Originally posted by DeepThought
I don't agree with the first part of that but do the second, if I've understood what you are saying. My problem with the first part is that as elements of the set the points at infinity are specific extended real numbers, with the admittedly odd property of being infinitely far from any specific element of the real subset. I think the argument in the s ...[text shortened]... nt at minus infinity is to add a beginning so the procedure defeats the purpose of the argument.
Perhaps I was not clear, but I am not disputing that you can specify two members of the extended reals that yield the result you want (say, minus-infinity and zero). Rather, my point is that in the context of the original argument, it is probably excusable on the part of the objector if he takes this to be some sort of ad hoc stipulation.
To the extent that the number line is here a useful analog for the timeline, the objector's initial rebuttal is clear: the original argument implies that one can specify two times on the timeline that are infinitely remote; yet, one cannot specify two numbers on the number line that are infinitely remote. If you respond in turn with the extended number line, then this is the sort of dialectic symmetry I see:
Objector's response: "But minus-infinity is not a member of the number line; it's not a number. It's just some sort of nebulous element that you have stipulatively adjoined to the number line."
Is analogous to:
Objector’s response: “But that starting point is not a member of the timeline; it's not a time. It's just some sort of nebulous element that you have stipulatively adjoined to the time line."
Of course, I take it that the objector is sort of missing your point: your point is that perhaps time is analogous to the extended real number line, not the real number line; whereas the objector is stubbornly insisting on an analog with the real number line, which sort of begs the question. At some point I guess this results in a clash of different intuitions about the nature of infinity. But, I would think that moving beyond an analog of the number line to one of the extended number line can be viewed as an argumentative sleight of hand. It is a stipulation that just happens to yield the result you want and in a particularly robust way: the introduced adjoining element minus-infinity stands infinitely remote from any member of the real number line. So, it amounts to stipulating into being a starting point to the timeline that, by definition, it is impossible to move away from without having to require an infinite amount of traversal. Whether the analog is apt or not, it has the looks of a cheap argumentative trick, is my point.
EDIT: I forgot to add something else, as well. The original argument puts forth the idea that what is problematic is the idea of traversing an infinite amount of time. Then, prima facie, the proponent of the original argument cannot adopt your analog between time and the extended number line. After all, as discussed above, the analog amounts to inserting a beginning point of the timeline, from which it is impossible to move away from without traversing an infinite amount of time. So, to the extent that the proponent of the original argument thinks traversing an infinite amount is problematic, then this analog is also problematic for him or her. Otherwise, what does his view of time amount to? Just being perpetually stuck at minus-infinity?
I suppose this point relates to what you mean when you say that adding a beginning defeats the original purpose of the argument.