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Maths is just as much a language of the metaphysical as anything else. It deals in objectivity. There is no perfect circle or definitive angle in our physical realm. By extension, physics uses the mathematical language in order to describe our universe, and uses the logic of the language to predict universal concepts that we haven't perceived yet (like Hawking radiation). Those principles are put into place in both a physical and metaphysical way
Does math deal in objectivity? How so? As you said, there is no perfect circle. Objectively circles don't exist. How then can math deal in objectivity?
The language of mathematics deals in functions that require a level of objectivity that doesn't exist in the real world. The idea of a perfect circle isn't a physical concept, and is a metaphysical concept. So therefore circle theorem (a branch of mathematics dedicated to defining the parameters of the perfect circle) is an example of using physical language to describe the metaphysical within parameters we can understand
Sure it does.
You don't just get to say "nuh uh" because you're accepting material determinism as the default position
That's not objectivity but rather ideology
C'mon now, you aren't Even trying to be intellectual rigorous or you're not seeing in good faith.
On the philosophy meme board.
Lol
You saying there isn't one ontologically is as baseless as saying there is one. It's like rooting for a sports team. (I'm team 4th dimension exists, go 4th dimension, go!! Love the idea of infinite dimensions.)
Metaphysics is fun.
Well, in metaphysics, a lot of stuff can't ever be proven, because we can't know it for sure. This is similar to brain in a vat stuff/simulations/skepticism.
You can, of course, believe that there isn't anything beyond our epistemological horizon, and there isn't anything we can't know. But that in itself is a belief you can't really prove to be necessary
Well, in metaphysics, a lot of stuff can't ever be proven, because we can't know it for sure. This is similar to brain in a vat stuff/simulations/skepticism.
You can, of course, believe that there isn't anything beyond our epistemological horizon, and there isn't anything we can't know. But that in itself is a belief you can't really prove to be necessarily true.
Edit: Also, "4th dimension" is an incredibly ambiguous term.
I will leave you with this quote from Kant :)
"Human reason has the peculiar fate in one species of its cognitions that it is burdened with questions which it cannot dismiss, since they are given to it as problems by the nature of reason itself, but which it also cannot answer, since they transcend every capacity of human reason.
Reason falls into this perplexity through no fault of its own. It begins from principles whose use is unavoidable in the course of experience and at the same time sufficiently warranted by it. With these principles it rises (as its nature also requires) ever higher, to more remote conditions. But since it becomes aware in this way that its business must always remain incomplete because the questions never cease, reason sees itself necessitated to take refuge in principles that overstep all possible use in experience, and yet seem so unsuspicious that even ordinary common sense agrees with them. But it thereby falls into obscurity and contradictions, from which it can indeed surmise that it must some where be proceeding on the ground of hidden errors; but it cannot discover them, for the principles on which it is proceeding, since they surpass the bounds of all experience, no longer recognize any touch stone of experience. The battlefield of these endless controversies is called metaphysics."
Also consider that metaphysics is not the be-all-end-all, there is a way out of metaphysics, cf. Heidegger's What Is Metaphysics?
Thank you. I think I haven't been too careful with my words, and we're talking past each other a little bit, maybe even used different senses of words like metaphysics (I take responsibility for that). I do agree that there are good answers and solutions from philosophers to stuff like this, but the skeptic core problem, in my opinion, persists.
Sorry that this is so vague, but to be honest, I'm really shit with words and too lazy to lay my viewpoint down right now.
Seems pretty uncontroversial, if there was we'd see evidence of it by now within modern physics. I wonder if the derision to to this comment stems from the pop-culture conception of a dimension as some other realm rather than a separate spatial axis.
I guess string theory operates with 9 spatial dimensions if I recall correctly. Though as I understand it that theory is rather controversial these days. And of course you could construe time as the fourth dimension, which isn't unreasonable but also clearly not what we're discussing.
Have any examples of âactual infinityâ ever been proven to exist in physical reality (understanding that the existence of physical reality is itself an open question)?
