Well, begbert2, my point was, you believe that 1/2 a second exists, 1/3 a second exists, 1/4 a second exists, 1/5 a second exists, etc., without limit. (The term we choose to use to label this fact does not particularly matter)
But could not someone else, employing your same form of argument, say “No. You are wrong. We can ask the question ‘How many points in time were contained in the last second?’ and receive a particular finite integer as an answer. It’s all right there in the pile; count it up. Clearly, there is some limit to the ability to dividing time into smaller and smaller pieces, since, if the number of things in the pile is, say, 907, then we cannot divide a divide a second into 908 pieces, each with its own particular distinct starting point in time; there won’t be any such thing as 1/908 of a second.”
I consider your argument form broken, and I am hoping that such examples can illustrate why. It can be used to support conclusions which even you would not accept.
Begbert, do you believe that it is impossible to have anything that is infinite in the real world? Is the idea of space extending infinitely a problem in your mind also? Or is it just time?
If you have a problem with anything being boundless, then it makes sense why you would include time in that also, but if you are making a special case of time then I don’t understand why you would do that.
Er, my last post seems to have been infused with some editing errors. Here, let me try that again:
Well, begbert2, my point was, you believe that 1/2 a second exists, 1/3 a second exists, 1/4 a second exists, 1/5 a second exists, etc., without limit. (The term we choose to use to label this fact does not particularly matter; I think you may have been perceiving me as making an argument other than the one I intended to. So, I apologize for using the term “infinitely divisible”, and will accept your alternative of “divisible without limit” or “divisible into as many pieces as you like”)
But could not someone else, employing the same form of argument as you did regarding the finiteness of the past, say “No. You are wrong about the divisibility of time. We can ask the question ‘How many points in time were contained in the last second?’ and receive a particular finite integer as an answer. It’s all right there in the pile of moments which have elapsed in history; count it up. Clearly, there is some limit to the ability to divide time into smaller and smaller pieces, since, if the number of things in the pile is, say, 907, then we cannot divide a second into 908 pieces, each with its own particular distinct starting point in time; there won’t be any such thing as 1/908 of a second.”
I consider your argument form broken, and I am hoping that such examples can illustrate why. It can be used to support conclusions which even you would not accept. Now, I am guessing that you will tell us that my use of your argument form is flawed, but I am also guessing that no one here, other than you, can predict on what grounds you will dismiss it. It’s hard to know what to make of such a style of argument; surely, if it were convincing in virtue of its mere logical form, we would not need to ask you of each potential application whether it would be kosher or not.
“So. Looking at the universe, I can ask the question ‘How many seconds have passed total?’ And, ‘an infinite number of seconds have passed’ is not a valid answer - that’s not a number!”
If you’re asking how many seconds have passed, it’s implicit that you’re asking how many seconds have passed since something, which is the beginning of your timeline. Then you take the idea that this “something” was when it all began, and logically reason that it had to have a beginning. Well, duh.
That’s why everybody here is pointing out how your argument is begging the question - your statements implicitly assume that there was a beginning, then you derive that there was a beginning! If there was no beginning, then it doesn’t even make sense to ask your question about how many seconds have passed.
There’s a difference between being boundless and being infinite in quantity. For example, the number line is boundless, but despite that, nobody has counted up to infinity, and nobody has an infinite amount of shoes in their closet. (Not even zero-sized shoes!)
It is of course the case that I’m drawing a distinction between ‘time’ and ‘potential time’; in that one is the total span in which anything has occurred or could be known about, and the other is the abstract direction into which the timeline of events could have extended, were the timeline longer. If I was not drawing such a distinction, of course, the idea of the future timeline “growing” would be ridiculous.
I have no problem with “infinite” (that, is, boundless) empty extents existing beyond the limits of all actualized time or matter. Of course, this would have be be actual empty extents, not just places we haven’t been to. So far as I know, space exists is one of those extents; the limit of the actual space in the universe is (I gather) an expanding edge equidistant from the point of the big bang and expanding at the speed of light; the ‘potential space’ into which it is expanding in all directions is without limit. Presumably, assuming it’s not fixed in length and we’re just running along it, time is extending similarly in the positive direction - but causality argues that it is not extending similarly in the negative direction. As noted, this is subject to debate.
