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THE PARADOXES OF DENYING INFINITY

3/24/2021

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[Originally published at Debunking Christianity]

It is common for theists — especially those familiar with the Kalam Cosmological Argument and William Lane Craig's defense of it — to deny the existence of actual infinities. And since the question of infinity recently came up in one of the comment threads at Debunking Christianity, I thought I'd re-publish an old blog post on this, with minor modifications.

It consists of two parts — the main blog post, plus (for those who want to delve a bit deeper into the issue) an addendum on the solution to Zeno's paradox:



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Although it may be surprising, no claim I've made has been criticized more by the religious than the claim that there are actual infinities. Every time I so much as mention infinity, someone goes out of their way to "inform" me of the errors of my ways. And yet there appear to be clear cases of infinity all around us. For example, every time you move, you go through an infinite number of subintervals: You first go half of the way, then 3/4 of the way, followed by 7/8, 15/16, and so on, covering what is obviously an infinite series. Nevertheless, you are able to complete the motion.


Using this as an example of an actual infinity raises the question of Zeno's paradox, however, and some of the aforementioned critics have done just that. According to Zeno, since this series of intervals never ends — since there are an actual infinite number of them between any two points A and B — one can never go from A to B.

Now, other than agreeing with Zeno that all motion is an illusion, there are two basic alternatives available to us here. The first is to agree that there are an infinite number of intervals but to claim that we can in fact move across all of them. The other is to deny that there are an infinite number of intervals.

It is often said that an infinite series is one that by definition cannot be completed, since (after all) it has no end. But that isn't the case. Consider the principal claim that the religious finitists (i.e., infinity deniers) insist on making, namely that the past cannot be infinite. They do so by arguing that an infinite series of events could not have already taken place, as that would mean it has been completed. But why can't an infinite series of past events occur given an infinite amount of time? If there is a time for each event, and there are an infinite number of times (or moments), then there can be an infinite number of events. (My favorite example here is one that is credited to Wittgenstein: imagine coming across someone who says “...4, 3, 2, 1, 0. Finished!” and then explains he has been counting backwards from eternity.)

We can resolve Zeno's paradox, it seems to me, once we see how the infinite series that it involves is also one that can be completed. However, the solution is a bit complicated, so I'll leave it to the end of this post. First, I want to consider the other alternative, that of the finitists.

There is more than one way to be a finitist about the number of intervals in a line segment, but the one that appears to be favored by the religious is to argue that the number of subdivisions involved is only potentially, rather than actually, infinite. That is, these individuals claim that there are an infinite number of possible subdivisions in that one could in principle keep dividing a line segment forever. However, since one could never finish the task, there never could be an actual infinite number of intervals there.

The problem with this is that it treats the intervals as something created by our actually dividing the line (at least mentally). That is, it assumes that when one is talking about each part of the line in question (its first half, the 1/4 that follows that, and so on), one is talking about parts that have already been marked off from the rest. But that is not what Zeno's paradox is about. The first half, the one fourth that follows that, and all the other intervals that Zeno is referring to are geometrical parts that are present in the line whether anyone takes notice of them or not. In other words, they are already there. It's true that we could never actually mark off all of them, as it would take forever, but that's irrelevant.

A second way of arguing that a line does not contain an infinite number of intervals is to claim that the line is not infinitely divisible, not even potentially — that is, to claim that there are smallest parts that cannot be divided further even in principle. But that too is highly problematic. For then what one is claiming is that there are segments of a line that are not zero sized and yet that have no further parts. But if one of these spatial "atoms" isn't zero sized, that's because there is some distance between one end of it and the other — which in turn means the two ends must be distinct. And that contradicts the claim that it has no parts.

Finitists object to infinity by claiming that it leads to paradoxes. And yet what these examples show is that the rejection of infinity introduces paradoxes as well. One individual I debated online on this topic — a defender of the above potential infinity view — actually maintained that the two halves of a line do not exist until someone divides the line in half. Now, I admit that there are unresolved paradoxes of infinity. Philosophers and mathematicians haven't yet worked out all of the issues on the subject. But I find the existence of infinity paradoxes more acceptable precisely because they involve infinity — a concept that our minds have a hard time grasping. It's very easy for or intuitions to go wrong whenever the infinite is involved. I'm not willing to accept, however, that my intuition that the two halves of a line exist prior to our dividing the line could be wrong!


