Infinity, actual and potential

[KantianAgnostic]

Infinite: Without limit, after a certain kind toward which the term refers (ex. infinite in space, infinite duration, infinite number of, etc.)

Potentially infinite: A term applying to an object where, when any limit is explored for that object, the object exceeds the limit.

Actually infinite: A term applying to an object where the object is at once expressed actively in all its limitlessness.

Example: A geometric line is potentially infinite in divisibility (it can be divided n number of times, and still be divided once more). If it were actually infinite in divisibility, it would be impossible to move along the line, because such movement would require an infinite number of smaller movements along each division.

My assertion:

It seems to me that, though an actual infinite may be possible, human reason is incapable of comprehending it.

(My speculation: because of this, space and time as we think of them are necessary categories in reason that process what we observe, but don’t have their own substantial reality; logic and observation, such as in Zeno’s paradox and quantum mechanics, respectively, illustrate this well)

A few questions:

1. Do you agree with my definitions (do you think it is useful)?
2. Do you agree with my assertion (do you think it is right or wrong)?
3. Do you think God is actually infinite?
4. If yes on (3), why not the universe being actually infinite?
5. If no on (3), why not the universe being potentially infinite, like God might be?

If you are interested in commenting on my speculation, please start a thread about this, or ask me to. I’d like to concentrate on the assertion, definitions, and on your views on this complex philosophical issue.

 

[Eucharisted]

1. God is infinite because He is without beginning or end
2. The Universe is not infinite because it has a beginning and an end

 

[liquidpele]

1. God is infinite because He is without beginning or end
2. The Universe is not infinite because it has a beginning and an end

1. You were told that by things that are finite… unless you are schizophrenic I suppose.
2. That is a major assumption. Maybe true, maybe not, but in any case the universe is thought to be infinitely large so it meets the criteria in at least one way regardless of how you define it’s age.

 

[KantianAgnostic]

1. God is infinite because He is without beginning or end

This must be agreed upon because this is definitional. Now, such an object of thought may not exist (and may turn out not to be a proper object of thought, even if it does exist, because of the inability of the mind to grasp infinities).

1. The Universe is not infinite because it has a beginning and an end

As for “end”, it would be hard to state that the Universe has one, since it’s still around.

As for “beginning”, it does seem as though the Universe had a point of origin (big bang), but this may render the universe at least aeviternal. Still, one only then has to assert that there is something akin to a multiverse, or some other impersonal force that is infinite in duration (and maybe a potentially infinite number of objects with finite durations), and it would be difficult to see why one would favor the answer “God” over one of these other objects.

 

[masterjedi747]

1. Do you agree with my definitions (do you think it is useful)?

…not sure about your definition for “potentially infinite” (though I’m happy with how you used it in the example).
But the other two seem OK… nothing horrible jumps out at me, anyway.

1. Do you agree with my assertion (do you think it is right or wrong)?

I’m kind of stuck in a middle ground at the moment, to be honest.

Aristotle and Aquinas both seem to be convinced that an actual (existing) infinite magnitude and an actual (existing) infinite number are both impossible. As far as number goes, I think I can see that, since it seems obvious that any given number will be a finite number, and that there is no “last” or “infinite” number past which it is impossible to count one higher. In other words, number is infinite in the sense of “potentially infinite”… any finite number can be achieved, but an infinite number (almost by definition) can never be achieved or made actual.

As far as magnitude goes, however, I’m a bit more torn. On the one hand, I do recognize that something like an infinite circle is impossible, since a circle (or any other geometrical figure) is necessarily defined and limited by its circumference… so I think it’s kind of stupid to try talking about an infinite area that’s bounded by anything. Similar and related to this would be the fact that there is a given/finite distance between any two given points. But could you ever simply grant the existence of an actual infinite (unbounded) line or surface? I’m not entirely sure, because I fail to see the contradiction that would be involved in doing so. However, Aquinas argues pretty much the same as he did in the case of number, saying that any given magnitude has boundaries, and is therefore finite – of course it could always be made larger, but the fact remains there is no largest magnitude that can be achieved. And it makes sense to a certain point, so I can’t entirely disagree… but as it is, I still don’t see why an infinite line/surface/volume would necessarily be absurd, as it clearly would be in the case of an infinite-yet-still-enclosed geometrical figure.

1. Do you think God is actually infinite?

Yes… but it’s not quite the same. When Aquinas argues that God is infinite, what he essentially means is that God is unlimited in being/perfection/power. But calling God “infinite” in that way is not quite the same as the way that we would call a magnitude “infinite” (in length/area/volume)… and furthermore, for what it’s worth, since God is not material, He does not have those dimensions of length/area/volume anyway.

