An Idea of Infinity

Bill, you wrote, “You are saying that the number of entities in the universe is infinite. In so doing, you are saying that there is an actual number of things in the universe and that the number is infinite.”

No, I am not saying there is an actual or definite number of things. Just like you do not say there is, has been, or will be an infinite amount of time, (rather, you say that the universe is eternal,) so I am saying that that which exists may be without definite extent, it may be “inextensive.” I am no more claiming to have the number than you are claiming to have the length of time.

Post 41

Friday, August 18, 2006 - 9:43pm

Bill, you wrote,
“You are saying that the number of entities in the universe is infinite. In so doing, you are saying that there is an actual number of things in the universe and that the number is infinite.”
No, I am not saying there is an actual or definite number of things. Just like you do not say there is, has been, or will be an infinite amount of time, (rather, you say that the universe is eternal,) so I am saying that that which exists may be without definite extent, it may be “inextensive.” I am no more claiming to have the number than you are claiming to have the length of time.

Okay, so then you're not saying that the number of entities in the universe is infinite, because I thought you were. At least, you were in several of your preceding posts. Have you changed your mind?

Remember that when Peikoff says that universe is "eternal," he doesn't mean temporally infinite; he means out of time or non-temporal. So, are you saying that the universe is non-numerical, that there is no actual number of things in the universe -- no actual number of planets, stars, galaxies, etc.? If so, on what do you base this? If there is a specific number of planets in our star system and a specific number of stars in our galaxy, then why isn't there a specific number of galaxies in the universe?

- Bill

Post 42

Friday, August 18, 2006 - 11:11pm

Bill,

I haven’t changed my mind but my terms, in an attempt to get around your hang-ups over certain ones, such as “actual number.” I am rejecting your description of infinitude as a designation of an “actual number.”

Your approach to this question is rationalistic. It pivots around, “it must be some specific number,” and, “Either there is an actual number or there isn't.” You keep asserting this but I don’t see why reality must conform to it.

You ask, “If there is a specific number of planets in our star system and a specific number of stars in our galaxy, then why isn't there a specific number of galaxies in the universe?”

There probably is a specific number of galaxies in this expanded singularity. Might there be other expanded singularities, though? And why couldn’t the universe consist of expanded singularities in all directions without end? I don’t assert this, but I do leave the possibility. I continue to find odd any positive assertion to knowledge that this cannot be so.

Post 43

Sunday, August 20, 2006 - 6:20pm

Jon, to say that the universe is "without definite extent" is not analogous to saying that the universe is "outside of time" or "non-temporal." What you mean in referring to the universe as "without definite extent" is that that its constituent parts have an unlimited extension in space. What Peikoff means in referring to the universe as "eternal" or "non-temporal" is not that it has an unlimited extension in time, but that the concept of time doesn't apply to the universe. You're not saying that the concept of number doesn't apply to the entities in the universe, are you? You're saying that the number is without definite extent -- that it's unlimited. You wrote,

I haven’t changed my mind but my terms, in an attempt to get around your hang-ups over certain ones, such as “actual number.” I am rejecting your description of infinitude as a designation of an “actual number.”

Your approach to this question is rationalistic. It pivots around, “it must be some specific number,” and, “Either there is an actual number or there isn't.” You keep asserting this but I don’t see why reality must conform to it.

I wouldn't say that my approach is rationalistic. Quite the contrary. It avoids the dangers of a floating abstraction by requiring an anchor in reality. You are saying that the number of entities in the universe is infinite. What I am saying is that the concept of number requires specificity -- that there can be no such thing in reality as a nonspecific number of entities -- because it is precluded by the very concept of number. A number means a definite quantity. Granted, the quantity can change; it can increase or decrease, but it must always be some specific number. Otherwise, it is a meaningless concept. The same is true for the attribute of length or extension. In order for length or extension to exist, it must be some particular length or extension -- five feet, five hundred feet, five thousand feet, etc.

You ask, “If there is a specific number of planets in our star system and a specific number of stars in our galaxy, then why isn't there a specific number of galaxies in the universe?”

There probably is a specific number of galaxies in this expanded singularity. Might there be other expanded singularities, though? And why couldn’t the universe consist of expanded singularities in all directions without end? I don’t assert this, but I do leave the possibility. I continue to find odd any positive assertion to knowledge that this cannot be so.

