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.”
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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