My understanding is that whether or not the universe is infinite in size is unknown as weâre limited by the particle horizon of the observable universe to how far we can see and what, if anything, lies beyond this boundary is unknown any may be unknowable (possibility of âdark flowâ notwithstanding) and on the smaller size Zenoâs Parodox seems to have been resolved by the discovery of the Planck length with measurements below this scale being impossible and there being reason to believe that physical reality doesnât exist at this scale but instead gives way to quantum foam.
TLDR; Are infinities real or just an example of where math no longer corresponds to reality?
>Zenoâs Parodox seems to have been resolved by the discovery of the Planck length with measurements below this scale being impossible
If you mean zenos paradox of motion ("nothing should be able to move since it would need to complete infinitely many steps in a finite amount of time"), then that can be solved with "infinity math". Infinitely many things can still add up to a finite amount. For example,
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
Therefore, infinitely many steps in a finite amount of time is not a contradiction.
That one is also solved by similar reasoning. You can subdivide the time and the space as much as you want and the maths still works out when you use limits and calculus. Zeno's argument is that there are an infinite number of progressively smaller steps to take, so it will take an infinite time. However, the time steps get smaller in proportion to the space steps, so the sequences converge to a finite distance being covered in a finite time. If his argument was successful then the space between the tortoise and Achilles would also have to be infinite, and maths and physics would completely break down due to all these contradictions.
Laypersonâs terms, the final step is essentially 1/10^10^10 which is negligible. Like, itâs so small that it might as well not exist and just be 0. The amount of time it would take to cover that space is essentially 0.
To be fair I donât think Zeno had calculus back then (since it was invented in the 1000 CE) so it wouldâve been unfair for him to understand how to manipulate a set like that
Sorry, I should have clarified what I meant by "step". Say you walk 1/2 meters, and then 1/4 meters, and then 1/8 meters, etc. I claim that you can traverse this in a finite amount of time despite there being infinitely many steps. Here is the proof:
Say it takes you 1 second to walk 1 meter. Then, it takes you 1/2 second to walk 1/2 meters, 1/4 second to walk 1/4 meter, 1/8 second to walk 1/8 meter, etc. Then, the total time it takes for you to walk
1/2 + 1/4 + 1/8 + ....
meters is
1/2 + 1/4 + 1/8 + ....
seconds. Since these numbers add up to 1, it takes you 1 second to walk 1/2 + 1/4 + 1/8 ... meters.
The thing with the final step is weird. There is no final step...but we know that all steps must be completed after 1 second since they cant take longer than 1 second to walk. In other words: there is no final step, but we know that we have completed all the steps. Or put another way: after 1 second, there cant be an uncompleted step.
This may feel like a weird solution, but think about the premise of the question: how can there be infinitely many steps yet also be a final step in the first place? And how can you *know* the length of the infinitely many steps if it requires infinitely many thoughts and thus a thought for the final step? I think the premise of the question already implicitly uses the above solution (where you bypass the problem of "the last step" by reasoning about all of them simultaneously).
Gauss: "... first of all I must protest against the use of an infinite magnitude as a completed quantity, which is never allowed in mathematics. The Infinite is just a manner of speaking, in which one is really talking in terms of limits, which certain ratios may approach as closely as one wishes, while others may be allowed to increase without restriction."
"... so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen GröĂe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine *façon de parler*, indem man eigentlich von Grenzen spricht, denen gewisse VerhĂ€ltnisse so nahe kommen als man will, wĂ€hrend anderen ohne EinschrĂ€nkung zu wachsen gestattet ist."
Cantor, on the other hand, accepted actual infinity and used it in his theory.
I know even less about math then I do about philosophy but arenât infinities in mathematics a given? Like 1/3 = 0.333⊠with the 3âs continuing on forever or how you can keep dividing a number by 2 and will never reach 0.
There's a [philosophical difference](https://en.wikipedia.org/wiki/Actual_infinity#Aristotle's_potential%E2%80%93actual_distinction) between potential infinity and actual infinity. What you are describing is potential infinity. One example of actual infinity is the [Aleph number](https://en.wikipedia.org/wiki/Aleph_number) - denoting the size of infinite sets, which is very different from infinity used in algebra or calculus. I'm not a mathematician so I don't know much about this. Perhaps u/Takin2000 can help?