Of course, the existence of matter is limited to actual time - and no piece of matter is infinity years old, because it could count the seconds of its existence and that amount must be finite. (It doesn’t matter if it didn’t count the seconds; that it could have is enough.) Similarly, if and god exists, it could only be aware of the events in the finite range of actualized time; It cannot not know how many seconds it knows about (presuming it’s not forgetful of course, which seems a reasonable assumption).
I suppose some people will take this as some kind of backing down or turning around of my position, if they have been misunderstanding my position the whole time. If they do, that’s their problem.
My argument to you has simplified to “There’s a difference between heading upwards and getting there; the first is possible, the second is nonsense.”
Your little analogy is flawed in that division size is arbitrary, but seconds are not. Because division size arbitrary you can ‘bait and switch it’ without changing the total; you can say (like you did) the number of items in the pile is 907 - oh wait, now that I’ve said that, it’s not 908 - ah ha, gotcha!
I honestly don’t care to assert how many seconds have passed. (Science my have an opinion, but that’s another matter.) But however many have passed, that many have passed; unless the amount is changing of course. The ‘actualized’ timeline (that is, the timeline where any history or events or existence of any sort takes place, as opposed to potential time (described above)) could easily be growing the the forward direction; one reasonable assumption would be that the ‘present’ is the leading edge, advancing forward through ‘potential time’ and growing the timeline at one second per second. And, I suppose, the back end of time could be growing too, though you’d have to argue around causality to assert that.
But stating that it has already grown to infinity is like saying that you went up so far that you reached “upinity”. No matter how far back the timeline extends, or if it’s growing or how fast it’s growing, it does not already span back over an infinite number of years of history. That’s patent nonsense.
By the way, you misrepresented the form of my argument. Here it is with corrections, and transferred to an applicable ‘number-line-like’ analogy, which is not a false analogy like subdividing numbers.
“No. You are wrong. We can ask the question ‘How many feet of altitude have been achieved?’ and receive a particular finite integer as an answer. It’s all right there in the contrail; count it up. Clearly, there is [NOT necessarily] some limit to the ability to fly upward, [of course, but no matter how far up you’ve gone, you’ve still only gone some finite portion of the way, and have not managed to transcend distance and measurement itself in some magical fantasy way].”
I wasn’t “changing the division size with a bait-and-switch”; I wasn’t asking “How many 1/907 second intervals lie within a second?”, I was asking “How many particular points in time lie within a second?”. [There’s the point in time at 0 seconds, and at 1 second, and at 1/2 a second, and at pi/4 seconds, and at sqrt(2)/2 seconds, and so forth; each of these is a point, not an interval. How many such points are there within a second?]. And my only intention in doing so was to illustrate the glibness of “Every quantity in the universe is finite, and this includes, in particular, every measure of the past”
Can the universe be spatially infinite? If so, how did it grow to such an infinite size? Or has it perhaps always been spatially infinite? If a quantity is capable of having always been infinite, then why not the pile of past seconds?
Let us be more clear about your premises and your conclusion. You outlined an argument above which I think ran like this:
Premise 1: The amount of time which lies in the past cannot be infinite
Premise 2: If there is no beginning of time, then the amount of time which lies in the past must be infinite
Conclusion: There must be a beginning of time
Is this correct? But why would any of us who don’t already agree with you accept premise 1? Particularly, which is really the only matter of debate, why would we take it as logically necessary? Particularly when there are so many other quantities in the universe which are logically capable of being infinite (e.g., the number of points on a ruler, the number of points in a second of time, the total volume of the universe, etc.).
Here is my position: whatsoever is capable of being mathematically modelled is a live possibility as far as pure logic is concerned; a logical inconsistency must also be a mathematical inconsistency, so to speak. If an argument is logically valid, then any mathematical model satisfying its premises must also satisfy its conclusions; the existence of a single counterexample model demonstrates the invalidity of the argument.
As concerns this discussion, I find it bizarre that anyone would disagree.