You're not grasping the principle here. It makes no sense to deny that (at any point in time) there is a specific number of galaxies in the universe, if you grant that there is a specific number of stars in a galaxy -- just as it makes no sense to deny that there is a specific number of stars in a galaxy if you grant that there is a specific number of planets in a solar system. There is no difference in principle between these two examples. The concept of number, size or extension requires specificity, without which it is nothing more than a floating abstraction.

- Bill

Post 44

Sunday, August 20, 2006 - 8:02pm

Bill,

“You're not saying that the concept of number doesn't apply to the entities in the universe, are you?”

No. Just as you are not saying that the concept of time doesn’t apply to the entities and events in the universe.

“You're saying that the number is without definite extent -- that it's unlimited.”

Correct. That it could be. That it’s a scientific question
that logic and reason cannot preclude rationalistically. You beg the question by insisting that the universe be finite because only then will we have an “actual number.”

“…nothing more than a floating abstraction.”

Not a floating abstraction. Just that we could never find the end as there is none. That has a definite meaning and smuggles in no concepts severed from their basis. You may not like it for its untidiness, but it is not a floating abstraction.

I admit to have run out of ways of expressing myself. Thanks for the efforts you have made, Bill.

Post 45

Monday, August 21, 2006 - 12:43am

How many more events will occur?

Post 46

Monday, August 21, 2006 - 6:31pm

You're saying that the number is without definite extent -- that it's unlimited.
Correct. That it could be. That it’s a scientific question that logic and reason cannot preclude rationalistically. You beg the question by insisting that the universe be finite because only then will we have an “actual number.”

On the contrary, it's a philosophical question, not a scientific one. Now, I see what's behind your criticism of my response as "rationalistic." Yes, it would be rationalistic, if this were a scientific question, but it isn't. Nor could it ever be answered, if it were.

I think you may be missing the point of my criticism, which is that the number of entities in the universe cannot be unlimited by the very nature of the concept of number. There is not, nor can there be, such a thing as an infinite number, whether of things or in the abstract; there can only be the (unlimited) potential to add to an already existing number.

…nothing more than a floating abstraction.
Not a floating abstraction. Just that we could never find the end as there is none. That has a definite meaning and smuggles in no concepts severed from their basis. You may not like it for its untidiness, but it is not a floating abstraction.

It's not a matter of tidiness. The concept of number is arrived at from observing instances of quantity in reality. The number 5 means: | | | | |. In other words, it refers to a certain quantity of units: five lines, five stones, five apples, five things. A number of things can be any quantity, but it must be some quantity -- some specific quantity, like 5, 500, 5000, etc. To say, therefore, that it is possible there is an unlimited number of things is simply another way of saying that it is possible there is a number of things that is not some specific number, because if there were a specific number, then the number wouldn't be unlimited. However, since the universe is composed of a number of things, it must be a specific number and therefore cannot be unlimited. Now, you can argue that this is rationalism, if you want, but it is no more rationalistic than to say that two two's must equal four.

I admit to have run out of ways of expressing myself. Thanks for the efforts you have made, Bill.

You're entirely welcome, Jon. Always happy to oblige! :)

You also asked in a subsequent post:

How many more events will occur?

Within what period of time? The number of future events is potentially unlimited, but not actually unlimited. It's always possible for more to occur; however, within any period of time, only so many will have occurred.

- Bill

Post 47

Monday, August 21, 2006 - 8:40pm

How many events have occurred?

Post 48

Monday, August 21, 2006 - 11:05pm

How many events have occurred?

Since when? Since the beginning of this century? Since the beginning of the earth? Since the beginning of our galaxy? You have to specify a time period in order for the question to be meaningful. And if you do, the number will be finite.

- Bill

Post 49

Tuesday, August 22, 2006 - 1:30am

Bill,

Regarding future events you write, “It's always possible for more to occur…” And regarding past events you don’t suggest there will be any limit to how far back a period I may name. So, there is no limit in the past or the future for more events to occur. Your favorite caveat that a finite number results from naming any specific period aside, you are saying that there is no limit to the number of events. This is what “infinite” means.

Post 50

Tuesday, August 22, 2006 - 5:38pm

Jon, It's always possible for me to count to a higher number than the one I've counted to. So, in that sense, there is potential infinity, because there is no intrinsic limit on how high I can count (barring a limited lifespan and other such restrictions). But there is no actual infinity, because whatever number I've counted to will always be a specific number. Ditto for the number of events that can occur.