1. If yes on (3), why not the universe being actually infinite?

Hopefully this is somewhat clear from above.
Aquinas, at least, would argue that since any actual infinite magnitude is impossible, God* could not* have created an infinite universe – not because of any defect in His power (on the contrary, He could always have made a larger universe), but simply because it is absurd/self-contradictory, just like an “infinite circle” would be.

1. If no on (3), why not the universe being potentially infinite, like God might be?

I don’t think we would ever refer to God as being “potentially infinite”.
…what would that even mean? That He could always be better than He is, perhaps?

P.S. – most all the stuff I’ve taken from Aquinas can be found in Question 7 of the Summa, here:
nd.edu/~afreddos/summa-translation/TOC-part1.htm

My speculation: because of this, space and time as we think of them are necessary categories in reason that process what we observe, but don’t have their own substantial reality

I don’t think I would quite agree with that, but I’m not super-comfortable going into much detail on Kant, and it seems a bit beyond the scope of this particular thread anyway, so… hopefully you won’t mind if I pass on this one?

It seems to me that, though an actual infinite may be possible, human reason is incapable of comprehending it.

I do think it’s true that the human intellect is finite, and thus necessarily incapable of comprehending any actual infinite… so that isn’t the problem at all. It’s just whether or not (or rather, in what way) an actual infinite is possible.

Example: …If it were actually infinite in divisibility, it would be impossible to move along the line, because such movement would require an infinite number of smaller movements along each division.

This is just me being a nit-pick, but I think you would be better off saying “if it were actually divided infinitely” rather than “if it were actually divisible infinitely”… just because it seems there is a way in which you can speak about “actually having a potential” for something (as opposed to not even having the potential at all). So you could say that a line actually has the potential to be divided infinitely (without limit), but that it does not actually have the potential to be actually divided infinitely… if that makes sense. In much simpler terms, the line is able to be divided any number of times that you wish, but you will always be able to divide it again, so you can’t actually divide it infinitely.

needs to take a break :whacky:

 

[geometer]

Example: A geometric line is potentially infinite in divisibility (it can be divided n number of times, and still be divided once more). If it were actually infinite in divisibility, it would be impossible to move along the line, because such movement would require an infinite number of smaller movements along each division.

This is a restatement of Zeno’s paradox, and his paradox is false…

 

[MindOverMatter]

This is a restatement of Zeno’s paradox, and his paradox is false…

Thats a very attractive statement you have there. Can you please tell me why zeno’s paradox is false?

 

[MindOverMatter]

This is just me being a nit-pick, but I think you would be better off saying “if it were actually divided infinitely” rather than “if it were actually divisible infinitely”… just because it seems there is a way in which you can speak about “actually having a potential” for something (as opposed to not even having the potential at all). So you could say that a line actually has the potential to be divided infinitely (without limit), but that it does not actually have the potential to be actually divided infinitely… if that makes sense. In much simpler terms, the line is able to be divided any number of times that you wish, but you will always be able to divide it again, so you can’t actually divide it infinitely.

needs to take a break :whacky:

Your abilities are vast. I pray that you never take a break!!!

 

[mstone]

Thats a very attractive statement you have there. Can you please tell me why zeno’s paradox is false?

You can’t divide a quark. (Yet.)

-Modified from a Richard Feynman memoir.

 

[Dameedna]

Hopefully this is somewhat clear from above.
Aquinas, at least, would argue that since any actual infinite magnitude is impossible, God* could not* have created an infinite universe – not because of any defect in His power (on the contrary, He could always have made a larger universe), but simply because it is absurd/self-contradictory, just like an “infinite circle” would be.

I don’t understand this statement.

If God is “actually infinite” then how could any Actual infinite be impossible?

Genuine question, I probably missed something in translation.

I like this thread, even if I’m not sure I’ve thought about it too much. Alway’s new stuff to learn

 

[MindOverMatter]

I don’t understand this statement.

If God is “actually infinite” then how could any Actual infinite be impossible?

Genuine question, I probably missed something in translation.

I like this thread, even if I’m not sure I’ve thought about it too much. Alway’s new stuff to learn

Its not any easy think to grasp. It took me a while to understand it to.
Basically, its impossible to create an actual infinite, either as a being, or a number. That does not mean that it is impossible that an actual infinite actually exists; but it would have to be a timeless-eternal, with out numbers or a beginning. In other words, a true actual infinite is not a number, for an actual infinite is a true objective whole. It cannot be defined by dimensions or parts, for it transcends those realities.