In comments on an earlier thread, both Ocham and Brandon express perplexity over my defense of the impossibility of traversing an actually infinite past. Before responding, I must first elaborate a bit on the potential \/ actual infinite distinction. The distinction goes back to Aristotle and is pivotal in his response to Zeno’s paradoxes. Basically, what it amounts to is this.<\/p>\n
A potential infinite<\/span> is a collection or quantity that can increase without bound but is always actually finite. For example, I can begin counting 1, 2, 3, … and keep going, but I’ll never reach an ‘infinitieth’ number. At every point I am at some finite number. Similarly, I can take a line segment and divide it, thus yielding two line segments. I can then divide it again (3), and again (4), and so forth. At every point what I have is an actually finite number of line segments. The notion of a potential infinite is the one involved in the mathematical notion of a limit<\/span>. Thus, we say, for example, the limit of 1\/x as x approaches infinity<\/span> (i.e., increases without bound) is zero. We don’t evaluate the limit by setting x equal to infinity – 1 over infinity is undefined – but by imagining the value of x getting larger and larger, indefinitely. In sum then a potentially infinite collection or quantity is indefinitely large, but always actually finite<\/span>.<\/p>\n An actual infinite<\/span> (of a given order of magnitude), by contrast, is a collection or quantity that has a proper subset of equal magnitude as the parent set. Since Cantor, the standard way to determine whether two (denumerable) sets are equal in magnitude is to place them in 1-1 correspondence with each other. Thus, the set of natural numbers {1, 2, 3, …} is an actually infinite set because it has a proper subset (a set containing only, but not all, members of the parent set) that can be put into 1-1 correspondence with it. For example, the set of even numbers {2, 4, 6, …} is a proper subset of the set of natural numbers, and both sets can be mapped on to each other without remainder. Thus, 1 maps onto 2, 2 onto 4, 3 onto 6, etc. The important point to note here is that an actually infinite collection is given all at once<\/span> – it doesn’t start out as a finite collection and then, by finite addition, become<\/span> actually infinite. No, we posit actually infinite collections as infinite from the get-go. Thus, we refer to the set of natural numbers<\/span>, or the set of prime numbers<\/span>, and so forth, as completed totalities.<\/p>\n Now, one way of developing the kalam cosmological argument is as follows:<\/p>\n The rationale for the first premise is that no beginning to time means that before every state of world is another, different state. If only a finite number of events has elapsed, then there would be a state without a prior state. There would be a first state of the world, and a first event. By hypothesis, however, there is no first event; hence, the number of elapsed events must be infinite, not finite.<\/p>\n The rationale for the second premise is simply the idea that one can’t turn a potentially infinite magnitude into an actually infinite magnitude by finite addition. This seems obvious: I can keep counting till the cows come home and beyond, count faster and faster and faster, but I’ll never arrive at an infinitieth number. At every point in the process, the quantity is finite.<\/p>\n Now, Brandon objects:<\/p>\n If every traversal requires a beginning and an end, and an infinite past has no beginning, this is a problem only<\/i> if we already assume that traversal of an infinite past would require traversal of infinite days. But on the infinite past view, every day in the past is finitely distant from the present; it’s just that for every finitely distant day there’s a day that is more distant. Thus this is true: For every day in the past, traversal of the days from that day to today is traversal of a finite number of days.<\/i><\/b> The fact that there are infinite such days doesn’t change this. This is true just as much as it is true that the fact that every integer is a finite distant from 1 is not affected by the fact that there are infinite integers. The crucial assumption of this argument is this:<\/p>\n (A) Every day in an infinite past is finitely distant from the present.<\/p>\n The problem with this assumption is that it conflates the distinction between potential and actual infinites and thereby fails to take the idea of an actually infinite past seriously<\/span>. (A) is clearly true when we’re talking about a potential infinite. We start at the present and run through the time series in reverse, moving farther and farther into the past. Nevertheless, at any point we stop at, we’re only a finite <\/span>remove from the present. But if the distance from past event E to the present is actually finite<\/span>, then we haven’t yet captured the idea of an actually infinite past.<\/p>\n Similarly, the notion of ever larger integers being still a finite remove from 1 is that of a potential infinite, of a magnitude increasing without bound, not of an actual infinite.<\/p>\n The problem with an actually infinite past is that it requires us to posit the impossible, some event in the past that is at an actually infinite<\/span> remove from the present. The proper analogy is not one of starting in the present and then receding ever further into the past, but of something like trying to count up from negative infinity to zero. It can’t be done. And that’s the point.<\/p>\n","protected":false},"excerpt":{"rendered":" In comments on an earlier thread, both Ocham and Brandon express perplexity over my defense of the impossibility of traversing an actually infinite past. Before responding, I must first elaborate a bit on the potential \/ actual infinite distinction. The distinction goes back to Aristotle and is pivotal in his response to Zeno’s paradoxes. Basically,\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-148","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts\/148","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/comments?post=148"}],"version-history":[{"count":0,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts\/148\/revisions"}],"wp:attachment":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/media?parent=148"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/categories?post=148"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/tags?post=148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
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