THE SOLUTION TO ZENO'S PARADOX


It is often claimed that Zeno's paradox is dissolved once we realize that the sum of the infinite series 1/2 + 1/4 + 1/8 + …. is 1. But that would be the solution only if Zeno's claim were that the series does not add up to 1, or that, given that the series is infinite, the distance from A to B must itself be infinite — and that doesn't seem to be what Zeno was claiming. Moreover, even if he was claiming that, it doesn't matter, since we can see a different problem — namely, that in order to reach B (never mind that it is clearly only a finite distance away), one must complete an infinite series. That is what appears impossible.

But it isn't. Above, I claimed that at least some infinite series can in fact be completed. For instance, given an infinite amount of time, an infinite number of events can occur. So some infinite series can be completed given infinite time in which to do so. But in fact it doesn't necessarily take an infinite amount of time. Aristotle pointed out that, obviously, the time it takes to move from A to B can also be divided into its first half, its next fourth, and so on. Thus, there are an infinite number of subintervals of time in which to cover the infinite number of subintervals of length between A and B. The situation is therefore analogous to the one in which an infinite number of events occur in an infinite amount of time: in both cases, there is a time available for each event in question to take place.

Now, this at first may not seem sufficient to solve the paradox. We can imagine Zeno objecting that, just as there is a problem with completing the infinite series of intervals between A and B, there is a problem (given that the span involved contains an infinite number of temporal intervals) with the passage of time from moment T1 to moment T2. But it is easy to answer such an objection. We can begin by noting that the infinite number of subdivisions (whether of temporal or spatial intervals) adds up to the span in question. That is (to consider the spatial case first), we already know that it is the distance between A and B that contains the infinite number of intervals we're discussing. So there is no question about there being a point B, only one about how one can arrive at it. Similarly, we know that it is the span T1 to T2 that has an infinite number of temporal intervals. So once again, there is no question about the existence of T2, only about how it ever can arrive. But now one can see that the above objection really makes no sense. Because for there to be a time T2 is the same thing as for it to eventually arrive! It is incoherent to claim that there is such a time and to also claim that it can never get here. (It would be as incoherent as to claim that there is a point B and to insist that it is nowhere.) Therefore, Zeno cannot reasonably object that it is impossible for the time between T1 and T2 to transpire. It would be like saying that the distance between A and B does not exist.

(I am not, incidentally, claiming that Zeno wouldn't say that. In fact, as a disciple of Parmenides, he would say precisely this about the distance — and might, faced with the above considerations, say the analogous thing regarding the time span. But none of this is relevant to the paradoxes, which take as their starting point the fact that there are distances and durations, before attempting to show that such an assumption creates a problem.)

Given that T2 will eventually be here, it follows that there is in fact one temporal interval available for moving across each interval between A and B (all the way to and including B). The fact that both the number of line segments and the number of temporal intervals is infinite is no more problematic than the fact that in a universe without a beginning, the number of past events and the number of past moments are both infinite. In both situations, the availability of a time for each event to occur in is all that is needed to avoid paradox.

One final note: Above, I stated that one could never finish the task of dividing the distance between A and B into an infinite number of intervals. However, in principle, someone could in fact do so — just as we can, and do, go through all of these intervals in moving from A to B. The someone in question would simply need to perform the task at an ever increasing rate — e.g., by making the first division in 30 seconds, the second in 15, the third in 7.5, and so on, so as to complete the task in one minute flat. As Bertrand Russell once said, the impossibility of performing an infinite number of tasks in a finite amount of time is merely “medical.” ​
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AQUINAS'S ABJECT FAILURE

11/13/2018

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There are quite a few things wrong with the first cause argument, but the worst thing about it — Aquinas’s attempt to show that the chain of efficient causation cannot extend back to infinity — is ignored by most critics. The claim that there cannot be an infinite causal regress is often disputed, of course, but Aquinas’s bizarre reasoning to the contrary is usually passed over — maybe for fear it would just be confusing to readers. Whatever the case may be, I think it’s worthwhile to be aware of it, especially given that Aquinas’s old argument is still touted by many. 

