Stupid board timeouts. Ignore the above post. That post got cocked-up; here’s what it should have been:
Let me expand/modify that definition of “infinite” a bit, being fairly formal. I do not claim this is the only way to do it (after all, I do not claim any concept has a single, fixed definition), but I do claim that any mathematician would immediately recognize this as a natural, reasonable formalization as far as this discussion goes.
We assume there are some preexisting notions of “an interval in time”, of one interval being “(wholly) contained within” another, and of the “magnitude” of an interval in time (it would not be difficult to formalize these in terms of other concepts, but I do not think it would necessarily be productive for the purposes here either).
We also have notions of “empty”, “disjoint”, and “decomposable”. We can take them as primitive, or define them as follows: We say that an interval is “empty” if it is contained within every interval. We say two intervals are “disjoint” if every interval contained within both of them is empty. We say that an interval A is “decomposable into” intervals {B, C, D, …} if, for every interval X, X contains each of {B, C, D, …} if and only if X contains A. We could define these concepts in other, perhaps more intuitive ways as well, but presumably the definitions would be equivalent to these.
There is some select unit interval (e.g., one second or one year or what have you), whose magnitude we give the defined name “1”. We also have the magnitude “0”, referring to the measure of an empty interval. When one interval is decomposable into several pairwise disjoint others, we say the magnitude of the former is equal to the sum of the magnitudes of the others, defining “addition” of magnitudes (at least as a partial function). Now, call a property P of magnitudes of intervals of time “inductive” if 0 satisfies P AND whenever x satisfies P and y = x+1, then y satisfies P. We say that n is “a natural number (magnitude of an interval of time)” if n satisfies all inductive properties.
We define an interval of time to be “infinite” if, for every natural number n, it wholly contains an interval of magnitude n.
Not that I think you need this level of formalization to contribute to this discussion, but simply because begbert2 keeps extrapolating things from what he considers the definition of “infinite” to be, I want to put out there a particular, fairly concrete definition.
OK, I’m sorry for trying to put words in your mouth. I was trying to make sense of what you’ve been saying, and the idea that you were implicitly assuming a starting point was the only way I could get it to make sense.
So now, if you ask how much time has passed on our hypothetical clock, I’ll answer “an infinite amount.” I have yet to see you support your assertion that this can’t be the case.
This statement makes no sense to me. How can a specific point in time be after an infinite quantity of time? That sure seems like you’re talking about the mirror image of your implicit beginning of time, which I thought you were saying, but you just denied, but now seem to be implying it again by symmetry.
If you don’t equate ‘having passed’ with ‘having passed from a certain point’, though, your argument (or at least, the general form of the argument as it was used in the OP) does not lead to a contradiction.
You’re right in saying that there’s meaning in the idea that ‘a certain amount of time has passed’ without this necessitating a beginning point, I was wrong about that – to illustrate, the open interval between 12:00 and 12:01 has a length of one minute whether or not there ever was a 12:00, thus you can say at 12:01 that a minute has passed regardless of that interval having a beginning.
Let’s consider the original form of the argument as presented in the OP:
This is what I was referring to in my last post, taking it to be the argument in question (though now you have clarified the actual form of your argument). It establishes that there are no two points in time between whom there is a distance of infinite length, then uses this to argue that now, in an infinite timeline, could never have been reached, since it ought to be the second point of such an interval, which can’t exist if the first point exists; however, since there is no necessity for the first point to exist, the argument isn’t sound.
Not to try and overrule Pleonast, but I’d be just as happy if you used everything at your disposal to answer the question ‘do you have any issue with the statement “for any point x in time, there exists a point x - 1 (1 being some fixed finite interval of time)”?’
Well, obviously ‘simultaneously’ is going to be a bit of a problematic term here, but since we’re all grown ups, I think we can just accept it and carry on.
The position above seems to restrict the future, as well, to being finite. Am I reading that right? Would you have any problem with an infinite future existing ‘simultaneously’?
See above.
It wasn’t a synonymous phrase, it was a logically equivalent statement, which it is perfectly valid to swap.
