Do you think the human notion of 'infinite'/'infinity' is actually real infinity?

[Kyriakos]

Oh, so you are just talking about something being infinite in one of the dimensions - whether it's time or space? Then yeah, we might never know. But the mathematical concept of infinity doesn't really rest on us ever finding out; it's just a quite natural concept that would have arisen one way or another.

It's easy to go from "bounded" to "unbounded", you've just got to think a bit and there you go.

I once wrote a short story where a successor-race to humans was not having a theoretical concept of infinity. To them it was a practical concept, given their world was very small and bounded by a massive free-fall/chasm, which they sensed as infinite in some manner.

I think that humans are very much defined as thinkers by the ability to have notions like (seemingly) very defined (eg One, in some level, in others it is not that defined), and very vague (eg Infinity).

 

[Leoreth]

I don't get the distinction between a human notion of infinity and "real" infinity. By putting it out here for discussion, "real infinity" is a human notion, isn't it?

It seems that the answer to this question can only ever be "yes" or "mu", but never "no".

 

[Kyriakos]

I don't get the distinction between a human notion of infinity and "real" infinity. By putting it out here for discussion, "real infinity" is a human notion, isn't it?

It seems that the answer to this question can only ever be "yes" or "mu", but never "no".

Yes

Well, it still is different from set human infinity. For example:

A) Human-used concept of infinity (as in limits, etc)

B) Ideal infinity (hypothetically existent), ie humanly idealised non-human concept-dependent (ie thing-by-itself) infinity.
Which is a one-off allusion to infinity-by-itself, which cannot be theorised upon other than brought up as a singular notion not further examinable.

So there is still a distinction between A and B (and a C, infinity-in-itself, if it existed, but that is not part of our notions), it is just that both can only be human notions, the first being expandable, the second being a single point like all 'thing-in-itself' concepts are. But if C exists it won't be human-tied.

 

[Leoreth]

Oh, okay. So its more about infinity as an independent entity versus infinity of series?

It's worth mentioning that not everything that is called infinity in math is the same. There's a nice Vihart video that touches on some of these things.


Link to video.

 

[warpus]

I once wrote a short story where a successor-race to humans was not having a theoretical concept of infinity.

These guys would then be limited to finitism.

I'm actually not very familiar with this way of thinking and doing math, so I can't tell you what the limitations of such a way of doing math would be. But I don't think calculus would be possible? Or real the full set of numbers? And seemingly many other things.

So sooner or later someone would figure infinity out, (un)fortunately. It would open up the door to many new ideas in math and allow you to for example formulate all real numbers, which no doubt an advanced enough civilization would eventually figure out.

 

[Kyriakos]

Oh, okay. So its more about infinity as an independent entity versus infinity of series?

It's worth mentioning that not everything that is called infinity in math is the same. There's a nice Vihart video that touches on some of these things.


Link to video.

That was pretty cool a video

Pls tell me that she is not ugly (nice video, again)

These guys would then be limited to finitism.

I'm actually not very familiar with this way of thinking and doing math, so I can't tell you what the limitations of such a way of doing math would be. But I don't think calculus would be possible? Or real the full set of numbers? And seemingly many other things.

So sooner or later someone would figure infinity out, (un)fortunately. It would open up the door to many new ideas in math and allow you to for example formulate all real numbers, which no doubt an advanced enough civilization would eventually figure out.

Hey, given they are in my story, i can speak about them

It is not that they haven't yet discovered the notion of infinity. It is that it is sensed in their world by them, and is not a notion but a sense. Try to imagine that we had no sense of integers in the world of material things (eg a book, a human, etc), and we had to form that sense as a highly theoretical concept. For us then the 'integer' would be really a vague concept (we might have been a sort of hive-mentality being, or at any rate not a distinct individual in a species). While infinity might have been pretty evident sense-wise, and then not easy to define mathematically as something which is not axiom-bound (or rather likely the axiom itself, given it would be deemed as basic for us).

 

[Leoreth]

She's smart and different and has a pleasant voice, so you can already tell from the video that she is not ugly.

Spoiler :

But you can see what she looks like in this video for example:


Link to video.

It's hard to imagine an alien intelligence conceptualizing math in a way that doesn't lead them to at least some kind of infinity (especially countable infinity). But then again it's hard to imagine an alien intelligence with a different math already, so maybe that is once again just a symptom of how well math and the human mind seem to be suited for each other.