(Of course, if one of your premises is “time does not extend infinitely into the past”, then you will indeed be able to prove from this that time can only extend finitely into the past, and the mathematical models won’t quibble with you. But then, your argument does little useful work either; no one who disagrees with you would consider that an obvious premise. If you want to make the argument more useful, you would have to establish that very premise under debate, using other, more agreeable premises)
I don’t think begbert2 will agree with this - he doesn’t realize that your Premise 1 is one of his premises. The way I see it, his starting premise is “time had a beginning reference point.” His point about modeling the passage of time with a clock that’s been running since the universe began illustrates that. If a clock has been running since the universe began, then the universe had a beginning.
I think the way to get a breakthrough is not with the subdividing-time sidetrack, which I didn’t understand the point of, but by getting him to realize that infinite future time (which he already accepts) is just as logically possible as infinite past time.
Ah, but he will never agree to that equivalence. Only he alone knows when “All questions must be answerable with particular finite numbers” applies and when it doesn’t. (My point with the subdividing time sidetrack was to show that even he believes that some questions about quantities measuring the past do not have particular finite numbers as answers)
And now, the post I was typing as you said that, addressed to begbert2:
Bah, it doesn’t matter. We’re both just wasting our time; we’ll never make headway, not on what is logically demanded of temporal structure, not on what mathematicians are willing to countenance as numbers, not on the legitimacy of discussion of the infinite, not on anything.
I find joy in mathematical possibilities, including the ways that logic is able to analyze the infinite without reducing to trivial rejection. But you have some superstition regarding the infinite and have convinced yourself that logic shares your bias; in so doing, you have blinded yourself to potentially fascinating subjects of study (and I do not refer here primarily to “Set Theory”, for which, you may be surprised, I share some of your antipathy). You are missing out on some areas of great beauty. C’est la vie.
Sorry; was that a bit getting up on the cross? I got embarrassingly frustrated. The main point was “You have some superstition regarding the infinite and have convinced yourself that logic shares your bias”; logic does not, in the same way that mathematics does not. You can ignore the rest of that post if you like.
The way I see it (“it” of course, being my argument, which I’m sure means that you’re more of an authority on it than I) is that my starting premise is that that the definition of infinity rather explicitly is not a quantity, it is basically the concept “there is no limit*”. That is, there is no limit to the number of unique values you can find between 0 and 1. There is no limit to the number of positive numbers. There is no limit to the number of negative numbers. There is no limit to the length of a (geometric mathematically-modeled construct known as a) line.
Of course, a key fact about infinities is that you can never get through an infinity of things from one end to the to the other if the things you have an infinity of are of a constant non-zero length of time. Like, you know, seconds.
That is, anything (like a universe) that tries to sit through an infinity of anything will never finish.
You lot are claiming the universe has sat through an eternity of time and finished. It’s fascinating how that doesn’t seem to bother you.
*There are of course other ways for there to be no limit than to trend to infinity; this just seemed better than interrupting the sentence with “there is no limit and the tendency as you go onward is for things to get bigger”.
But there isn’t an infinite amount of future timetime stretched out ahead of us either. There may be infinite (that is, unlimited) potential for growth of the timeline, but then, I allow that to the past direction as well, if somebody can explain how to reconcile negative-directional time growth with causality.
And here I was thinking that you clearly have some strange mental block that keeps you from even accepting that the terms “infinite” and “infinity” have definitions, and those definitions are not that they’re quantities; they’re a failure to resolve to a quantity. “Infinity is so a number”, and all that.
It’s the math that kills your argument - specifically the definition of the mathematical concept ‘infinity’. It’s not the logic. Logic just organizes the proof. As you can’t even nail down the crux of our disagreement, I don’t see how you’re likely to correct in your ad hominem psychoanalysis, either.
I apologize for my ad hominem psychoanalysis. But I am familiar with “the definition[s] of the mathematical concept ‘infinity’”, and “to be infinite” does not generally simply mean “to fail to resolve to having a quantity” (and thus to fail to exist, so that we could conclude nontrivial facts about the universe simply from such a definitional investigation). Certainly, to be infinite is to fail to resolve to having a finite quantity, but that is of no great matter.
I don’t believe you ever mentioned whether you believe it is logically necessary that the universe be spatially finite, but I would be interested to know. It is unclear to me whether your position is that “There is no such thing as an infinite quantity” applies to this or not. I imagine you probably feel the universe must be spatially finite, the same way you probably feel the universe must have only a finite number of hydrogen molecules in it, and so forth, simply because the alternatives are illegitimate quantities. But I don’t want to make assumptions without checking first.