- Bill

Post 51

Wednesday, August 23, 2006 - 3:05pm

Bill,
I've come into this thread late, so if I am repeating previous arguments, please say so. I read most of the other posts so I don't think I am.
You said:

To say, therefore, that it is possible there is an unlimited number of things is simply another way of saying that it is possible there is a number of things that is not some specific number, because if there were a specific number, then the number wouldn't be unlimited. However, since the universe is composed of a number of things, it must be a specific number and therefore cannot be unlimited.

It seems that by saying "the universe is composed of a number of things" you're assuming what you are arguing for. Why is the universe composed of a number of things; why does the universe have a specific number of things? I've heard the argument before, and Bob mentioned it in post #25, that

The universe is finite; it has to be or it would lack identity.

But this assumes that the universe is an entity; only entities have identity. Why is the universe an entity? Isn't this an example of the "fallacy of composition"; that is, claiming that what is true of a part must be true of the whole?

I would claim that the objects in the universe can have a one-to-one correspondence with the real numbers. You can count objects and no matter how many you count, you will find more. Whether this is true is a question for science not philosophy.
Thanks,
Glenn

Post 52

Wednesday, August 23, 2006 - 6:49pm

Hi Glenn,

You wrote

It seems that by saying "the universe is composed of a number of things" you're assuming what you are arguing for. Why is the universe composed of a number of things; why does the universe have a specific number of things? I've heard the argument before, and Bob mentioned it in post #25, that The universe is finite; it has to be or it would lack identity. But this assumes that the universe is an entity; only entities have identity. Why is the universe an entity? Isn't this an example of the "fallacy of composition"; that is, claiming that what is true of a part must be true of the whole?

I wouldn't say that only entities have identity. Why do you say that? Remember, existence is identity, and that means that whatever exists has an identity, including the universe -- unless you want to argue that the universe doesn't exist.

You ask, "Why is the universe composed of a number of things?" Well, would you say that it is not composed of things -- that there are no things in the universe? And if you would not say this -- if you would grant that there are things in the universe -- then why isn't there a quantity of things in the universe. It seems odd to me to say that there are things in the universe but that there is no quantity of them in the universe. And if there is indeed a quantity of them, then why isn't it a specific quantity? Isn't every quantity specific?

I would claim that the objects in the universe can have a one-to-one correspondence with the real numbers. You can count objects and no matter how many you count, you will find more. Whether this is true is a question for science not philosophy.

Well, if it's a question for science, then how would science answer it? After all, it can't be answered empirically. No, I think this is a philosophical question. We define the universe as "all that exists." And if the universe is all the things that exist, then there aren't any more. But if the quantity of things in the universe is infinite, then there is always more. So either the universe is all that exists, in which case, it isn't infinite; or it is infinite, in which case, it isn't all that exists. But obviously, the universe is all that exists; therefore, it isn't infinite.

-- Bill

(Edited by William Dwyer
on 8/23, 7:08pm)

Post 53

Thursday, August 24, 2006 - 6:23am

Bill,
How many numbers are there in the universe?
You said:

Well, would you say that it is not composed of things -- that there are no things in the universe? And if you would not say this -- if you would grant that there are things in the universe -- then why isn't there a quantity of things in the universe.

This is a false dichotomy. You are saying that there either are no things in the universe or there is a (finite) quantity of things in the universe. You have thrown out the third possibility, which is the point of this discussion: the possibility that there is a countably infinite number of things in the universe.
Thanks,
Glenn

Post 54

Thursday, August 24, 2006 - 7:48am

I admire your tenacity on this, Bill - but they just "don't get it".......
(Edited by robert malcom on 8/24, 7:49am)

Post 55

Thursday, August 24, 2006 - 11:59am

I'm surprised know one here is familiar with the essay by Alex Silverman about the integration of "finite" with "unbounded" regarding measurement of time and matter in the Universe. Check it out: http://www.geocities.com/rationalphysics/Unbounded_Finite.htm

Post 56

Thursday, August 24, 2006 - 12:20pm

Glenn, you asked,

Bill,
How many numbers are there in the universe?

You mean, how many things are in the universe, don't you? It would depend on how you define "thing," of course. But assuming that you've specified it, I would say that I don't know and will never know, because I'm not omniscient.

