I left this comment at Jack’s blog, which I was going to wait to write something here, but since the subject came up there…

You know, ever since I stumbled in my philosophy readings and discovered there is such a thing as infinity math (the cardinality thing, right?), I came to wonder if the concept of infinity is really sound. Can we truly conceive of the infinite? I have my doubts at this moment, but it’s nothing conclusive, just something I’ve been wondering now and then. I don’t have much time to spend elaborating on why the concept itself may be beyond our minds or our frame of thinking, but it doesn’t sit very well with me.

What image(s), if any, comes to your mind when you think about infinity?


Update April 13:

Jack replied:
We can certainly imagine things that are infinite (from the Latin: without an end). So I would say that we can “conceive” them. There is certainly an immense amount of mathematics that has been done regarding infinity. In fact, without a good understanding of infinity, one doesn’t have Calculus, and Calculus must be valid in some respect, because its applications work so well in the real world!

On the other hand, I would agree that we cannot really “comprehend” the infinite. We are finite, and we can only comprehend finite things, and in fact most mathematics involves reducing questions about infinitely-sized sets (which we can’t answer) to questions about finitely-sized sets (which we can).


I replied:
I was thinking along the lines of the problem that infinity is outside the scope of rational thought, or logical thought.

With infinity, everything we know does not make sense anymore.

Like in your math example, I think we always employ this reduction to the finite as if we were still being able to handle the infinite, like fooling ourselves, but the infinite seems outside our capability of precision. I’m not sure your conception/comprehension distinction is what I am trying to posit as the problem. I’m having difficulty wording it.

You said:
‘We can certainly imagine things that are infinite (from the Latin: without an end). So I would say that we can “conceive” them.’

For example, what can you conceive as infinite?

Update April 15:

Math is not physical, that’s why I think it makes sense to talk about infinite things in math (like numbers and quantities, they are abstract things).

I understand what the definition of infinite material things is (unbounded), but I question that physical things can be infinite. And it all has to do with space. No material thing can be infinite in any aspect if there is no infinite space. And I don’t see how space can be infinite going outwardly, unless that “outwardly” turns around back to itself. Even the closed universe model (pictured like a globe) is impossible to make sense visually.

I wish I could take this Cosmology course.

Update April 16:

My problem with the mathematical concept of infinity applied to the physical world (someone wrote it better):

What I believe to be the basic misconception of modern mathematical physicists – evident, as I say, not only in this problem but conspicuously so throughout the welter of wild speculations concerning cosmology and other departments of physical science – is the idea that everything that is mathematically true must have a physical counterpart; and not only so, but must have the particular physical counterpart that happens to accord with the theory that the mathematician wishes to advocate. [Herbert Dingle, Science at the Cross-Roads, pp. 124-5.]

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Below is the sequence of comments exchanged between Jack and me on his blog – with a couple excluded because not relevant:

Alessandra said…

You know, ever since I stumbled in my philosophy readings and discovered there is such a thing as infinity math (the cardinality thing, right?), I came to wonder if the concept of infinity is really sound. Can we truly conceive of the infinite? I have my doubts at this moment, but it’s nothing conclusive, just something I’ve been wondering now and then. I don’t have much time to spend elaborating on why the concept itself may be being our minds or our frame of thinking, but it doesn’t sit very well with me.

10:15 PM, April 12, 2005

Alessandra said…

mistake:
I don’t have much time to spend elaborating on why the concept itself may be being our minds or our frame of thinking, but it doesn’t sit very well with me.

should be:
I don’t have much time to spend elaborating on why the concept itself may be beyond our minds or our frame of thinking, but it doesn’t sit very well with me.

10:16 PM, April 12, 2005

jack perry said…

We can certainly imagine things that are infinite (from the Latin: without an end). So I would say that we can “conceive” them. There is certainly an immense amout of mathematics that has been done regarding infinity. In fact, without a good understanding of infinity, one doesn’t have Calculus, and Calculus must be valid in some respect, because its applications work so well in the real world!

On the other hand, I would agree that we cannot really “comprehend” the infinite. We are finite, and we can only comprehend finite things, and in fact most mathematics involves reducing questions about infinitely-sized sets (which we can’t answer) to questions about finitely-sized sets (which we can).

I gotta run, but I hope that answers your question somewhat.