So, what’s with black holes, then? The reason for the hypothesized non-existence of an infinite mass density are absolutely of physical nature; I#m not aware of any a priori logical arguments that preclude it.
I’m not entirely sure I agree with this – I think it can be said that there are infinitely many ‘now’ points, at all points along an infinite timeline, according to relativity. But I’ll accept this one on the grounds of the ‘speciality’ hypothesis – as long as it works for us, the rest of the universe can go f*ck itself.
It’s not, as Indistinguishable has shown – the interval between negative infinity and zero is certainly an infinite quantity, but from it’s last finitely many points, the interval’s end can be reached by successive addition of a fixed quantity. So, there are infinite quantities such that there exists an amount x for which the operation P + x leaves the interval for infinitely many points P. Your definition imposes the restrictions needed for the argument to work.
Anything regarding reality is merely a description, never a definition, except as it pertains to the concepts used to argue about reality – i.e. you can well define sheep to be white animals, but that doesn’t negate the existence of black sheep; you’d just have to find another name for those.
Similarly, you can well define infinity as not existing in reality, yet it wouldn’t negate the actual existence of non-finite quantities (should such exist).
I guess it boils down to this: if there can be a moment in the past, I see nothing logically precluding a moment a little further in the past, for all moments.
No, my question is not identical. (We can make Half Man Half Wit’s adjustment if need be.)
You have said that it is logically impossible for an infinite amount of time to have passed. Ok, let’s say I except that. Instead, I posit that every time has an earlier time. No infinities involved. I’ve merely removed a “beginning”.
This is how mathematicians avoid getting entangled in infinities. The term is imprecise. I’ve made a precise hypothesis. It is sloppy to equivocate it with “infinity”. I believe you’ve made a similar point earlier about process vs quantity. So I’m asking you to reformulate your argument without the term “infinity”. If your argument is correct, you should be able to put it in more precise terms.
As for your “thermodynamic” argument, I’m not seeing it.
This is more of a cosmological argument. You seem to be confusing total energy and energy density. Simple measurements indicate that there is a finite energy density. Current evidence (changes in the rate of expansion of the observable universe) indicates that the universe is open. That is, we have no upper limit on the total quantity of matter-energy in the entire (not just observable) universe.
Yeah, I know you never accused me of circularity; I’m sort of defending on several fronts here and not everything will apply to everybody.
I am not assuming that there is no such thing as an infinite amount; I have been trying to point out that the mathematical usage of ‘infinites’ is not such that allows infinite amounts of constant units of time to have passed. This has been complicated by the fact that we have had difficulty agreeing on a definition for the thing,
So every infinite stream or series does have a beginning, yes? The apparently two-unbounded-ended ones just being two one-unbounded-ended ones stuck back-to-back.
Note that if it’s a ‘quantity’, though, order is irrelevent, since quantities are abount strictly amount and not order. So I’m not required to move through it in any particular direction if we’re using it as a quantity.
Denied. No changing of my premises allowed, regardless of what you’d "rather’. Is it all good as written or isn’t it?
No, it doesn’t follow from your ‘reinterpretation’ of P3, but it does follow from my original P3. If all points P in the ‘infinite quantity’ have a successor in the ‘infinite quantity’, then there is no walking out of it.
I contest that it was invalid as I actually wrote it, prior to your convenient reinterpretation. Are you going to contest the premise then?
Such legerdemaine to avoid saying that it’s growing backwards from the origin! Of course, from this it doesn’t follow that either every number has a successor or that every number has a predeccessor, because in being vague enough to be described as a quantity and to describe all ‘infinite quanties’ of time, it withholds the information about what direction it is infinite in. Which makes Your P3b unacceptable as well - under this definition, we can’t talk about traversing time at all!
Apology accepted.
All infinite quantities, as Indistinguishable has noted, start at some point and count up from there. There is a direction to this counting and if you reverse it, you aren’t looking at an infinity anymore; you’re looking at counting down from the point of reversal to your starting point, to 0. There is no way to be counting down without making this reversal and abandoning the infinity - and your clock is counting down (as in, in the opposite direction of the supposed infinity).
