“These arguments against an actual infinity, however, are all based on a confusion between two notions of “bigger than.” One notion is numerical: a set is bigger than another if it has a greater number of members. The other notion is in terms of part-to-whole relations: a whole is bigger than any proper part. When dealing with finite quantities, anything that is bigger in the part-to-whole sense is also bigger in the numerical sense. But this is not so in the case of infinite quantities. Although in the part-to-whole sense there are more people in the hotel after a new guest arrives and there are more members of the original series of events, in the numerical sense there are not. Indeed, mathematicians take the failure of the part-to-whole sense of “bigger than” to imply the numerical sense to be the defining feature of infinity.” - Martin, Michael. The Cambridge Companion to Atheism. New York: Cambridge UP, 2007. p. 93
Perplexity: Here’s the problem with the above account: the specific difference that determines or measures multitude into, say, the number ten is the final unit in this number. (
Comm. on the Met., Bk. VIII, les. 3, n. 1725) Varying the final unit in any such way varies the specific difference, i.e, the
species of any number, i.e., all numbers. And, when we vary the final unit, whether in a “set” or in a series of real numbers, we are changing its species, its
specific difference. (This is in no way a confounding of “bigger or smaller.” Mr. Martin’s contention is utter Non-sense.)
IOW, we would then be allowed to equivocate the word, “infinity” (or “infinite”), with each and every new modification, or speciation, that it was supposed to represent. Can you imagine what such an indefinitely changing set of rules would do to the exigency of
understanding? There would definitely be no more need for communication. What we would end up with then is on-going gibberish.
We have things that waiver from this to that, such as wave lengths, the crests and valleys of frequencies, but these things “exist,” i.e., they exist in mediums. On the other hand, number does not have this kind of existence. Number has only immaterial existence. As St. Thomas defines mathematical abstraction:
“The mathematician, scientific in Aristotle’s sense, makes two abstractions: one, to go from the sensible world to the order of individual intelligible matter; the other, to go to the universal or common consideration of intelligible matter itself.” -
Com. on the Trin., q. 5, a. 3
What St. Thomas means by his use of the word, “intelligible,” is precisely, “imaginable” - nothing more and nothing less. So, let’s not get wrapped around that axle.
It seems to me then, what Craig is doing is holding infinite sums to the same expectations he holds finite sums to.
Of course. If not, then finite mathematics is also absurd.
No wonder he thinks this is absurd. But, when you stand the argument on its head and say it’s that oddity which is a defining feature of infinity, it’s not that surprising.
Likewise; standing your argument on its head as the defining attribute of ‘infinity’ is not merely fiction, it is infinitely more Non-sense.
Bill Craig, in The Blackwell Companion to Natural Theology, p. 118, objects to an infinite past by saying:
“In order for us to have “arrived” at today, temporal existence has, so to speak, traversed an infinite number of prior events.”
He here references The Cambridge Companion to Atheism, citing a response to this by Richard Gale (a response I find quite convincing):
“Richard Gale protests, “This argument depends on an anthropomorphic sense of ‘going through’ a set. The universe does not go through a set of events in the sense of planning which to go through first, in order to get through the second, and so on” (Gale 2007, pp. 92–3). Of course not; but on an A-Theory of time, the universe does endure through successive intervals of time. It arrives at its present event-state only by enduring through a series of prior event-states. Gale’s framing the argument in terms of a “set of events” is maladroit since we are not talking about a set but about a series of events which elapse one after another.”
I think Gale states what I was getting at. On B-theory, there is ontological parity between all temporal events (i.e., it’s not like time is actually moving from future to present to past). The only “series” of events on B-theory is our experience of time, but there is no such temporal progression in reality. So, Gale and I accuse this argument of depending on an anthropomorphic sense of ‘going through.’ We needn’t speak of sets, so Craig’s last remark is more of a quibble. But as you’ll note, Craig doesn’t actually dispute this, in fact he says “Of course not.”
This is unequivocally (even) more Non-sense. If time were not an asymmetrically advancing aggregating exigency, then we would be witness to all sorts of contradictory states! You need to think this through a bit more, I think. There must be a “temporal progression to reality.” If not, then my old dog would not be dead, he’d be alive and dead!

I must say that I’m very happy that this is not the case, and truly do not mind such a conception being called, “anthropomorphic.”
God bless,
jd