10:49 AM, April 13, 2005

Alessandra said…

I was thinking along the lines of the problem that infinity is outside the scope of rational thought, or logical thought.
With infinity, everything we know does not make sense anymore.
I think we always employ this reduction to the finite as if we were still being able to handle the infinite, like fooling ourselves, but the infinite seems outside our capability of precision.

You said:
‘We can certainly imagine things that are infinite (from the Latin: without an end). So I would say that we can “conceive” them.’

For example, what can you conceive as infinite?

9:41 PM, April 13, 2005

jack perry said…

What can you conceive as infinite?

The presence, and ability, of God. The space of the universe. The sets of natural, integer, rational, real, and complex numbers, along with all sorts of vector spaces and polynomial ideals. The number of measurable distances between two points. In some respect, all these things can be conceived as infinite.

Since it was so easy to write these things, I am sure that I must have misunderstood your question πŸ™‚

With infinity, everything we know does not make sense anymore.

I don’t agree. In mathematics at least, I would say that we only began to understand many things once we threw in the concept of infinity β€” the calculus, for example β€” just as we only began to understand many things once we threw in the concept of zero, or of negatives, or of imaginary numbers, or of irrational numbers, or of any of the major advances in mathematical thought.

11:14 PM, April 13, 2005

Alessandra said…

I am having difficulty conceptualizing and articulating what I find is the problem with infinity. So I asked you this question about the examples to help me think things through. Tell me what you understand of what I’ve written below:

Leaving out metaphysical subjects, such as God, I think you gave other examples that have helped me conceptualize the problem. This is attempt two, I think I’ve been able to make some distinctions but I’m still confused. Regarding the conception/comprehension distinction:

1) To conceive – is to conceive simply to think of an idea regardless if this idea makes sense? If so, then I agree we can conceive “infinity.” For example, if I say ” a triangle with 4 vertixes” this is an idea, but it doesn’t make sense, it’s not logical. I can put words together that gramatically may be correct, but the concept itself makes no sense. We can’t put together “triangle” with “4 vertixes”. Or is a conception something that necessarily must make sense?

2) Let’s examine the concept of infinite space. Can you actually conceive of infinite space? Is “conceiving of infinite space” simply coming up with a definition of what that is, even if the definition may not make sense or be real? Let me see if I can clarify the issue:
Take a picture of finite space. You have a region ( 3 dimensional) and it extends in all directions until it stops. Finite space is inside a larger space. Let’s conceptualize with objects. You have a cube, it occupies a finite amount of space. The cube sits in a room with certain dimensions. So I think when people talk about the infinite, they fool themselves that the room is infinity, when the room sits inside another larger space region. Because visually and spacially we cannot conceive of something that extends *infinitely* in all directions. To picture visually infinite space is not possible with our minds. Your mind stops at certains limits, rational thought forces you to stop somewhere. Which is why I really appreciated what you said regarding how we reduce questions of infinity to finity and pretend we are still talking about infinity. Is this what you meant when you said we cannot comprehend infinity?

3) Take an infinite line. A person who thinks they can think about an infinite line will say the line starts here and it just goes forever. Where? To infinity. So infinity is posited as a place you can go to but that you can never reach. See the logical contradiction? If it is infinite, it doesn’t stop going, if it doesn’t stop going, it no longer makes sense, because places we can go to are finite.

4) Is it too complicated to explain to me what Calculus has to do with infinity? (I don’t know much math).

5) “The number of measurable distances between two points.”
This is an interesting example, because this I think is something that is not illogical regarding infinity.

I think what I am trying to get at is the problem of infinity applied to concrete things, such as space, and probably time.

Let me know if you have any thoughts on the above.

10:40 AM, April 14, 2005

jack perry said…

To conceive – is to conceive simply to think of an idea regardless if this idea makes sense?

I guess I can agree with that definition, although I think the notion of “infinity” makes perfect logical sense. A triangle with 4 vertices is self-contradictory, whereas an infinite number (=”unbounded” number) of galaxies is not.

So I think when people talk about the infinite, they fool themselves that the room is infinity, when the room sits inside another larger space region.

I don’t agree with this image of infinity. Imagine infinite space this way instead: a comet is ejected from our solar system, and proceeds outward from the galaxy without ever encountering another galaxy or other object. Assume it continues traveling and never meets another object. Assuming this is possible (which, due to gravity, it probably would not be), the space it could travel would be infinite.

Because visually and spacially we cannot conceive of something that extends *infinitely* in all directions. To picture visually infinite space is not possible with our minds.

There appears to be an assumption what you’re saying: that what we know is only what we can imagine. I disagree with that assumption.