The fact that your clock is counting down implies a reversal point to make that counting down possible. That is, from our perspective, a beginning point.
Assuming that time is passing doesn’t make the infinite quantity impossible - but, with a little bit of argument (which I don’t believe I have laid out before; these threads are great for forcing you to eventually clarify and organize your thoughts, if belatedly) it can be shown that the direction of that passage through time does.
The sad fact is, we are at a specific point in time: now. Yes, it can be shown that the specific point cannot follow an infinite amount of time; if I show it then I am not assuming my conclusion. In the argument you quoted that statement from, it was shown by the fact all points in the infinite quantity had a successor (a premise which is now under dispute by Indistinguishable).
Keep in mind, I am saying that the current time cannot be preceded by an infinite amount of time; I’m just not asserting it as a premise. It’s perfectly correct to make statements as inferences and conclusions in your argument - if not, we wouldn’t be able to make arguments at all.
The issue is that we’re not going backwards in time; we’re going forwards. As Indistinguishable has clarified (and then obfuscated), all infinities are derived inductively and are basically one-directional. (Putting aside n-dimensional infinities, which still have multiple perpendicular directions but a single origin; for the current problem we’re only talking one dimension after all.) I therefore have objection to all infinities being declared as all elements having predecessors; this is only true if the direction of travel is the opposite the direction of growth of the infinity.
Also, in cases where you are traveling ‘against the grain’, as I argued to CurtC above that traveling in reverse implies a ‘reversal point’, beyond which you could have succeeded further into the past, but didn’t. (Which is equivalent to the ‘finite beginning of time’ model.) So, I think that I have the infinite predecessors thing beat both coming and going; the only way you can get your infinity’s worth out of an infinity is to be traveling forward, with the grain.
Yes, I would; there are no ‘simultaneous’ infinities with non-zero sized elements, in any direction; there are no infinite quantities.
Note though that I am not opposed to the future end ‘growing’ (in the same metatime that we need to describe a ‘now point’ moving), which if it’s growing as fast or faster than the ‘now point’ is moving will result in a functionally infinite future as far as the now point is concerned, because it will never catch up and try to go beyond the end - but at any given metatime the future will have a finite length and definite end.
Note that, aside from causality problems and my personal preference for the notion of an unchanging past, I’m not really opposed to the past growing in this manner too - but that’s not the same as the ‘infinite past’ that’s under discussion here.
I don’t know enough about black holes to comment, really, though I will say in my current ignorance I believe that, though they may be very, very, very dense, with possibly even a density that is constantly increasing at an exponential rate as their own mass and gravity continues to crush itself, I do not believe that they have “infinite” mass at any given point. (Trending towards infinity, sure - but that’s not the same thing.)
I thought about mentioning this, but didn’t want to muddle the argument. If there is a nonzero set of ‘now points’, then we can just pick one at random, and do the argument on it (as you note). The presence of other not-under-discussion ‘now points’ doesn’t hurt the argument any.
Right - though that definition (that is, Premise 3) may come under fire very shortly. As has been noted, you can’t get from “all of infinity” to “its last finitely many points” without at some point reversing your direction from “traveling towards infinity” to “traveling away from infinity”, at which point you create that beginning point that some people here are trying to avoid.
Are you assailing the definition or ain’t ya?
Well, for the sake of the argument it sure would. Of course, the correct response for that is to attack the definition.
Neither do I - presuming the ‘now’ point is heading into the past. If not, well, it turned around (or started) at some point, and went back no further.
I think I have - in the given logical argument, and in this post, if nowhere else.
I never stated the thermodynamic argument; I assumed it would be obvious. Entropy happens. Given infinite time and lacking an infinite source of external energy, I had vaguely understood that at the very least, one observable effect of this would be that everything in the universe would be the same temperature. As that has observably not occured (at a much more readily observable level than the evidence for the big bang is), it can be empirically known by everyone everywhere that the universe has not passed through infinite time.
