Two kinds of infinity- how can we understand them?

[Kyriakos]

It has been a while since i am occupied in some thoughts and a couple of literary works, with the double nature of the infinity understandable by humans.

The two infinites are the following:

1)The infinite seen as an endless collection of smaller parts, giving when added up a finite sum

An example of this is the progression of numbers like 1+1/2+1/4+1/8+...1/n=2. It almost equals 2, but we have added infinite numbers to reach this finite result.

This type of infinity understanding is the basis for Zeno's paradoxes, where one never gets to the end of a finite space or time, since in order to do that he would have to go through infinite parts.

2) The infinite seen as a collection of parts, giving an infinite sum

An example of this is the addition of all the natural numbers (1+2+3+4+....n). It is at the same time an infinite sum, and one giving an infinity as a result.

This type of infinity understanding is what enables us to draw infinite shapes, and realize them as finite. For example any kid can draw a line on a blackboard, because it views it as finite. In reality it is a sum of infinite parts. The circle can also be drawn, while it is at the same time infinite and periodically repeated.

Since i am thinking of concluding a larger literary work with this subject, i felt like asking you if you have any thoughts about this double nature of our understanding of infinity. While infinity is studied in math, i think the question as to how we can have these two antithetical examinations of it is not answered at all.

Looking forward to your views. I could have posted this on the science forum, but i think many people who go there post here as well, and there is nothing to say people without a hard scientific background cannot reflect on this issue.

 

[Leoreth]

I'm not sure what difference you see between these kinds of infinity: is it the infinite number of elements in a sequence compared to the infinite value of a variable or the result of a series? It's nothing surprising that some series converge at a finite number while others have infinite limits. At least not to me, but I'm a math person - I usually accept things if they logically follow from mathematical definitions.

Oh, and the "infinite sums with finite result" is exactly the reason why Zeno's paradox isn't a paradox at all (poor Zeno didn't know infinitesimal calculus).

(By the way, from the title I thought this thread is about natural vs. real numbers.)

 

[Kyriakos]

I am interested particularly in the double ability we have to view what is an infinite as both infinite and finite:

In the example i gave, a schoolboy is told to draw a line on a blackboard. Sure he can do that, because he views it as finite, it has a beginning, a body, and an end. But you know it is in reality an infinite amount of tiny particles, which always can be subdivived even more, to infinity.

My question therefore is: how can we have the ability to see what is in essense infinite, as finite? And how can we at the same time have the ability to view something infinite (eg sum of natural numbers) as infinite? Those are surely two very different abilities, about mostly the same thing.

The question is more about philosophy of the mind, than pure mathematics, although it does not negate mathematic perspective on it either

 

[Thorgalaeg]

In physics there are basically two infinities but not exactly the ones you described in your post. The first one is called "Aleph-null" and it is an infinite set of discrete numbers. (both sets described in your posts would be Aleph-null so basically of the same size aka "cardinality").

The second kind is the "Continuity" (the infinity of real numbers). While both infinities are well... infinite, the Contniuity (C) is "larger" than Aleph-null (X), In fact it can be proved that C=2^X

 

[Gigaz]

In physics there are basically two infinities but not exactly the ones you described in your post. The first one is called "Aleph-null" and it is an infinite set of discrete numbers. (both sets described in your posts would be Aleph-null so basically of the same size aka "cardinality").

The second kind is the "Continuity" (the infinity of real numbers). While both infinities are well... infinite, the Contniuity (C) is "larger" than Aleph-null (X), In fact it can be proved that C=2^X

There are in fact infinitely more infinities because from each of them one can easily construct a bigger one. But the number of infinities is Aleph-null-infinity

 

[Gatsby]

I highly recommend you check out Charles Seife's "Zero: The Biography of a Dangerous Idea", which gives a really interesting and non-mathematician-friendly history of the concept of zero and its "twin" infinity.

One of the things that book discusses is 19th century mathematician Georg Cantor's idea that there were actually an infinite number of infinities nested in each other, starting with the infinity of rational numbers (called "aleph-nought"), then the infinity of real numbers (aleph-one, the first uncountable infinity), aleph-two and so on. I bring this up because I find it to be an excellent illustration of how infinity is not a number but a characteristic that number sets can exhibit: no matter how large or small a (finite) number is, a larger or smaller number can always be conceived of.