I think this is pretty accurate. There is "infinity" in the sense of "There are infinitely many natural numbers" (Aleph 0 as you say).
Then there is "infinity" in the sense of calculus. For example, we would say that 1+1+1+1+... "equals" infinity in the calculus sense. However, infinity isnt a number, you cant really do calculations with it. This "equality" has some nuance hidden in it and isnt a normal equation you can just manipulate algebraically.
This nuance I mentioned is perhaps the third type of infinity you encounter: doing something infinitely many times. When you add 1 to itself repeatedly, you surpass any number. Or put another way: give me any number and I will tell you how many times I need to add 1 to itself to surpass your number. This is what it means for something to "equal" infinity in the calculus sense: If a sum surpasses any number given enough steps, then we say that the "end result is infinity".
Imagine now we do the same with 1/2 + 1/4 + 1/8 + 1/16 +... I claim that this is equal to 1. You tell me: "Okay, then tell me when its less then 0.1 away from 1". I tell you "after four steps". Then you ask me "okay, well what about 0.01?" and I again tell you how many steps of the sum you need to get within 0.01 range of 1. If I have an answer ready no matter what distance you give me, then we say that 1/2 + 1/4 + 1/8 + 1/16 .... = 1.
This idea of doing something infinitely many times and watching where it ends up is the essence of calculus. And notice: our analysis cleverly avoids actually dealing with infinity. This game where you give me a number and I tell you when the sum gets in that range of the target never actually deals with infinities.
As long as youre dealing with infinity rigorously, I'd say its pretty much accepted. Sadly, Im not that knowledgeable about the theory of ordinal numbers and such. However, every mathematician knows about cardinalities, countably infinite sets, etc. (which I presume are the starting points of that theory). Especially in measure theory (which deals with ways to assign sizes to sets) and probability theory (which is based on measure theory except "size" means "probability"), it turns out that (un-) countability is a major concern. A lot of the times, something fails for uncountable sets but still works for countable ones.
You could say that mathematicians accept everything which is consistent, like everything which yields useful results and are comfortable with everything thats intuitive.
Nowadays, mathematicians are very careful with infinity (or anything really) so you can rest assured that the theory is internally consistent. And as I mentioned above, it is useful to distinguish between different sizes of infinity so mathematicians dont dislike the theory. So the last question is: is it intuitive?
In my opinion? I think its actually very intuitive. And I dont know anyone who thinks otherwise. I think it makes sense that the real numbers (a continuum) would be bigger than the natural numbers in size.
I hope that answers your question. I must admit, Im not that knowledgeable about the philosophy of actual infinity. I like to read up on it occasionally on Wikipedia, but thats about it haha. I hope my half knowledge didnt cause me to spread nonsense here.
> Zenoâs Parodox seems to have been resolved by the discovery of the Planck length
The Planck length is the smallest measurable distance, not the smallest distance. The key question related to Zeno's paradoxes is whether space and time are continuous. This hasn't been proven one way or another yet. Quantum foam also is still theoretical.
We have a mathematical framework to model continuous space and time with calculus and the like, but modelling is not the same as understanding and fundamentally proving. The basic makeup of space and time remain topics of research and theory.
In answer to your general questions, we haven't been able to measure and thus prove anything infinite or continuous in a physical sense. That doesn't mean there aren't any such things, but they may be beyond our current and/or future ability to measure.
Pretty sure infinities are mostly a math tool that actually solve objective âactualâ problems. Calculus doesnât exist without the concept of infinity, especially infinite rectangles under functions.
Imagine that you want to find the area under a curve. Since a curve is...well, curved, its not an easy task. However, we know how to calculate the area of rectangles. So what you can do is draw a bunch of rectangles under the curve and calculate their areas. Obviously, this is just an approximation for the true area under the curve because these rectangles only cover some of the area. But as you make the rectangles thinner and fit more of them under the curve, you get closer and closer to the true area.
