Moderated Infinity!!!

[LarryS]

Looking back at the opening post, I realize that its first line has multiple possible interpretations.




In my first reading of it, I construed "Christian" as a noun (which should have been plural) giving the meaning: Christians (all of whom are apologists and casuists) and other non-Christian apologists and casuists.... However, if I accepts "Christian" as an adjective, I get: Christian apologists and Christian casuists as well as non-Christian apologists and non-Christian casuists....

Other variations are possible as well. There is also a question of whether it is all of the sentence subjects that "keep telling us", just some, most, or maybe only one.

If the opening poster wouldn't mind clarifying his original intent, I would be grateful.

and getting back to the OP, don't most folks (especially atheists) think that infinity is a concept and not found in reality?

 

[Myriad]

and getting back to the OP, don't most folks (especially atheists) think that infinity is a concept and not found in reality?

There are a bunch of different concepts. In earlier posts I thought I could distinguish infinity as an abstract mathematical concept from "actual infinity" but it turns out "actual infinity" is also an abstract mathematical concept already in use. The difference in mathematical concepts is exemplified by the classic troll-on-troll argument over whether 0.99999... infinitely repeating "really" equals 1. Actual infinity says it does, while the counter-argument "potential infinity" regards the addition of 9's as an iterative process that is never complete and only has a finite (but indeterminate) number of 9's upon any examination. Actual infinity has been the preferred mathematical interpretation since Cantor re-centered the foundations of mathematics on set theory. For another example, infinite ratios count as actual infinities but they're still just abstractions. A vertical line segment is still just a line segment, and doesn't break geometry, even though its "slope" is an infinite ratio.

Computing theory appears to retain implicit potential infinity. For instance the tape of a Turing machine is defined as infinite in length, but at no point in the operation of any Turing machine, halting or not, is an infinite amount of tape required. The tape, like the number of steps in a computation, is unlimited but only "potentially" infinite.

Aversion to actual infinities in mathematics, in earlier eras, is apparently the reason calculus is taught as the analysis of limit processes, keeping all the infinities safely "potential," instead of the analysis of infinitesimals which implicitly embraces actual infinities right off the bat. Either way leads to the same results though. Some potential infinites can turn into actual infinites using hacks like "Suppose each 9 added, or each operation of the Turning machine, took half the amount of time as the previous one. Then at some future time, an "actually" infinite number of 9's/steps would have occurred..." This is probably another reason why actual infinity predominates in present-day philosophy of mathematics, outside of the introductory calculus curriculum.

In any case, neither of these mathematical concepts match what I originally wanted to call "actual infinity" in this thread, which I suppose I could call "material infinity" instead. That would be, either an infinite number of distinct material objects or distinct events, or an infinite extent of some usually measurable quantity like time or distance or mass, actually existing in reality.

Zeno's paradoxes require two assumptions to be actual paradoxes. The first is that the actual infinity inherent in the abstract description of a phenomenon e.g. an arrow's flight (half the distance, then half the remaining distance, then half the remaining distance, and on) implies a material infinity of distinct events occurring during the flight of an actual arrow. The second is that, if so, such a material infinity is impossible. As far as I know, the first question, which boils down to whether or not time in reality (as opposed to e.g. in the equations of our physics models) is infinitely divisible, is still open scientifically. If time has quanta (e.g. physically real Planck units) then there can only be a finite number of distinct events in the arrow's flight and the second question is irrelevant. The interesting implication here, though, is that the impossibility of a material infinity was assumed from the start, or else there's no paradox. So we can count Zeno, in the third century BCE, as among the "apologists and casuists [who] keep telling us that [some definition of] infinity is a nonsense concept that is not in reality."

As far as questions about material infinities are concerned, such as whether time is infinitely divisible or whether our observable universe might exist within a larger universe that's of infinite size, the trend in science has been against material infinity so far. To the point where we interpret singularities where our models predict some materially infinite quantity as peculiarities or weaknesses in our models ("we can't use the ratio of vertical to horizontal to model the steepness of gradients when the trajectory is vertical") rather than as actual material infinities ("OMG a vertical cliff breaks the universe!") Back when we modeled electrical charges as points, F = k * q1 * q2 / d2 predicted infinite force when the charges come in contact, that merely pointed out some limitations of that model and helped guide us toward particle physics and quantum mechanics.

Thus, there is a rational basis for doubting whether material infinities are possible. Such doubt in no way claims or implies that potential or actual infinities in mathematics are "nonsense."

Where Christian apology intersects these questions, as far as I can tell, is in the argument over whether or not there must be a "first cause." That argument is antiquated. Our current scientific models place the Big Bang as the first knowable cause, the first event for which information about it can exist in our universe, and that besides that, we have no idea. No one any more is seriously arguing that the Kalam argument is invalid because the universe has already existed for a materially infinite length of time or that a materially infinite quantity of past causes might have occurred. So there's not much use or relevance for any "material infinity is impossible" counter-argument on the apologists' side either.

What Christian apologists can and sometimes do argue is that God is the unknowable cause of the Big Bang. It kind of fits with "Let there be light!" but it also kind of doesn't because by that account the Earth supposedly already existed at that time. Nothing there to do with potential/actual/material infinities, of course.

 

[Myriad]

I noticed an interesting example for illustrating actual versus potential infinities, and some of the complexities of describing them: space-filling curves. Space-filling curves are defined as the limits of iterative processes by which each segment of a multi-segment curve on a plane is replaced by a smaller copy (usually rotated and often flipped) of the entire original curve. In the limit, after infinite iterations, the curve includes every point on a plane region.

If you want to describe this in terms of potential infinity, as a limit that is approached but never actually reached, you can reach conflicting descriptions. If you measure the distance between any arbitrary point on the plane region, and the nearest point of the space-filling curve, that distance decreases with each iteration such that it's valid to say it approaches zero. (For an infinitesimal fraction of the points, those with rational coordinates, the distance reaches exactly zero at some finite number of iterations, but what matters here is all the other points, for which the distance gets arbitrarily close to zero but doesn't reach it in any finite number of iterations.)

But if you measure the fraction of the points of the plane region that are on the curve after each iteration, that remains precisely zero after any finite number of iterations, so it's not at all obvious that it will become 1 after an infinite number of iterations. There doesn't seem to be any "approaching" that limit. That the space-filling curve really fills the plane region completely is not apparent as a potential infinity. It only appears in the actual-infinity outcome of an actually infinite number of iterations.

That kind of reminds me of supertask paradoxes, where for instance you start with {1} and iterate the process of adding ten new higher integers to the set, while removing the lowest integer from the set. After one iteration the set would contain {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The set gets larger after every iteration but after an actual infinite number of iterations it must be empty, because every integer n is removed at iteration n.