Right. Similarly, I think that the timeline is of finite length at any given ‘meta-time’ (note that it might stay a constant length, partially removing the need for ‘metatime’), and may or may not be growing into the ‘empty timespace’ preceding and following it.
Well, the argument in the OP fell apart rather quickly, as it was based in an assumption about atheists that turned out to be pretty much wrong. (And an assumed causal link between the universe starting and God that was not universally accepted.)
I guess what it comes down to is that you hold the following positions:
We should distinguish between the “potential infinite” (finite at any instance, but capable of growing with no fixed upper bound) and the “actual infinite” (a completed infinity already present as such; something like that)
While quantities in the universe may be examples of the potential infinite (as illustrated by, for example, the forward march of time, the number of revolutions we can run round a circle, the distance we can travel in any direction, etc., as well, I suppose, as by the granularity into which we can divide a unit of time or space), it is logically impossible for any quantity in the universe to be an example of the actual infinite
Whatever the quantity of history past, it is a “completed” one
Thus, by combining these three positions, we see that were the past infinite in extent, it would be an example of the barred “actual infinite”; we are forced to conclude that the past must, of logical necessity, be finite.
Is that a fair description of your views? I have tried to avoid snark or prejudice in this post.
By even considering the question of whether time is finite, we distinguish between the “potential infinite” and the “actual finite/infinite(?)”. Because to speculate that there was a beginning of time, say, a hundred billion years ago, then we have to tacitly accept that under that scenario there would be an endess stream of prior times that we can provide names for, but which would precede the beginning of time and thus ‘not be real’ despite being nameable.
My ultimate position is that there are no actualized infinite spans. Infinites appear in the abstract and in theory all the time - that infinitely extending geometric line and those infinite sets, for example. But in the real, non-abstract world, there are no infinite two-by-fours or bottomless bags of holding.
Definitionally, mathematically, if the items you are accumulating are not constantly diminishing at a rate sufficient to reach a finite limit (like a ball bouncing to a stop, or the ‘substeps’ of Zeno’s Paradox) then you cannot possibly accumulate them all. Notably, if the time it takes to accumulate the items does not decrease in that manner, you by mathematical definition will never stop accumulating these seconds once you start - and you will also never have accumulated an infinite quantity of seconds, no matter how long you go. No matter how many seconds you accumulate, no matter how many pass, it’s still a finite number.
You can keep running around and around the circle, but no matter how many times you go around, you’ll still only have gone that many finite times around; not infinitely many times around. That’s just how it works.
So, mathematically impossible, not logically impossible, but still, impossible.
From 2, we know that whatever the quantity of history past, it’s finite - there is no mathematical way to accumulate an infinite amount of time, no matter how much of it you accumulate. This (irrefutably, I would have thought) proves that an infinite amount of time has not passed.
The notion of a ‘completed’ infinity is just dissonance to me, when not talking about diminishing towards a finite limit. The definition doesn’t allow it. So when people talk of an infinite amount of time having already passed; it’s like saying that they’ve been to ‘up’ and back; it just doesn’t make sense.
Now, as I’ve said, the timeline might or might not be growing at one or both ends - but that’s not the same as history already being infinitely long. I see no problem with the timeline being either fixed-length or constantly growing in the future direction, but having a constantly growing timeline in the negative direction strikes me as contradictory to our views of the past as being fixed and unchanging, and also causality becomes problematic at the negatively growing end. So, I personally find it hard to believe our timeline is growing at the back end, which means I presume a fixed beginning point (that is, a ‘completed’ finite past) as the (by far) most plausible case of the two.
Well, I think I crammed everything into point 2 - it really does follow straight from the way that infinities are used in mathematics at the level of counting things like seconds. But yes, of mathematical necessity, an infinite number of seconds cannot have already passed - if were were trying to accumulate an infinite number of seconds, we would still be in the middle of trying to do it, and still not have succeded in doing so; that’s what it means for a series to be infinite (presuming it takes a constant amount of time to accumulate each element).