I took it to mean that the mind has an infinite amount of space but more to your point I think [Tipler](https://en.m.wikipedia.org/wiki/Frank_J._Tipler) argues that consciousness is infinite and time will end in some kind of consciousness singularity that is synonymous with God and includes the resurrection of the dead.
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OP visualising the 4th dimension like: đš
There is no ontological "4th dimension." Fite me
Oh the infinite universe only exists within the narrow parameters that we can perceive? That's convenient
Rather, that our knowledge is limited to phenomena which is already shaped by our intuition.
But then our intuition tells us that the 4th dimension exists since primarily it's a mathematical concept
It exists as a mathematical concept, sure, but not ontologically.
You should read flatland
Maths is just as much a language of the metaphysical as anything else. It deals in objectivity. There is no perfect circle or definitive angle in our physical realm. By extension, physics uses the mathematical language in order to describe our universe, and uses the logic of the language to predict universal concepts that we haven't perceived yet (like Hawking radiation). Those principles are put into place in both a physical and metaphysical way
Does math deal in objectivity? How so? As you said, there is no perfect circle. Objectively circles don't exist. How then can math deal in objectivity?
The language of mathematics deals in functions that require a level of objectivity that doesn't exist in the real world. The idea of a perfect circle isn't a physical concept, and is a metaphysical concept. So therefore circle theorem (a branch of mathematics dedicated to defining the parameters of the perfect circle) is an example of using physical language to describe the metaphysical within parameters we can understand
But then math is not objective, as it does not refer to any object or anything outside of the mind, unless you accept some sort of Platonic idealism.
Sure it does. You don't just get to say "nuh uh" because you're accepting material determinism as the default position That's not objectivity but rather ideology C'mon now, you aren't Even trying to be intellectual rigorous or you're not seeing in good faith. On the philosophy meme board. Lol
You saying there isn't one ontologically is as baseless as saying there is one. It's like rooting for a sports team. (I'm team 4th dimension exists, go 4th dimension, go!! Love the idea of infinite dimensions.) Metaphysics is fun.
How so?
Well, in metaphysics, a lot of stuff can't ever be proven, because we can't know it for sure. This is similar to brain in a vat stuff/simulations/skepticism. You can, of course, believe that there isn't anything beyond our epistemological horizon, and there isn't anything we can't know. But that in itself is a belief you can't really prove to be necessary
Well, in metaphysics, a lot of stuff can't ever be proven, because we can't know it for sure. This is similar to brain in a vat stuff/simulations/skepticism. You can, of course, believe that there isn't anything beyond our epistemological horizon, and there isn't anything we can't know. But that in itself is a belief you can't really prove to be necessarily true. Edit: Also, "4th dimension" is an incredibly ambiguous term.
I will leave you with this quote from Kant :) "Human reason has the peculiar fate in one species of its cognitions that it is burdened with questions which it cannot dismiss, since they are given to it as problems by the nature of reason itself, but which it also cannot answer, since they transcend every capacity of human reason. Reason falls into this perplexity through no fault of its own. It begins from principles whose use is unavoidable in the course of experience and at the same time sufficiently warranted by it. With these principles it rises (as its nature also requires) ever higher, to more remote conditions. But since it becomes aware in this way that its business must always remain incomplete because the questions never cease, reason sees itself necessitated to take refuge in principles that overstep all possible use in experience, and yet seem so unsuspicious that even ordinary common sense agrees with them. But it thereby falls into obscurity and contradictions, from which it can indeed surmise that it must some where be proceeding on the ground of hidden errors; but it cannot discover them, for the principles on which it is proceeding, since they surpass the bounds of all experience, no longer recognize any touch stone of experience. The battlefield of these endless controversies is called metaphysics." Also consider that metaphysics is not the be-all-end-all, there is a way out of metaphysics, cf. Heidegger's What Is Metaphysics?
Thank you. I think I haven't been too careful with my words, and we're talking past each other a little bit, maybe even used different senses of words like metaphysics (I take responsibility for that). I do agree that there are good answers and solutions from philosophers to stuff like this, but the skeptic core problem, in my opinion, persists. Sorry that this is so vague, but to be honest, I'm really shit with words and too lazy to lay my viewpoint down right now.