The part I’m referring to is the following: 

“Now it is not possible to proceed to infinity in efficient causes. For if we arrange in order all efficient causes, the first is the cause of the intermediate, and the intermediate the cause of the last, whether the intermediate be many or only one. But if we remove a cause the effect is removed; therefore, if there is no first among efficient causes, neither will there be a last or an intermediate. But if we proceed to infinity in efficient causes there will be no first efficient cause, and thus there will be no ultimate effect, nor any intermediate efficient causes, which is clearly false. Therefore it is necessary to suppose the existence of some first efficient cause…” 

Translating from medievalese, this just says that if there is no first cause, then neither will there be a second, third, and so on — and thus no ultimate effect. But if the chain extends back infinitely, then obviously there is no first cause; there will always be others before any that we care to consider. Therefore, if the chain extends back infinitely, there will be no ultimate effect! 

This is such a bad argument that it is surprising anyone ever fell for it. Aquinas is simply confusing the existence of a cause that he calls “the first cause” with its property of being first. That is, what we might label the “first” cause cannot be missing from the chain, otherwise the “second” cause would also be missing, and so on. Given Aquinas's assumptions, that much is true. But it does not follow that this event that we're calling “the first cause” cannot have a predecessor. If the chain of causation is infinite, then there is no first cause. But no link in the chain will be missing on account of that. 

One can also understand Aquinas's mistake this way: he is either equivocating or begging the question, depending on how what he refers to as the “first” cause is interpreted at the beginning of the above passage. He is begging the question if by the “first” he means the cause that has the property of being first, for then he is simply assuming the very thing he is attempting to demonstrate. If, on the other hand, he is merely naming a particular cause in the chain the “first”, then he equivocates when he later uses the same term to mean the cause that has the property of being first.

Aquinas therefore has failed to show that the chain of causation must have had a beginning. As a result, his entire argument falls apart. 



Summa Theologica excerpt from the Laurence Shapcote trans. (London: O. P. Benziger Bros., 1911) 


[Originally published at Debunking Christianity]

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THE PARADOXES OF DENYING INFINITY

9/23/2016

1 Comment

 
As I mentioned last time, no claim I've made has been criticized by the religious more than the claim that there are actual infinities. And yet there appear to be clear cases of infinity all around us. For example, every line segment contains an infinite number of subintervals, as can be seen from (among other things) the fact that in going from its beginning to its end one first goes half of the way, then 3/4 of the way, 7/8, 15/16, and so on ad infinitum.

Using this as an example of an actual infinity raises the question of Zeno's paradox, however, and some of the aforementioned critics have done just that. According to Zeno, since this series of intervals never ends – since there are an actual infinite number of them between any two points A and B – one can never go from A to B.

Now, other than agreeing with Zeno that all motion is an illusion, we have two main alternatives here. The first is to agree that there are an infinite number of intervals but to claim that we can in fact move across all of them. The other is to deny that there are an infinite number of intervals.

It is often said that an infinite series is one that by definition cannot be completed, since after all it has no end. But that isn't the case. Consider the principal claim that the religious finitists (i.e., infinity deniers) insist on making, namely that the past cannot be infinite. They do so by arguing that an infinite series of events could not have already taken place, as that would mean it has been completed. But why can't an infinite series of past events occur given an infinite amount of time?

We can resolve Zeno's paradox, it seems to me, once we see how the infinite series that it involves is also one that can be completed. However, the solution is a bit complicated, so I'll leave it to the end of this post. First, I want to consider the other alternative, that of the finitists.

There is more than one way to be a finitist about the number of intervals in a line segment, but the one that appears to be favored by those who have corresponded with me is to argue that the number of subdivisions involved is potentially, rather than actually, infinite. That is, these individuals claim that there are an infinite number of possible subdivisions in the sense that one could in principle keep dividing a line segment forever, but that, since one could never finish the task, there never could be an actual infinite number of intervals there.