This is of course an empirical argument, and is thus unsatisfying to those seeking an argument of pure logic. And it’s completely unrelated to my main argument anyway.
But, if P + 1 necessarily means “the point in time 1 unit of time later than P”, then this is an intolerably bizarre restriction of what kinds of quantities can be infinite, gerrymandering out any ray which stretches infinitely backwards but not forwards. In that case, I would have to reject your proposed definition of an “infinite quantity” of time. (Not that the definition with “P - 1” isn’t overly restricted as well; I only meant for it to be sufficient to cover the specific proposed cases under argument. For a general definition, I would use something along the lines of the lengthy one I gave in my other post)
The problem with your “P + 1” definition is that it would declare something like {1 year ago, 2 years ago, 3 years ago, 4 years ago, 5 years ago, …} to not be infinite, even if it stretched on without end (because not every negative integer has a successor which is a negative integer). But this is precisely the kind of thing which we have all been defending as an example of the logically tenable infinite past, and is undoubtedly an example of a past without a (chronologically) first point.
I’m not sure where you read me as making these statements, but I suppose you’re getting it from my post on a formal definition of infinity for this discussion. But I do not agree with this interpretation; the definition of “infinite interval” in that post is one which the negative integers (which have a last (i.e., highest) point, but not a first one) can satisfy: for every natural number n, there is an interval of negative integers of that magnitude [e.g., the interval [-n, -1]]. Granted, in that post, the notion of natural number/finite magnitude is derived inductively, and I suppose you could say there is some directionality to this process, but then, no one would argue with there being a smallest natural number/finite magnitude [clearly, 0, the magnitude of an empty interval]. But that does not force chronologically earliest points into intervals of time themselves, any more than the definition of the natural numbers forces the real numbers or integers to have a least element…
The argument I gave does not refer to mere infinities - it refers specificially to this odd beast of yours called an ‘infinite quantity’. Being a quantity, it doesn’t matter which order I enumerate its elements; I can presume then to be re-numbered in any order I like. For the purpose of this discussion, you may presume that they are ordered in increasing order.
If you wish to back down from it being a quantity, and to recognize that we are talking about an ordered series, that post also comments on the functional effects of going the wrong way down an infinite series - it ceases to be an unlimited count towards infinity and becomes a finite countdown towards 0.
You can pick if you want your infinity to be a quantity or an ordered series counted bacwards from the ‘now point’, but it can’t be both to be swapped quickly in and out so as to dodge the arguments against either. Pick one and stick with it.
Alright, if the label number doesn’t have to correlate with the chronological ordering, then what do you care if I use +1 or -1? Anyway, like I said, it is not the case that your Statement 4 follows from your Premise 3.
To recap:
“Premise 3 (by definition): an ‘infinite quantity’ of time is defined such that no matter what your time P, there is always another time P+1 that is within (as in, not having passed the end of) the ‘infinite quantity’. (Note that this doesn’t reference or refer to a beginning - it’s true for all points in the infinite quantity.)”
“Statement 4 (from P3): If the ‘now point’ was ever anywhere within an ‘infinite quantity’, it would never advance past the end of the ‘infinite quantity’.”
“If all points P in the ‘infinite quantity’ have a successor in the ‘infinite quantity’, then there is no walking out of it.”
After all, we might suppose the labels are as follows, in chronological order:
… 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 …
Then the period of time labelled by positive integers is “infinite” by your definition, but there is no obstacle to the now point having once been in them (say, at 1 yesterday) and making its way out (at 0 today).
Alternatively, if you will accept as a suitable definition of “infinite quantity” that for every point labelled P within it, there is a point labelled P + 2 within it (and why not? The labelling order is detached from the chronological ordering, you say), then we can label in the following manner, using only nonnegative labels:
… 11 9 7 5 3 1 0 2 4 6 8 10 …
The odd integers satisfy closure under +2, and yet there is no problem with the now point having been at 1 yesterday but at 0 today.