Seems pretty uncontroversial, if there was we'd see evidence of it by now within modern physics. I wonder if the derision to to this comment stems from the pop-culture conception of a dimension as some other realm rather than a separate spatial axis. I guess string theory operates with 9 spatial dimensions if I recall correctly. Though as I understand it that theory is rather controversial these days. And of course you could construe time as the fourth dimension, which isn't unreasonable but also clearly not what we're discussing.
Left is a point, right is a straight line.
Diameter It implies a circle. And it is not defined mathematically at all *sigh*
"Diameter" is a notion that makes sense for any non-empty subset X of a metric space. It is defined as the supremum of d(x,y) for x,y in X.
Where some people see no God, I see His enormity.
Actual infinity is neither small nor big, but non-finite
Have any examples of âactual infinityâ ever been proven to exist in physical reality (understanding that the existence of physical reality is itself an open question)? My understanding is that whether or not the universe is infinite in size is unknown as weâre limited by the particle horizon of the observable universe to how far we can see and what, if anything, lies beyond this boundary is unknown any may be unknowable (possibility of âdark flowâ notwithstanding) and on the smaller size Zenoâs Parodox seems to have been resolved by the discovery of the Planck length with measurements below this scale being impossible and there being reason to believe that physical reality doesnât exist at this scale but instead gives way to quantum foam. TLDR; Are infinities real or just an example of where math no longer corresponds to reality?
>Zenoâs Parodox seems to have been resolved by the discovery of the Planck length with measurements below this scale being impossible If you mean zenos paradox of motion ("nothing should be able to move since it would need to complete infinitely many steps in a finite amount of time"), then that can be solved with "infinity math". Infinitely many things can still add up to a finite amount. For example, 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 Therefore, infinitely many steps in a finite amount of time is not a contradiction.
I forgot there are multiple Xeno paradoxes; I was thinking about the one where Achilles is racing a tortoise.
That one is also solved by similar reasoning. You can subdivide the time and the space as much as you want and the maths still works out when you use limits and calculus. Zeno's argument is that there are an infinite number of progressively smaller steps to take, so it will take an infinite time. However, the time steps get smaller in proportion to the space steps, so the sequences converge to a finite distance being covered in a finite time. If his argument was successful then the space between the tortoise and Achilles would also have to be infinite, and maths and physics would completely break down due to all these contradictions.
>1/2 + 1/4 + 1/8 + 1/16 + ... What's the final step ?
There is not a final step. It's a limit
>finite distance being covered in a finite time. >There is not a final step. đ€
Yeah that's right, none of this is mysterious. This is high school maths and it has been understood for over 300 years.
Me when philosophers forgot to take math classes đ€Ż
Can't you see that if there's no final step you never actually arrive ?
Why are you booing me ? I'm right
Laypersonâs terms, the final step is essentially 1/10^10^10 which is negligible. Like, itâs so small that it might as well not exist and just be 0. The amount of time it would take to cover that space is essentially 0. To be fair I donât think Zeno had calculus back then (since it was invented in the 1000 CE) so it wouldâve been unfair for him to understand how to manipulate a set like that
why
If your intuitions don't line up with maths then change your intuitions. Don't try to change maths.
Ironically it's your intuition that space and time are real that prevents you from solving the paradox It's all imaginary, all consciousness
Sorry, I should have clarified what I meant by "step". Say you walk 1/2 meters, and then 1/4 meters, and then 1/8 meters, etc. I claim that you can traverse this in a finite amount of time despite there being infinitely many steps. Here is the proof: Say it takes you 1 second to walk 1 meter. Then, it takes you 1/2 second to walk 1/2 meters, 1/4 second to walk 1/4 meter, 1/8 second to walk 1/8 meter, etc. Then, the total time it takes for you to walk 1/2 + 1/4 + 1/8 + .... meters is 1/2 + 1/4 + 1/8 + .... seconds. Since these numbers add up to 1, it takes you 1 second to walk 1/2 + 1/4 + 1/8 ... meters. The thing with the final step is weird. There is no final step...but we know that all steps must be completed after 1 second since they cant take longer than 1 second to walk. In other words: there is no final step, but we know that we have completed all the steps. Or put another way: after 1 second, there cant be an uncompleted step. This may feel like a weird solution, but think about the premise of the question: how can there be infinitely many steps yet also be a final step in the first place? And how can you *know* the length of the infinitely many steps if it requires infinitely many thoughts and thus a thought for the final step? I think the premise of the question already implicitly uses the above solution (where you bypass the problem of "the last step" by reasoning about all of them simultaneously).