The problem with this is that it treats the intervals as something that is created by our actually dividing the line (at least mentally). That is, it assumes that when one is talking about each part of the line in question (its first half, the 1/4 that follows that, and so on), one is talking about parts that we have already marked off from the rest. But that is not what Zeno's paradox is about. The first half, the one fourth that follows that, and all the other intervals that Zeno is referring to are geometrical parts that are present in the line whether anyone takes notice of them or not. In other words, they are already there. It's true that we could never mark off all of them, as it would take forever, but that's irrelevant.

A second way of arguing that a line does not contain an infinite number of intervals is to claim that the line is not infinitely divisible, not even potentially – that is, to claim that there are smallest parts that cannot be divided further even in principle. But that too is highly problematic. For then what one is claiming is that there are segments of a line that are not zero sized and yet that have no further parts. But if one of these spatial "atoms" isn't zero sized, that's because there is some distance between one end of it and the other – which in turn means the two ends must be distinct. And that contradicts the claim that it has no parts.

Finitists object to infinity by claiming that it leads to paradoxes. And yet what these examples show is that the rejection of infinity introduces paradoxes as well. One individual I debated online on this topic – a defender of the above potential infinity view – actually maintained that the two halves of a line do not exist until someone divides the line in half. Now, I admit that there are unsolved paradoxes of infinity. Philosophers and mathematicians haven't yet worked out all of the issues on the subject. But I find the existence of infinity paradoxes more acceptable precisely because they involve infinity – a concept that our minds have a hard time grasping. It's very easy for or intuitions to go wrong whenever the infinite is involved. I'm not willing to accept, however, that my intuition that the two halves of a line exist prior to our dividing the line could be wrong!


THE SOLUTION TO ZENO'S PARADOX


It is often claimed that Zeno's paradox is dissolved once we realize that the sum of the infinite series 1/2 + 1/4 + 1/8 + …. is 1. But that would be the solution only if Zeno's claim were that the distance from A to B must be infinite – and that doesn't seem to be what Zeno was claiming. Moreover, even if he was claiming that, it doesn't matter, since we can see a different problem – namely, that in order to reach B (never mind that it is clearly only a finite distance away), one must complete an infinite series. That is what appears impossible.

But it isn't. Above, I claimed that at least some infinite series can in fact be completed. For instance, given an infinite amount of time, an infinite number of events can occur. (My favorite example here is one that is credited to Wittgenstein: imagine coming across someone who says “...4, 3, 2, 1, 0. Finished!” and then explains he has been counting backwards from eternity.)

So some infinite series can be completed given infinite time in which to do so. But it doesn't necessarily take an infinite amount of time. Aristotle pointed out that, obviously, the time it takes to move from A to B can also be divided into its first half, its next fourth, and so on. Thus, there are an infinite number of subintervals of time in which to cover the infinite number of subintervals of length between A and B. The situation is therefore analogous to the one in which an infinite number of events occur in an infinite amount of time: in both cases, there is a time available for each event in question to take place.

Now, this at first may not seem sufficient to solve the paradox. We can imagine Zeno objecting that, just as there is a problem with completing the infinite series of intervals between A and B, there is a problem (given that the span involved contains an infinite number of temporal intervals) with the passage of time from moment T1 to moment T2. I myself made this objection in the past – as did Wesley Salmon (Space, Time & Motion, p. 36). However – and regardless of whether this is exactly what he had in mind – Aristotle's simple observation contains within it the solution to the problem.

We can see this by first keeping in mind that the infinite number of subdivisions (whether of temporal or spatial intervals) adds up to the span in question. That is (to consider the spatial case first), we already know that it is the distance between A and B that contains the infinite number of intervals we're discussing. So there is no question about there being a point B, only one about how one can arrive at it. Similarly, we know that it is the span T1 to T2 that has an infinite number of temporal intervals. So once again, there is no question about the existence of T2, only about how it ever can arrive. But now one can see that the above objection really makes no sense. Because for there to be a time T2 (that is not infinitely in the future) is the same thing as for it to eventually arrive! It is incoherent to claim that there is such a time and to also claim that it can never arrive. (Similarly, it would be incoherent to claim that there is a point B and to also claim that it is nowhere.) Therefore, Zeno cannot reasonably object that it is impossible for the time between T1 and T2 to transpire. It would be akin to the claim that the distance between A and B does not exist.

