation of "well that was it folks!". Later if you know what an eclipse is, it loses most of its capacity. Unless you're not in a horde of morons (humans) but in a field where suddenly every being is holding its breath, and you realize what you thought was silence was actually deafening noise.

wtf - I can now begin to undermine your claim that all infinity is just basically the same thing. All I need to do for this is follow the threads. A philosopher is a detective.
Let us start in the context of bijection on good ol wikipaedia....

"Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers."

But I can add to this to make it stronger. Different infinities are also distinguished in terms of how they can be overlaid. If they can be overlaid, like the set of all even numbers with the set of all natural numbers, then it is the same type of infinity. The same goes for rational numbers, they too can be overlaid with the natural numbers. But irrational numbers can not. They form a different class in which even the distance between 1 and 0 contains more numbers than all of the natural or (thus) rational numbers.

This is orthodox math, so where do you want to go from here?

[wtf]

An irrational number. That speaks volumes. Its infinity is not a neat row or axial system, but more like snow on a tv.

Extremely good insight. An arbitrary real number is essentially a number whose decimal digits are random, in the sense of incompressibility. There is no formula or computer program anyone could ever come up with that can crank out its digits.

By the way note that pi is NOT a random real number in that sense. Its digits are the output of many formulas and algorithms. Pi is a computable real number. Most real numbers are not computable.

wtf - I can now begin to undermine your claim that all infinity is just basically the same thing.

I have said no such thing and I believe no such thing. You have made up a claim out of thin air an attributed it to me.

"Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers."

I'm perfectly well aware of modern set theory and the idea of cardinalities. I'm glad you looked it up and perhaps learned something; and if you have any questions about it, I'd be happy to answer them. I'm fully conversant with the theory.

This is orthodox math, so where do you want to go from here?

Happy to discuss transfinite set theory. One of my favorite topics. In fact the cardinals are what everyone hears about, but the ordinals are even more interesting. Cantor discovered them too.

By the way, a historical note. What was Cantor doing when he discovered transfinite numbers? Did he wake up one day and say, "I think I'm going to revolutionize the foundations of math, piss off my mentor Kronecker, and have a nervous breakdown?"

No in fact that's not what happened. Cantor was engaged in studying trigonometric series, the very series that arose from Fourier's research into heat. In other words Cantor was led to discover transfinite cardinals and ordinals based on problems that arose directly from physical phenomena. Something to think about when contemplating the philosophy of the infinite.

Finally, again reiterating what I said earlier, the statement you attributed to me is NOTHING I said. Nothing at all. You just made it up then pretended to debunk it. I believe in the philosophy biz that's called a strawman argument.

[Serendipper]

A set is infinite if it may be bijected with a proper subset of itself. That's the working definition.

How would you describe an infinite set of oranges using that definition of infinite?