Like numbers, the concept of infinity only becomes apparent when an observer tries to categorise and quantify some aspect/s of reality. The type of infinity you encounter actually depends on your approach. To take your example of the two seemingly irreconcilable infinities, the first is effectively a consequence of taking a dividing approach, while the second is a consequence of taking an additive approach. Hence your two different types of infinity are not really antithetical or contradictory at all, but are just the results of different and mutually exclusive approaches to quantification and categorisation.

To me the relationship between zero and infinity is very reminiscent of Yin and Yang. There seems to be a fundamental trade-off between the two: for example, as the number of pieces of a finite object increases (i.e. approaches infinity), the average size of each piece of that finite object gets smaller (approaches zero), and vice-versa. Perhaps the inherent polarity between zero and infinity indicates that there is a fundamental rule of reality which only allows one thing to be enlarged by making another thing smaller.

Even more curiously, the number 1 also seems to be intimately intertwined with zero and infinity in a way that makes zero seem like the thesis, infinity the antithesis, and 1 the synthesis. It almost seems like zero and infinity are two archetypal gods of equal and opposite power waging war across eternity, and phenomenal reality with all its apparent finitude is the manifestation of the front line or no-mans-land in that war: a place where neither infinity or zero has absolute control but both have some degree of influence. Another more diplomatic way of looking at it is to think of phenomenal reality (represented in archetype by the synthesis of the number 1) as a compromise or mutual venture agreed to by zero and infinity. What happens when an irresistible force meets and immovable object? Perhaps the answer is "phenomenal reality"?

 

[Valka D'Ur]

Kyriakos, how about this: while the human brain has a finite number of cells at any specific point in time, there are some individuals who appear to be infinitely stupid (as there seems to be no end of the stupid things they do and say, things that have a lasting impact far beyond those specific points in time).

(The individual I have in mind for this is Stephen Harper, Canada's infinitely stupid Prime Minister, who is stupid enough by himself, but his cabinet ministers have kicked it up much farther beyond the capacity for human stupidity than I'd have thought possible. Please note that I'm not trying to drag Canadian politics into this thread; this is just the example that occurred to me to illustrate what I think the OP may be asking.)

 

[Kyriakos]

Thank you all for your very interesting answers

I am approaching this from a philosophical point of view. I am wondering which innate mental ability enables us to have this double intake of infinity (as infinite divisable parts, and infinite added parts, but the question extends to infinitly divisible parts being understood as a finite sum).
I have stated before that i do not think Zeno was making a false statement; that is i am not of the view that he thought he actually could not sense something infinite as finite, after all it is obvious he, like any other man, could. The arrow is moving, the tortoise will be surpassed by Achilles at some point in time, and so on.

I prefer to think that his paradox was nested on our dual ability in regards to examinining infinity, our ability to effectively create infinities with one stroke (such as in the drawing of a line). I am interested in imagining a person who has lost that ability, and cannot even draw a line. This is the basis of my literary work on this subject

 

[Atticus]

My question therefore is: how can we have the ability to see what is in essense infinite, as finite? And how can we at the same time have the ability to view something infinite (eg sum of natural numbers) as infinite? Those are surely two very different abilities, about mostly the same thing.

I don't think the word "see" is correct here. We don't see the line as infinite, we think it to be infinite. The idea of endless divisibility concerns ideal lines. On the other hand, we don't see infinite sums or sequences of natural numbers in any form, just symbols for them.

So how can we think something infinite to be finite would imo be more correct question. I'd say it's because of two meanings of the words: The line is finite when we talk about it's length. It isn't finite when we talk about number of smaller lines or points which it contains.

There are in fact infinitely more infinities because from each of them one can easily construct a bigger one. But the number of infinities is Aleph-null-infinity

It's at least as great as aleph_null, but do you have any reference for it being exactly that?

 

[Gatsby]

I am interested in imagining a person who has lost that ability, and cannot even draw a line. This is the basis of my literary work on this subject

I guess that person would just draw a finite number of discrete dots in a row, instead of a continuous line? Perhaps they would only be capable of thinking in terms of finite stocks, incapable of thinking in terms of flows, and every time they encountered a continuous line they would just see a large but countable number of dots in a row?

 

[Kyriakos]

That is one possibility; the work is formed as a detective story, in which the quest of the friend of that person is to understand how his mind works.