(I am not, incidentally, claiming that Zeno wouldn't say that. In fact, as a disciple of Parmenides, he would say precisely this about the distance – and might, faced with the above considerations, say the analogous thing regarding the time span. But none of this is relevant to the paradoxes, which take as their starting point the fact that there are distances and durations before attempting to show that such an assumption creates a problem.)

Given that T2 will eventually be here, it follows that there is in fact a temporal interval available for moving across each interval between A and B (all the way to and including B). The fact that both the number of line segments and the number of temporal intervals is infinite is no more problematic than the fact that in a universe without a beginning, the number of past events and the number of past moments are both infinite. In both situations, the availability of a time for each event to occur in is all that is needed to avoid paradox.
​

One final note: Above, I stated that one could never finish the task of dividing the distance between A and B into an infinite number of intervals. However, in principle, someone could in fact do so – just as we can, and do, go through all of these intervals in moving from A to B. The someone in question would simply need to perform the task at an ever increasing rate, e.g., by making the first division in 30 seconds, the second in 15, the third in 7.5, and so on, so as to complete the task in one minute flat. As Bertrand Russell once said, the impossibility of performing an infinite number of tasks in a finite amount of time is merely “medical.” ​

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 ACTUAL INFINITY AND THE INFINITE FUTURE

8/31/2016

5 Comments

 
No claim I've made has been criticized by the religious more than the claim that there are actual infinities. On the face of it, this is odd, given that God is supposedly infinite. But what motivates these critics is their belief that if no actual infinities exist, then the past cannot be infinite, and thus the universe had a beginning – which of course allows them to then more easily argue for the necessity of a creator.

These same people admit the infinity of the future, however – it's just that they don't see it as problematic. They believe that it doesn't represent an actual infinity, but only a potential one.

To understand this distinction, think of the process of counting. This process is potentially infinite – meaning it can in principle go on forever. However, no matter how long one counts, one will never actually reach infinity. Similarly, the future, it is said, is only potentially infinite: there will always be more tomorrows, but for all that the totality will always remain finite (reality never arrives at infinity), and so there never will be an actually infinite future.

An infinite past, on the other hand, is seen as problematic, because all of its yesterdays have already occurred – which means there have already been an actually infinite number of them.


This difference between the past and the future is highly intuitive – though interestingly, Aristotle, who came up with the actual/potential distinction, disagreed with it, as does mathematician and infinity denier James A. Lindsay (see his book Dot, Dot, Dot, pp. 52-53).

It's also true that the above difference cannot be maintained if one thinks of all of temporal reality as equally real, so that the future is every bit as much “there” as the present and past; the difference only works on the common sense view that time involves an actual coming into being of moments. But let's accept this common sense view (as maybe we should), and also leave aside the views of Aristotle and Lindsay (both of which I think can reasonably be denied). Let's grant, in other words, the difference between the past and the future in the above argument. Even in that case, I maintain, the infinite future presents us with an actual infinity, and therefore with a problem for infinity deniers.

Here's why. Although the future hasn't happened yet, it remains the case that there is a matter of fact regarding each and every single future event. Consider: if I say that tomorrow, x will happen, then my statement is either true or false. After all, either x will happen tomorrow or it will not. Even if the future is not predetermined, so that it has not yet been “decided” whether x will happen or not, it remains the case that it will either happen or not happen. Thus, my statement is either true or false. Moreover, if, as it turns out, x does happen, then my statement is true. (After all, in that case it certainly isn't false.) And the same thing applies to every tomorrow: no matter what future time you care to consider, there is a matter of fact regarding what will happen at that time. 

But this means that if the future is infinite, then there is an actual infinity of facts regarding what will happen. And what's worse – at least in that it makes the point more palpable – is that if you believe God knows all the facts about the future, then you must believe he knows an actual infinity of facts. (For more on the future and God's supposed knowledge of it, see my paper Is God's Foreknowledge Compatible with Free Will?, especially the addendum.)

This actual infinity of facts should be every bit as problematic as any other for infinity deniers. ​I conclude that infinity deniers who make the above argument are actually wrong.


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