Well, I tried to clarify. And the lack of snark is good; I tend to fall into that too. Half Man Half Wit, working forward from the definition of ‘infinite’, it explicitly follows that there is no way an infinite amount of time can ever have already passed. (That’s point 2 above, incidentally.) So, from that definition, the ‘infinite history with no beginning’ case is impossible - without even having considered the alternate case of ‘what if time had a beginning’ yet. I disprove the infinite case on its own contradiction with definition, rather than by looking at the other case and assuming the conclusion.
Admittedly, this argument is so basic and simple that it’s a little hard to separate the beginning from the end - it’s like proving whether a three-sided figure is a triangle or a square. It might look like I’m assuming the conclusion of trianglehood and not fairly considering the option of squareness, when in reality having three sides is so intertwined with the definition of trianglehood that I can’t even cite the definition without it being preferential to the one option.
I’m not sure what you mean by this… but I don’t feel it’s a fundamental point of dispute between us, anyway.
Hm… but is not the abstract a picture of a possible way for things to be? Why would something be mathematically coherent in the abstract and yet not capable of describing actuality without mathematical incoherence?
Well, we could say this: accumulating items at a constant finite rate over a finite period of time, I will end up with finitely more items than I started with. Specifically, accumulating one item a second for k seconds (where we shall always take k to be finite), I end up with k more items than I started with. If Q(t) is the number of items I have at time t, then we have that Q(t + k) = Q(t) + k. If Q(t) is finite, then so is Q(t+k).
But if I have an infinite number of items at time t, then I’ll continue to have an infinite number of items at time t + k. This is where everyone feels you are begging the question; it is only if you assume there is some starting t such that Q(t) = 0 that you can conclude Q(t+k) is finite forever thereafter. It is conceivable otherwise; that Q(t) is infinite now, has always been infinite, and always will be infinite forever more. Gaining items at a finite rate + at some point having no items = always having only finitely many items. But if you don’t assume “at some point having no items”, you can’t make this inference. Gaining items at a finite rate + always having infinitely many items is perfectly consistent.
This is what I meant to illustrate with that; it is an example of the distinction I felt you were trying to draw. I offered “There’s a circle in front of me [and I’ve never run round it before, but I’m thinking about starting]. How many times can I run round it?” as something that you might classify as “potentially infinite” but not “actually infinite”, meaning that there is no fixed upper bound to the number of runs I can make around the circle, but all the same, at any point, that number of runs is actually finite. It can grow without fixed upper bound, but it remains always finite.
Well, as I said, you only argued that I cannot increase my accumulated quantity by an infinite amount (over a finite span). But if I’ve never started from 0, if I’ve always had an infinite amount, then this is no contradiction to my currently having an infinite amount.
The definition of what doesn’t allow it?
Er, ok. Who has postulated a timeline growing at the back-end? (Whatever that would mean). Supposing we were to think of the past as a filmstrip, a giant home movie, then what we are postulating is that this filmstrip stretches back without limit, but new cels aren’t ever added to the back. They’re only added to the front, on the obvious way.
(Though, I should say, I am not inclined to think of “the timeline” as such a reified object in the first place.)
Well, if we started from a point finitely long ago where we had nothing, we could have accumulated infinitely many things (given that we are accumulating things at a finite rate). But who’s to say there was ever a point where we had nothing? Like I said, this is where most of us feel you are begging the question. If we’ve always had infinitely many things accumulated, then there is no obstacle to our continuing to have infinitely many things accumulated, even though we only accumulate them at a finite rate. The only conflict is with the presumption that we, at some point, had nothing.
Yes, thank you for your clarifications.
Supposing we ignored the word “infinite”, whatever its definition may be, and just asked about time stretching backwards without limit? Now what does the argument become?
I asked above about definitions. I provided, in an earlier post, some examples of mainstream mathematical definitions of the concepts of “finite” and “infinite”, and could provide some more, none of which are in conflict with postulating an entity (such as the past) which stretches on without finite limit. Some things in mathematics are uncontroversially held to be infinite (e.g., the interval containing all the negative real numbers), so it’s not as though math precludes any infinities. It’s not the mere definition of “infinite” which can be at play here; you must also have something else going on besides it (perhaps your definition of “the past” and so forth).