 

[Kyriakos]

^She is awesome. I will email her

 

[warpus]

Yeah, I'm not sure how they would get by without infinity. I mean, many of the number systems we use are infinite sets.

Could you do math with only integers and no fractions? Probably, but there would be limitations.. and everything would be incredibly cumbersome.

And it seems to me that if you allow for division, you will end up with fractions. So then how do you have math without division? Surely if you have a space fleet of 12 ships, you need a way to figure out how many ships will be in each sub-fleet, if you divide the fleet into 2 equal parts. But then division leads to fractions and fractions lead to infinite sets... so..

 

[Leoreth]

Why does division lead to infinity sets? Sure, the decimal notation of some fractions has infinite digits but that is only a representation. As long as you divide rational numbers, you can continue to represent the result as fractions of natural numbers. In fact, even in human mathematics decimal notation is usually avoided.

On the other hand, it's hard to imagine most of calculus without some kind of infinity involved. And it's hard to imagine spaceships without calculus.

 

[Kyriakos]

To reply to you both:

I was imagining a being for which there is no sense of a unity of something by itself/alone/distinct. Such a species would not have '12 ships' even if it has for us 12 ships. For it those ships are a continuum in some way, and not sensed as 12 things of a type.

Maybe a more direct way to grasp this is to imagine a case where an alien sees you, and to the alien you are a collection of an indistinct number of bands or Moebius strips. For the alien you are not a unity as in "that is evidently one human", but a multitude which cannot really be examined as a functional unity and a single being in a species.
Some of those parts may appear to mix with analogous ones the alien would see if more humans approached.

 

[Leoreth]

Yeah, I guess a good analogy for the way your alien species thinks is how humans perceive fluids as one continuous body instead of a set of discrete objects (even though it is).

However, I'm not sure how willing I am to accept this without some further detail on how such a species interacts with macroscopic discrete objects. Maybe for a life form that is itself fluid or gaseous, and has no set shape or even concept of individuals.

 

[warpus]

Why does division lead to infinity sets?

Because it is fairly easy to see (for a mathematician at least) that the number of fractions between 0 and 1 is infinite. So if they have fractions, they're going to have a concept of infinity, eventually. I don't see a way around it.

I was imagining a being for which there is no sense of a unity of something by itself/alone/distinct. Such a species would not have '12 ships' even if it has for us 12 ships. For it those ships are a continuum in some way, and not sensed as 12 things of a type.

I can't see a way for trading and then economics to work if you don't have an ability to count things. I have one, two, three chickens, I will give you them to you for one, two, three, four, five, six, shiny coins. Even if you go just by weight, you need to count a number of units - 500 grams, or 800 grams, or whatever. If everything is "one", and you can't keep track of how many things you have, I just don't see how you would even get math out of that. What sort of mathematical system would be possible? You need countable numbers for math, I think. Otherwise you can't even have variables, or addition, or subtraction.. so what would be left?

This was the state of human society before specialization.. I mean, way before specialization.. Eventually you're going to need to be able to count things, assuming there's a civilization in your future.

 

[Kyriakos]

Yeah, I guess a good analogy for the way your alien species thinks is how humans perceive fluids as one continuous body instead of a set of discrete objects (even though it is).

However, I'm not sure how willing I am to accept this without some further detail on how such a species interacts with macroscopic discrete objects. Maybe for a life form that is itself fluid or gaseous, and has no set shape or even concept of individuals.

I did not present the story from a POV requiring a translation of the alien POV to human, since that story has two parts. In the first part the narrator is of the new (eg alien, but supposedly just a future species that took over humans, and discovered remains of human civ) species, and he explains how he plans to create a work dealing with a specific "member of the previous species" (ie a human), who seems to be focusing on a concept the aliens have as a limit, but it is not entirely the same (this enables me to present the difference in how they think, and that for them it is sort of inverted as to what infinite and integer is to us).
But the alien species does not see itself as anything strange, which is natural at any rate. It sees the ancestors (humans) as strange. The language of the ancestors, for example, appear to the new species to be some sort of complicated glyph system, while we know we use alphabet

The second part is narrated by a human, cause it appears the previous part was a dream he had last night. He now tries to examine what it meant, and makes some points about infinity as a concept.