Suppose, analogously, that you said there are a finite number of grains of sand on a beach, and that I objected to this statement by asserting that the number is infinite. Would you consider it a fair question if I then demanded that you tell me how many? No, because you'd recognize that there is a difference between knowing that the number is finite and knowing what the number is.

You said:
Well, would you say that it is not composed of things -- that there are no things in the universe? And if you would not say this -- if you would grant that there are things in the universe -- then why isn't there a quantity of things in the universe.
This is a false dichotomy. You are saying that there either are no things in the universe or there is a (finite) quantity of things in the universe. You have thrown out the third possibility, which is the point of this discussion: the possibility that there is a countably infinite number of things in the universe.

Perhaps, I misunderstand what you mean by a "countably infinite number of things," but if the number of things is infinite, then how could it be countable? You can't count an infinite number of things, can you? You say I've thrown out the third possibility that the number of things is "countably infinite." Yes, I've thrown it out, but I've done so, based on an argument. All you've done is deny the conclusion of the argument by accusing me of a false dichotomy, when I've given an argument as to why I don't consider the dichotomy false. You need to address the argument.

I'd like to return to a claim that you made in your previous post, in which you wrote:

I would claim that the objects in the universe can have a one-to-one correspondence with the real numbers. You can count objects and no matter how many you count, you will find more.

It is true that if you start counting in the abstract, then no matter how far you've counted, you can always count more; you can always add another number to the total. But it doesn't follow from that that if you count actual things in existence, you can always count more. Observe that there is no infinite number even in the abstract. Every number that you count to, no matter how high, is a specific number and represents a specific quantity. To be sure, there can always, theoretically, be a greater quantity of something, just as there can always be a higher number that one can count to, but just as the higher number must be some particular number, so the greater quantity must be some particular quantity. Neither can be infinite.

What is infinite in the natural number series is not any particular number itself, but the potential to add to any given number in the series. There is no limit on that potential. You can always add more, but whatever you've added to your total will always be some specific number, and therefore the total will, in turn, be some specific number. For example, if I count to 1000, I will have counted 1000 abstract units; if I count an additional 9,000, I will have counted 10,000 such units. However far I go, I will always be left with a specific number of units. I will never be able to count to an infinite number, because there is none. And if there is no infinite number in the abstract, then how can there be an infinite number of things corresponding to it? You need to answer this question, if you are going to argue that there can be an infinite number of things in the universe, because "the objects in the universe can have a one-to-one correspondence with the real numbers."

You also need to address my argument that if "the universe" means all the things that exist, which it does, then there are no more things in existence because that's all there is, but that if the number of things in the universe is infinite, then there is always more. In short, the concept of 'infinity' is incompatible with the concept of 'entirety', because you can't subsume everything in an infinite series. You can't refer to the series in its entirety, because it doesn't have an entirety. So if the number of things in the universe is infinite, then "the universe," cannot mean all the things that exist, because the very concept of 'all the things that exist' becomes incoherent. It is this kind of contradiction that arises when you try to argue for an actual infinity of existents.

- Bill
(Edited by William Dwyer
on 8/24, 12:29pm)

Post 57

Thursday, August 24, 2006 - 12:45pm

"Perhaps, I misunderstand what you mean by a "countably infinite number of things," but if the number of things is infinite, then how could it be countable? You can't count an infinite number of things, can you?"

'Countably infinite' simply refers to any infinite set whose elements can be mapped 1-to-1 to the positive integers. You can trivially map even numbers, odd numbers, etc. to integers, and even all rational numbers to them, hence all these are countably infinite. Uncountable refers to infinite sets which cannot be mapped to integers - ie. they are infinitely more infinite than countably infinite sets :). The elements of these could not even theoretically be enumerated. The classic example is the real (including irrational) numbers.

Post 58

Thursday, August 24, 2006 - 3:19pm

Aaron,
That's exactly what I meant. Thanks.
Bill,
As usual, you have shown "infinite" patience. But as Robert sarcastically said: I just don't get it. Your arguments are clever, but so were Zeno's. I'll have to think about it some more. Nothing you have said has affected my intuition that it's possible that I can go in some direction and count the objects in the universe and keep going and keep counting and never reach the end. It's possible that for every integer there is a different object in the universe. Infinite sets can exist and the universe may be one of them.
Thanks,
Glenn

Post 59

Thursday, August 24, 2006 - 3:43pm