Which is why I really appreciated what you said regarding how we reduce questions of infinity to finity and pretend we are still talking about infinity. Is this what you meant when you said we cannot comprehend infinity?

I didn’t use the word “pretend”. πŸ™‚

We can say something about infinite sets, because we know characteristics of all the elements of these sets. These characteristics constitute a finite amount of knowledge. Then we reason on these characteristics.

For example: all even numbers are divisible by 2. Therefore, no even number is prime (except 2 itself). I have just said something about an infinite set: it is logically consistent, therefore true. However, I only used a finite number of statements.

So infinity is posited as a place you can go to but that you can never reach. See the logical contradiction?

Yes, except that infinity isn’t posited as a “place” at all. The definition you give is not one that mathematicians would usually use (for example) and I don’t think it’s one that theologians would use, either. One can define it as a place, and in some models it is defined like that (so-called “projective space” has a “point at infinity”, for example), but it’s done very carefully, so as to avoid contradictions.

Is it too complicated to explain to me what Calculus has to do with infinity?

Here’s an example: add 1 + 1/2 + 1/4 + 1/8 + … (Zeno’s paradox) This is an infinite number of summands; you can’t find the sum if you try to add the fractions manually. Calculus provides a logical framework that finds this sum successfully (the sum is 2).

11:25 AM, April 14, 2005

Alessandra said…

So infinity is posited as a place you can go to but that you can never reach. See the logical contradiction?

Yes, except that infinity isn’t posited as a “place” at all. The definition you give is not one that mathematicians would usually use (for example) and I don’t think it’s one that theologians would use, either. One can define it as a place, and in some models it is defined like that (so-called “projective space” has a “point at infinity”, for example), but it’s done very carefully, so as to avoid contradictions.
=======================
So if it’s not a place, than you can’t go to it. If you can’t go to it, it’s no longer concrete, it’s no longer real.
What is the definition of infinite regarding space?

1:03 PM, April 14, 2005

Alessandra said…

In mathematics at least, I would say that we only began to understand many things once we threw in the concept of infinity β€” the calculus, for example β€” just as we only began to understand many things once we threw in the concept of zero, or of negatives, or of imaginary numbers, or of irrational numbers, or of any of the major advances in mathematical thought.
=======================
You see mathematics is not concrete, space is, things which are physical are concrete.

I’m not sure concrete things are the same as non-concrete things. Maybe this is what I am grappling with.

1:06 PM, April 14, 2005

Alessandra said…

I don’t agree with this image of infinity. Imagine infinite space this way instead: a comet is ejected from our solar system, and proceeds outward from the galaxy without ever encountering another galaxy or other object. Assume it continues traveling and never meets another object. Assuming this is possible (which, due to gravity, it probably would not be), the space it could travel would be infinite.
==================
But see, you’ve made an assumption that it would be possible for this comet to travel endlessly. Why is this assumption a reality? It’s the endlessly that I have a problem with. How can something not have an end? If the comet were travelling along a path, we could potentially, by definition of what travel is (to be at a certain position at each instant), identify its position always. So how can something not end, not have a position somewhere? The definition of infinity just seems to equal a blank. You are here, then you go there and there and there and then… it’s infinity, like a big blank. How can the comet not stop somewhere? To me it falls off the plane of reality and goes into the imaginary. ThatΒ΄s why your example of infinite numbers I can kind of handle, it’s not concrete. (although I wish I knew Calculus to mathematically understand what you said πŸ™‚

To have a position is a concept that works within an identifiable finite space, not some imaginary space made on assumptions.

Does what I am asking make sense to you? Or do you think I have misunderstood something about the concept of infinity?

3:34 PM, April 14, 2005

jack perry said…

So if it’s not a place, than you can’t go to it.

Correct (as long as we’re in affine space).

If you can’t go to it, it’s no longer concrete, it’s no longer real.

You’re not expressing yourself well. I can’t go to Pluto, but that doesn’t make Pluto any less real. Now, I can say that Pluto is real because I could hypothetically go there, but that’s an assumption.

What is the definition of infinite regarding space?

Unbounded. That is, there is no finite (bounded) number that describes the volume of space.

You see mathematics is not concrete…

…therefore not real? πŸ˜‰

Anyway, I don’t agree. I think mathematics is very concrete. I would agree that mathematics is not material; maybe that’s what you mean.

I’m not sure concrete things are the same as non-concrete things.