 

[Leoreth]

Because it is fairly easy to see (for a mathematician at least) that the number of fractions between 0 and 1 is infinite. So if they have fractions, they're going to have a concept of infinity, eventually. I don't see a way around it.

Okay, but isn't that like realizing there are infinitely many natural numbers by looking at repeated +1 addition? Both are really obvious when you ponder the operations, but I would say for addition the consequence is even more immediately clear.

I'd say though that the difference is that you can basically ignore infinity without the math of these operations breaking down. Sure, it's implied somewhere in there, but you could still have this type of math if your mind is somehow wired not to think in terms of infinity.

That's why I contrasted calculus where infinity and infinitesimals are the tools of your trade.

I did not present the story from a POV requiring a translation of the alien POV to human, since that story has two parts. In the first part the narrator is of the new (eg alien, but supposedly just a future species that took over humans, and discovered remains of human civ) species, and he explains how he plans to create a work dealing with a specific "member of the previous species" (ie a human), who seems to be focusing on a concept the aliens have as a limit, but it is not entirely the same (this enables me to present the difference in how they think, and that for them it is sort of inverted as to what infinite and integer is to us).
But the alien species does not see itself as anything strange, which is natural at any rate. It sees the ancestors (humans) as strange. The language of the ancestors, for example, appear to the new species to be some sort of complicated glyph system, while we know we use alphabet

The second part is narrated by a human, cause it appears the previous part was a dream he had last night. He now tries to examine what it meant, and makes some points about infinity as a concept.

Okay, but you can't really work around the fact that the reader of your story is likely to be human So whatever alien concept is element of the story has to be communicated to the reader in a way a human can grasp.

 

[Kyriakos]

Okay, but you can't really work around the fact that the reader of your story is likely to be human So whatever alien concept is element of the story has to be communicated to the reader in a way a human can grasp.

Yes, but i cover my tracks like a fox does, erasing them with its tail:

-The alien being is alien, but also human given it is a human dreaming he is an alien

-To the alien the phenomena seem normal, and he even uses (mostly, not entirely) analogous notions we use when talking of such things. But it all is juxtaposed to the writings dug out in the alien society, found works of the human ancestor species. Those writings appear to be written in symbols that mostly consist of circles and half circles and numbers of lines surrounding them. But we already know (as humans) that our alphabet is not like that at all, so it is rather obvious that the non-human there already translates a phenomenon into its own alien form and is not going to prove it does so either.

 

[Leoreth]

Hm, I guess to give a more specific critique I'd need to actually read it. Time to brush up my Greek

 

[warpus]

Okay, but isn't that like realizing there are infinitely many natural numbers by looking at repeated +1 addition? Both are really obvious when you ponder the operations, but I would say for addition the consequence is even more immediately clear.

I agree, but it sort of drives the point home more effectively when you realize that there are an infinite amount of numbers just between 0 and 1. It's more intuitive to me, I suppose. You could always say "A very large number that we can't even imagine" without resorting to infinity, but you'd have no choice to accept that the set of real numbers, even a bounded set between two integers, is infinite in size.

 

[Kyriakos]

Hm, I guess to give a more specific critique I'd need to actually read it. Time to brush up my Greek

I agree, but it sort of drives the point home more effectively when you realize that there are an infinite amount of numbers just between 0 and 1. It's more intuitive to me, I suppose. You could always say "A very large number that we can't even imagine" without resorting to infinity, but you'd have no choice to accept that the set of real numbers, even a bounded set between two integers, is infinite in size.

Maybe infinity can be altered a bit as a concept, by using different (and cumulative, building up in some special way) limits within that infinity. I suppose this is already done in some math of infinity (don't know), but in my view a rigorous such expansion of the concept of infinity might place the issue in a rather easier to deal with basis. It might even give a (only in the bounds of that new system of examining infinities) solution of sorts to the 'set of all sets' issue and the non-completeness of set theory that is not axiomatically limited.

 

[Leoreth]

I agree, but it sort of drives the point home more effectively when you realize that there are an infinite amount of numbers just between 0 and 1. It's more intuitive to me, I suppose. You could always say "A very large number that we can't even imagine" without resorting to infinity, but you'd have no choice to accept that the set of real numbers, even a bounded set between two integers, is infinite in size.

Okay, I see what you mean.

(Of course the number of rational numbers between two natural numbers also becomes finite if you limit the set of natural numbers by some limit, but at this point we're leaving the area of intuitive arguments very quickly )