Of course, if the labelling order is detached from the chronological order, there’s no need to require labels to be numbers of any particular sort to begin with.
Incidentally, though this is a side question, given that you believe time is divisible without limit, how would you go about enumerating the points within a given second?
E.g.,
First: Start of the second
Second: End of the second
Third: Halfway into the second
Fourth: 3/4 in
Fifth: 1/4 in
Sixth: 0.62348234…
Seventh: pi/4
Eighth: sqrt(2)/2
…
?
So it’s not a quantity but rather a series extending negatively backwards from 0, then? (That’s ‘negatively’ as compared to the direction of movement of the ‘now point’, not ‘negatively’ with respect to any monkeying you have pointlessly done with the signs.)
In that case, at all points prior to the current now point, there was a finite distance to the current now point. Thus the now point has never been an infinite distance back in time; we are dealing with finite quantities and distances and timespans exclusively. Right?
Admittedly, pure logic can make no claim about what largest size of interval between ‘then’ and ‘now’ has been, so you can speculate it out to the largest number you can think of, multiplied by eight. Even so, though, at all points in time that you can come up with, there has been a finite countdown to now. At no point has the ‘now’ point been at or heading towards an infinite distance from the current ‘now’. Correct?
I’m not sure what “quantity” means to you, what the “quantity”/“series” distinction is, etc., so I cannot answer that.
(I’ve only done the monkeying with labels to fit your particular definition of “infinite” as “closure under +1”. If you had defined “infinite” as something else, rather than a definition which you had to justify with the potential arbitrariness of the labelling, then I would not have felt the urge to point out the possibilities present in arbitrary labellings.
But regardless of the labels used or the particular restricted definition of “infinite” adopted, it’s clear that that structure I was labelling lacks a chronologically earliest point.)
In the model I am proffering, any two points in time have a finite difference between them. That much is correct.
In the model I am proffering, at no point has the “now” point been infinitely distant from the current “now”, and never will the “now point” be infinitely distant from the current “now”. That much is correct.
So the claim is that an infinite amount of time has passed prior to now, despite the fact that at no time prior has there been an infinite amount of time remaining prior to now to pass?
It is the same as how I can say: There is an infinite amount of space directly to my left, despite the fact that every point directly to my left is only finitely distant from me.
False analogy - you’re not just claiming that there is an infinite amount of uncrossably vast empty ‘potential’ time extending into the past; you’re claiming that the ‘now point’ has moved through time from ?the other side of? the infinite span of prior time to get to the current now point.
That’s like claiming that you have have walked from left of infinitely far to the left to get to the current point; or that you have counted upwards from the, um, lowest negative number to get to 0.
In other words you are claiming simultaneously that infinite crossable spans do not exist (because all points are finitely far away from each other), and also that an infinite crossable span does exist, and has been crossed to get here. We’re back to the nonsense, the self-contradictory positions, in other words - you’re tring to push the ‘start of time’ back across all of infinity and off the other end of it, to make it disappear and for there never to have been a start of time. Which would give you the conclusion you want, except the problem is, no matter how far you push it back, you’ll never get it past or even to the end, by your own admission.
I think we’re back to that place where you can declare the impossible possible because it’s only an abstract model. It’s a crossable infinite distance because…I say so!
If this is where we’re at (again), I will surrender; if you hold the existence of crossable infinite spans as axiomatic despite contradictory facts, then, logically speaking, then no argument can convince you, because the extra premise of yours will lead all otherwise valid proofs to contradiction.
By “span” do you mean “interval between two points” or do you mean something else? Yes, there are no infinite intervals between two points; however, this does not contradict the fact that the period of time prior to now is infinite (for the period of time prior to now is not an interval between two points, because there is no earliest point in time).
It’s only like saying that if you assume that I started walking from the earliest point in time. But I didn’t start walking from any point in time. I’ve always been walking. I’ve been walking forever, with no beginning point. There are no two points along my journey which are infinitely distant. However, the journey, in total, has covered an infinite amount; there is no conflict because there is no starting point to the journey.