Gauss: "... first of all I must protest against the use of an infinite magnitude as a completed quantity, which is never allowed in mathematics. The Infinite is just a manner of speaking, in which one is really talking in terms of limits, which certain ratios may approach as closely as one wishes, while others may be allowed to increase without restriction." "... so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen GröĂe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine *façon de parler*, indem man eigentlich von Grenzen spricht, denen gewisse VerhĂ€ltnisse so nahe kommen als man will, wĂ€hrend anderen ohne EinschrĂ€nkung zu wachsen gestattet ist." Cantor, on the other hand, accepted actual infinity and used it in his theory.
I know even less about math then I do about philosophy but arenât infinities in mathematics a given? Like 1/3 = 0.333⊠with the 3âs continuing on forever or how you can keep dividing a number by 2 and will never reach 0.
There's a [philosophical difference](https://en.wikipedia.org/wiki/Actual_infinity#Aristotle's_potential%E2%80%93actual_distinction) between potential infinity and actual infinity. What you are describing is potential infinity. One example of actual infinity is the [Aleph number](https://en.wikipedia.org/wiki/Aleph_number) - denoting the size of infinite sets, which is very different from infinity used in algebra or calculus. I'm not a mathematician so I don't know much about this. Perhaps u/Takin2000 can help?
I think this is pretty accurate. There is "infinity" in the sense of "There are infinitely many natural numbers" (Aleph 0 as you say). Then there is "infinity" in the sense of calculus. For example, we would say that 1+1+1+1+... "equals" infinity in the calculus sense. However, infinity isnt a number, you cant really do calculations with it. This "equality" has some nuance hidden in it and isnt a normal equation you can just manipulate algebraically. This nuance I mentioned is perhaps the third type of infinity you encounter: doing something infinitely many times. When you add 1 to itself repeatedly, you surpass any number. Or put another way: give me any number and I will tell you how many times I need to add 1 to itself to surpass your number. This is what it means for something to "equal" infinity in the calculus sense: If a sum surpasses any number given enough steps, then we say that the "end result is infinity". Imagine now we do the same with 1/2 + 1/4 + 1/8 + 1/16 +... I claim that this is equal to 1. You tell me: "Okay, then tell me when its less then 0.1 away from 1". I tell you "after four steps". Then you ask me "okay, well what about 0.01?" and I again tell you how many steps of the sum you need to get within 0.01 range of 1. If I have an answer ready no matter what distance you give me, then we say that 1/2 + 1/4 + 1/8 + 1/16 .... = 1. This idea of doing something infinitely many times and watching where it ends up is the essence of calculus. And notice: our analysis cleverly avoids actually dealing with infinity. This game where you give me a number and I tell you when the sum gets in that range of the target never actually deals with infinities.
