Moreland offers an Aristotelian solution to one of Zeno's paradoxes as the basis of an argument against a beginningless past. Moreland sets up Zeno's Dichotomy paradox as follows:

"...Consider a runner who begins at some point A and who wishes to reach the midpoint between A and B. But before he can reach this midpoint, he must reach the midpoint of the midpoint. In order to move from any point to any other point, a runner must traverse an infinite number of points and this is impossible. Thus, [concludes Zeno] motion is an illusion."[1]

Moreland then argues that a structurally identical paradox applies to the hypothesis of a beginningless universe: he argues that if the past were beginningless, then the prospects of traversing all the events of the past to reach the present moment would be like those of Zeno's runner on the assumption that his task involved the traversal of an actual infinite: one couldn't even begin such a task, much less finish it.[2]

So Moreland thinks Zeno's Dichotomy paradox and his paradox for a beginningless past are structurally similar. His next step is to argue for a solution to the former, and then reason that, by analogy, the solution to the latter is thus similar. Thus, he argues that the most plausible solution to Zeno's Dichotomy paradox is to distinguish between an actual and a potential infinite, and to assert that the racer's task only involves the traversal of a potential infinite. And since all spatial distances that are merely potentially infinite are traversable in principle, the racer can traverse the whole track.[3] Similarly, the set of temporal distances in the universe's past is potentially infinite only, and thus finite. But that solution entails that the universe had a beginning. Therefore, thinks Moreland, the finitude of the past is justified.

What to make of this argument? One sort of worry about it is that it's not clear that the two paradoxes are sufficiently relevantly similar to justify Moreland's conclusion that their solutions are similar. For the runner's traversing task has a beginning or starting point; not so for a beginningless past. And the worry is that the feature that generates the problem in Zeno's Dichotomy parodox -- i.e., that the runner must start a task that has no starting point -- doesn't necessarily apply to a beginningless past. In other words, if the requirement of a start is merely a feature of Zeno's thought experiment, and not an essential property of traversals in general, then the grounds for thinking Zeno's runner's task is impossible do not provide grounds for thinking that traversing a beginningless past is impossible.

Now of course one might reply that it is an essential property of all traversals that they have a starting point. But the problem is that that's the very issue in dispute. For it's part of the very concept of a beginningless past that it involves traversing an infinite without a starting point. Therefore, whether or not such traversals are impossible, one cannot just assert the impossibility of a traversal that lacks a starting point without begging the question against the antecedently unconvinced.

By this time I'm no doubt belaboring the point, but here's a slightly more developed variation on the same worry: Roughly, the problem with the runner's task in Zeno's Dichotomy paradox is generated by the following inconsistent set of propositions.

1. The racetrack consists in a particular infinite open interval of spatial distances (IOISD).

2. The runner can traverse IOISD from the direction of the open "end".

3. The runner must start his traversal of IOISD if he is to engage in it at all.

4. All Infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

For given propositions (1) and (3)-(5), Zeno's runner is required to start the traversal if he is to engage in it at all. But given that there is no starting point or beginning to the open "end" of the interval, it follows that the runner can never finish his traversal, on the grounds that he can never begin. He will thus forever remain "outside" of, or "external" to, the segment represented by the negative integers. And if that's right, then (2) is false, i.e., the runner cannot traverse IOISD from the open "end".

Now, at first glance, Moreland's paradox about a beginningless past looks sufficiently similar in structure to warrant a similar solution:

1'. The past consists in a particular infinite open interval of temporal distances (IOITD).

2'. IOITD can be traversed from the direction of the open "end"

3'. The traversal of IOITD must have a start if the traversal is to occur at all.

4. All infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

However, at second glance, there are reasons for doubting that the runner's predicament in Zeno's Dichotomy paradox is sufficiently analogous to the case of traversing a beginningless past. For the reason why we're supposed to accept (3) in the former paradox -- i.e., that the runner's task must have a starting point -- is that Zeno stipulated that it have one. Now Moreland may be right that all traversals require a starting point. However, when it comes to the basis for accepting the parallel (3') in the latter paradox, something on the order of a mere stipulative ban on traversals without starting points is not going to cut it for those antecedently unconvinced of a finite past; they need a reason to think that it belongs to the very nature of a traversal that it has a starting point. But as we've seen, Moreland has so far failed to offer such a reason. Pending such a reason, then, traversals without beginnings remain epistemic possibilities for the unconvinced.

I conclude, then, that Moreland's use of Zeno's Dichotomy paradox fails to justify the claim that the past is necessarily finite.