You have hit on an important question of modern philosophy: ideas vs. the objects they represent. Ideas do not necessarily correspond to the objects they represent, and in some cases our ideas do not even represent “material” objects, but characteristics of such objects.

But see, you’ve made an assumption that it would be possible for this comet to travel endlessly. Why is this assumption a reality?

Why shouldn’t it be a reality? Is that any less reasonable than to assume that it can’t travel endlessly when nothing is in its way? You’ve made no argument that would support that statement, so it would have to be an assumption. The fact that I can’t experience something doesn’t mean that I can’t think about it, and therefore talk about it logically.

If the comet were travelling along a path, we could potentially, by definition of what travel is (to be at a certain position at each instant), identify its position always.

Correct.

So how can something not end, not have a position somewhere?

I don’t understand what you’re asking. To say that an object’s journey doesn’t end, does not at all suggests that it doesn’t have a position somewhere. I could identify the comet’s position at any time, and measure the distance from its stopping point. Unless some boundary (finis in Latin) obstructs the comet, it continues traveling, and those distances continue to increase. Because they can continue to increase without limit/boundary, the potential distance traveled is infinite: without boundary/end/etc.

To have a position is a concept that works within an identifiable finite space, not some imaginary space made on assumptions.

Actually, the concept of position works very well in infinite space; in fact, I think it’s safe to say that it works better than in finite space. Otherwise, you have to have a boundary on position: a “furthest” position, or a “largest” distance, from a fixed point. But that doesn’t make much sense, since you can always increase the distance by addition.

In any case, I would like you to name a single field of knowledge that is not based on at least one unproven assumption. The fact is that you can’t: all knowledge is based on some unproven assumptions.

Or do you think I have misunderstood something about the concept of infinity?

Yes. To start with, your confusing the noun and the adjective. Infinite describes a property of an object; it is not by itself a place or size. Infinity describes neither a place nor size; it is simply the notion that something can be infinite.

5:41 PM, April 14, 2005

Alessandra said…

Anyway, I don’t agree. I think mathematics is very concrete. I would agree that mathematics is not material; maybe that’s what you mean.
==========================
I understood how you used material, as in physical. So, yes, math is not physical, that’s why I think it makes sense to talk about infinite things in math (like numbers and quantities, they are abstract things).

But now I don’t understand your definition of concrete. What did you mean when you said math is concrete?

10:09 PM, April 14, 2005

Alessandra said…

If you can’t go to it, it’s no longer concrete, it’s no longer real.

You’re not expressing yourself well. I can’t go to Pluto, but that doesn’t make Pluto any less real. Now, I can say that Pluto is real because I could hypothetically go there, but that’s an assumption.
=======================
I understand what the definition of infinite material things is (unbounded), but I question that physical things can be infinite. And it all has to do with space. No material thing can be infinite in any aspect if there is no infinite space.

I’ll try to think of an analogy.

10:36 PM, April 14, 2005

Alessandra said…

I found this:
The horizon problem

OUR universe appears to be unfathomably uniform. Look across space from one edge of the visible universe to the other, and you’ll see that the microwave background radiation filling the cosmos is at the same temperature everywhere. That may not seem surprising until you consider that the two edges are nearly 28 billion light years apart and our universe is only 14 billion years old.

Nothing can travel faster than the speed of light, so there is no way heat radiation could have travelled between the two horizons to even out the hot and cold spots created in the big bang and leave the thermal equilibrium we see now.

This “horizon problem” is a big headache for cosmologists, so big that they have come up with some pretty wild solutions. “Inflation”, for example.

You can solve the horizon problem by having the universe expand ultra-fast for a time, just after the big bang, blowing up by a factor of 1050 in 10-33 seconds. But is that just wishful thinking? “Inflation would be an explanation if it occurred,” says University of Cambridge astronomer Martin Rees. The trouble is that no one knows what could have made that happen.

So, in effect, inflation solves one mystery only to invoke another. A variation in the speed of light could also solve the horizon problem – but this too is impotent in the face of the question “why?” In scientific terms, the uniform temperature of the background radiation remains an anomaly.

13 things that do not make sense

* 19 March 2005
* NewScientist.com news service
* Michael Brooks

==================
Does this idea that the universe is expanding makes sense to you? If the universe is space, how can it be expanding? It’s weird. If the universe is space, it can’t be sitting and expanding inside some empty space, otherwise, the latter empty space would be the universe.
So how can the universe expand? It doesn’t seem to have logic to me.

 

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