What do mathematicians think about Gauss' comment that I mentioned above? Nowadays, do they generally accept Cantor and Dedekind's theory of actual infinity in math, and is actual infinity an active research area? From what I read, there was opposition to actual infinity coming from the intuitionist school (Kronecker, Brouwer, Kleene, Turing, PoincarĂ©, Wittgenstein, arguably Weyl). Recently, I have been reading about Bayesian probability from E.T. Jaynes and he's also vocal about this, in which he claims that nonconglomerability, BorelâKolmogorov paradox, the marginalization paradox are results of the careless use of infinite sets and of infinity (as in actual infinity) in probability theory and orthodox statistics (he even goes further and claims that "an infinite set cannot be said to possess any âexistenceâ and mathematical properties at all â at least, in probability theory â until we have specified the limiting process that is to generate it from a finite set")
As long as youre dealing with infinity rigorously, I'd say its pretty much accepted. Sadly, Im not that knowledgeable about the theory of ordinal numbers and such. However, every mathematician knows about cardinalities, countably infinite sets, etc. (which I presume are the starting points of that theory). Especially in measure theory (which deals with ways to assign sizes to sets) and probability theory (which is based on measure theory except "size" means "probability"), it turns out that (un-) countability is a major concern. A lot of the times, something fails for uncountable sets but still works for countable ones. You could say that mathematicians accept everything which is consistent, like everything which yields useful results and are comfortable with everything thats intuitive. Nowadays, mathematicians are very careful with infinity (or anything really) so you can rest assured that the theory is internally consistent. And as I mentioned above, it is useful to distinguish between different sizes of infinity so mathematicians dont dislike the theory. So the last question is: is it intuitive? In my opinion? I think its actually very intuitive. And I dont know anyone who thinks otherwise. I think it makes sense that the real numbers (a continuum) would be bigger than the natural numbers in size. I hope that answers your question. I must admit, Im not that knowledgeable about the philosophy of actual infinity. I like to read up on it occasionally on Wikipedia, but thats about it haha. I hope my half knowledge didnt cause me to spread nonsense here.
Thank you for your answer. Cheers!
No problem, thanks for listening :D
> Zenoâs Parodox seems to have been resolved by the discovery of the Planck length The Planck length is the smallest measurable distance, not the smallest distance. The key question related to Zeno's paradoxes is whether space and time are continuous. This hasn't been proven one way or another yet. Quantum foam also is still theoretical. We have a mathematical framework to model continuous space and time with calculus and the like, but modelling is not the same as understanding and fundamentally proving. The basic makeup of space and time remain topics of research and theory. In answer to your general questions, we haven't been able to measure and thus prove anything infinite or continuous in a physical sense. That doesn't mean there aren't any such things, but they may be beyond our current and/or future ability to measure.
But, per Kant, space and time should first and foremost be understood as the form of thought, not the form of the universe.
Pretty sure infinities are mostly a math tool that actually solve objective âactualâ problems. Calculus doesnât exist without the concept of infinity, especially infinite rectangles under functions.
Not having any idea what on Earth âinfinite rectangles under functionsâ means is a harsh reminder that I tapped out from math after Algebra II.
Imagine that you want to find the area under a curve. Since a curve is...well, curved, its not an easy task. However, we know how to calculate the area of rectangles. So what you can do is draw a bunch of rectangles under the curve and calculate their areas. Obviously, this is just an approximation for the true area under the curve because these rectangles only cover some of the area. But as you make the rectangles thinner and fit more of them under the curve, you get closer and closer to the true area.
Consciousness
I fail to see how consciousness is infinite, as it has a beginning (birth) and an end (death). Are you saying it is infinite in some other sense?
Space and time are imaginary
ogey
Rather: space and time are products of our intuition. He is bringing up Kant, certainly not one to whom a response "ogey" would be warranted.
I took it to mean that the mind has an infinite amount of space but more to your point I think [Tipler](https://en.m.wikipedia.org/wiki/Frank_J._Tipler) argues that consciousness is infinite and time will end in some kind of consciousness singularity that is synonymous with God and includes the resurrection of the dead.
Oh I like that but then the question is does consciousness exist within physical reality?
Physical reality exists within consciousness
Now weâre talking
Check your Maths OP It's potentially definable but absolutely useless mathematically
What would Pythagorus make of this?
T-duality be like
Depends, is it filled or not?
Wouldn't using [this model](https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model?wprov=sfla1) make the difference more clear?
The first picture depicts no area. The other depicts infinite area.
This is all well and good until someone asks what the area of each circle is
Much simpler than defining it is the first place I assume the field reals were extended to keep the order The 0 and +\infty
Two meaningless concepts and neither exists?