===================

[1] Scaling the Secular City, p. 30.

[2] Ibid.

[3] Ibid.

"...Consider a runner who begins at some point A and who wishes to reach the midpoint between A and B. But before he can reach this midpoint, he must reach the midpoint of the midpoint. In order to move from any point to any other point, a runner must traverse an infinite number of points and this is impossible. Thus, [concludes Zeno] motion is an illusion."[1]

Moreland then argues that a structurally identical paradox applies to the hypothesis of a beginningless universe: he argues that if the past were beginningless, then the prospects of traversing all the events of the past to reach the present moment would be like those of Zeno's runner on the assumption that his task involved the traversal of an actual infinite: one couldn't even begin such a task, much less finish it.[2]

So Moreland thinks Zeno's Dichotomy paradox and his paradox for a beginningless past are structurally similar. His next step is to argue for a solution to the former, and then reason that, by analogy, the solution to the latter is thus similar. Thus, he argues that the most plausible solution to Zeno's Dichotomy paradox is to distinguish between an actual and a potential infinite, and to assert that the racer's task only involves the traversal of a potential infinite. And since all spatial distances that are merely potentially infinite are traversable in principle, the racer can traverse the whole track.[3] Similarly, the set of temporal distances in the universe's past is potentially infinite only, and thus finite. But that solution entails that the universe had a beginning. Therefore, thinks Moreland, the finitude of the past is justified.

What to make of this argument? One sort of worry about it is that it's not clear that the two paradoxes are sufficiently relevantly similar to justify Moreland's conclusion that their solutions are similar. For the runner's traversing task has a beginning or starting point; not so for a beginningless past. And the worry is that the feature that generates the problem in Zeno's Dichotomy parodox -- i.e., that the runner must start a task that has no starting point -- doesn't necessarily apply to a beginningless past. In other words, if the requirement of a start is merely a feature of Zeno's thought experiment, and not an essential property of traversals in general, then the grounds for thinking Zeno's runner's task is impossible do not provide grounds for thinking that traversing a beginningless past is impossible.

Now of course one might reply that it is an essential property of all traversals that they have a starting point. But the problem is that that's the very issue in dispute. For it's part of the very concept of a beginningless past that it involves traversing an infinite without a starting point. Therefore, whether or not such traversals are impossible, one cannot just assert the impossibility of a traversal that lacks a starting point without begging the question against the antecedently unconvinced.

By this time I'm no doubt belaboring the point, but here's a slightly more developed variation on the same worry: Roughly, the problem with the runner's task in Zeno's Dichotomy paradox is generated by the following inconsistent set of propositions.

1. The racetrack consists in a particular infinite open interval of spatial distances (IOISD).

2. The runner can traverse IOISD from the direction of the open "end".

3. The runner must start his traversal of IOISD if he is to engage in it at all.

4. All Infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

For given propositions (1) and (3)-(5), Zeno's runner is required to start the traversal if he is to engage in it at all. But given that there is no starting point or beginning to the open "end" of the interval, it follows that the runner can never finish his traversal, on the grounds that he can never begin. He will thus forever remain "outside" of, or "external" to, the segment represented by the negative integers. And if that's right, then (2) is false, i.e., the runner cannot traverse IOISD from the open "end".

Now, at first glance, Moreland's paradox about a beginningless past looks sufficiently similar in structure to warrant a similar solution:

1'. The past consists in a particular infinite open interval of temporal distances (IOITD).

2'. IOITD can be traversed from the direction of the open "end"

3'. The traversal of IOITD must have a start if the traversal is to occur at all.

4. All infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

However, at second glance, there are reasons for doubting that the runner's predicament in Zeno's Dichotomy paradox is sufficiently analogous to the case of traversing a beginningless past. For the reason why we're supposed to accept (3) in the former paradox -- i.e., that the runner's task must have a starting point -- is that Zeno stipulated that it have one. Now Moreland may be right that all traversals require a starting point. However, when it comes to the basis for accepting the parallel (3') in the latter paradox, something on the order of a mere stipulative ban on traversals without starting points is not going to cut it for those antecedently unconvinced of a finite past; they need a reason to think that it belongs to the very nature of a traversal that it has a starting point. But as we've seen, Moreland has so far failed to offer such a reason. Pending such a reason, then, traversals without beginnings remain epistemic possibilities for the unconvinced.

I conclude, then, that Moreland's use of Zeno's Dichotomy paradox fails to justify the claim that the past is necessarily finite.

===================

[1] Scaling the Secular City, p. 30.

[2] Ibid.

[3] Ibid.

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