Another explanation is that he sees everything being infinitely divided all of the time, and thus cannot accept any line (or any other form) as finite. I am imagining someone who, for example, uses an arithmetic notebook (the ones used in elementary school, with the division to squares) and has painted over the central square completely with ink. All others just see a filled with ink square, but he continues to pen it, seeing (or imagining) infinite smaller unpainted parts.

This all might have been the result of an accident in which he injured his head. At least this premise would rid me from much theoreticl explanation of the basis of this uniqueness.

 

[Thorgalaeg]

Well, he would think in Aleph-null therms only while the Continuity (the line) would be unimaginable for him, so he would draw plots until his hand fall to pieces. Poor fellow...

 

[Kyriakos]

Kind of like dividing the Greek (and other European) debt infinitely, while being unable to erase it with a single stroke

 

[Gigaz]

It's at least as great as aleph_null, but do you have any reference for it being exactly that?

For correctness: The question, whether a set exists with more elements then Aleph_0 but less then Aleph_1 is undecidable within Zermelo-Fraenkel-Choice.

 

[Leoreth]

Oh, and I would contest that if someone draws a line it consists of infinitely many infinitely small dots. If he draws a chalk line, it's made up of discrete chalk molecules, if he draws it on paper, the same for ink or graphite molecules etc. So there is a clear difference between idealized lines in mathematics and lines in reality.

 

[Kyriakos]

Oh, and I would contest that if someone draws a line it consists of infinitely many infinitely small dots. If he draws a chalk line, it's made up of discrete chalk molecules, if he draws it on paper, the same for ink or graphite molecules etc. So there is a clear difference between idealized lines in mathematics and lines in reality.

Which is why we can draw a line just fine. But the issue in the story is that one cannot, and it is being examined just why. So if you have any ideas about that i am anticipating them since they would help me write something better

 

[Kyriakos]

Also it seems that some degree of abstraction is needed so as to think anything. By this (which is having to do with this topic) i mean that if one was endlessly occupied with analysis, then he would sink to analysis of analysis of analysis of analysis and so on, without end, and lose the ability to view something as a whole, or at least as a larger part.
Likewise it seems to me that we can understand perfectly, with one glance, the shape that is called "a line", or a circle, a triangle etc, because exactly we can abstract our thought of it. We do not see a circle as an infinite and periodicaly repeating sum of dots all in equal distance from the center. We see it as a run of a periphery, which by itself is an abstract thought. (i guess that children who know no geometry would know the first thought, but the mere fact that later on they can grasp the second means that the ability to grasp it is innate) My quest in the story would be to try to examine just what happened to someone and disabled him from having abstract thoughts at least in regards to geometrical shapes, and if this had other implications for his thinking.

To my knowledge there is no other story with this theme. Borges' "Founes" has a type of loss of abstract thought as its topic, but it focuses on memory, and not so much in that antipode of general thought, the over-analytic.

 

[uppi]

In the example i gave, a schoolboy is told to draw a line on a blackboard. Sure he can do that, because he views it as finite, it has a beginning, a body, and an end. But you know it is in reality an infinite amount of tiny particles, which always can be subdivived even more, to infinity.

But the amount of particles is finite. If you draw a line with chalk on a blackboard you hit a limit once you reach the size of a chalk molecule. If you try to subdivide even further it stops being chalk. The amount of chalk molecules is huge, but it is a finite number.

In fact, one could argue, that infinity is a purely mathematical and/or philosophical construct, because nothing in nature is infinite. At one point everything could become quantized, so you would always hit a point, where subdividing becomes impossible. This is not conclusively proven (gravity is the sore point there), but might very well the case.

So someone not understanding the concept of infinity might actually have a more realistic view than anyone else.

Don't get me wrong: The concept of infinity is a very useful one, and without it understanding nature would be much harder. But it might not be one that is actually implemented in "reality" (whatever that is...).

 

[plarq]

The first one is called infinitesimal (if you mean the difference between 2 and 1+1/2... series), while the second one is called infinity.

 

[Ayn Rand]

Thanks Kyriakos, that's a very subtle observation. Have you also considered the infinity of potential? For example, on a sheet of paper you have an infinite number of choices regarding which shape/s you could draw, or story you could write. This infinite potential exists in any blank space, no matter how small or large.