‘Infinite history with no beginning’ is not the same as saying ‘an infinite amount of time has passed to reach now’; in fact, the latter is an infinite history with a beginning. That you run into conceptual troubles there isn’t any wonder.
Also, any proof by definition as it pertains to reality automatically seems spurious to me, as it appears to be likely that the definition imposes the very limits on reality it ostensibly proves.
Oh, I don’t know. I would naturally identify “history” with “what has passed prior to now”, and thus be happy to equate “infinite history” with “there is an infinite amount of time which has passed prior to now”. It just doesn’t imply “there is some particular moment X in history such that an infinite amount of time has passed from X until now”.
Accidentally elided negation reinstated in bold.
begbert2’s comments about negative time and causality confused me at first, but I think I’ve figured out what he means. He views the idea of time extending backwards an infinite amount, as someone, right now, adding seconds to the negative end of the timeline, and this process would violate causality. Is that what you’re saying, begbert2?
While what we’re saying is that the idea of time extending backwards an infinite amount, is that it has no limit in the negative direction.
begbert2, your concept of negative infinite time very much reminds me of the people who insist that 0.999999999… gets closer and closer to one, but is not actually equal to one. The idea is that infinity is not an ongoing process of adding more, it’s already complete.
Well, one could debate about whether the notion of ‘passing’ implies a beginning point, semantically; to me, it does, but that doesn’t seem to be what’s at issue here – the way begbert2 uses it in his arguments seems to me to require a beginning point, hence that’s the meaning I was referring to.
Everything else just carries the risk of getting all tangled up in equivocation.
Ah, I see where you’re coming from.
As a sideline, can I just say that this thread is a lush oasis of rational discourse and civility amidst the election-related foofawraw. And also deeply fascinating.
I have nothing mathematical or philosophical to contribute that has not already been said in greater depth or in more compelling terms than I could provide, however. Do carry on.
Most abstracts are not a coherent model of possible ways for things to be, because there are untold scores of rules and conditions in reality that aren’t built into the abstract (which is probably why they call it an abstract). Take division - you can divide a number by any massive number you like, and get that many identical sub-portions, but if you take an apple and try and divide it similarly into more and more subdivisions you’re eventually going to get into a point were there are not enough atoms to divide it evenly. Similarly, there is no such thing in the real world as a perfect sphere, because our matter is composed of little bits, which after zooming in on them hard enough cannot possibly form a perfect arc.
And, why should a mathematical model reflect reality? Mathematical models need only be internally consistent, and needn’t have anything to do with reality. Non-euclidian geometry, for example, is not particularly applicable to any real-world situation where euclidian geometry is handy. Symbolic logic is a pretty far cry from anything real; it only works on other abstract concepts (arguments), and sometimes we have to torture those pretty hard to get them to fit into it even.
And of course, just because something is mathematically possible, certainly doesn’t mean that that’s the way things happen to be. It’s mathematically possible for me to have ten thousand hundred-dollar bills sitting in my living room. There’s space and no mathematical rules are broken by the model. But guess what: it’s not happening.
And if you had NaN many items, you could keep throwing things on the pile and still have NaN number items too, right? And if you had a pile of objects with the quantity ‘up’, that would mean that all other things in existence were lower than you (other then other ‘up’-sized quantities, and especially ‘bigger’ ‘up’-sized numbers, with a larger aleph number)?. And, of course, if you had ‘wish’ many items, then by virtue of having the quantity ‘wish’, you would have a pony, right?
I know you like the idea of “infinite” being a quantity, but by my understanding of the word, by every definition of it I know, that’s not how it works. Despite it being the answer to some quantity-inquiring questions, it’s not a quantity itself, any more than its close cousin NaN is.
An infinite quantity is magic. It cannot possibly be created. Once created (which it cannot be), it cannot possibly be contained (assuming constant-size units). It cannot be measured. You cannot put anything of finite duration into it - all such things instantly disappear completely (it would take up n/infinity time, and thus have 0 duration, and thus never have existed).
Fascinating thing, this infinite quantity of yours. Where can I get one?
Everybody who claims that if I believe that time is unbounded forwardly, I logically must believe that it’s infinite backwardly, has postulated a timeline growing at the back-end, even if they didn’t know it because they erroneously assumed that I was moronically holding an impossible belief about the future while simultaneously arguing it’s impossibility in the past.
(And how do you know that new cels aren’t added to the back? Or that they are being added to the front - the timeline could be of fixed length, with us gliding along to our inevitable end, where the tape will ‘run out’. Who’s to say? All these models are theoretically possible. Unlike, um, certain others.)
re·i·fy (r-f, r-)
tr.v. re·i·fied, re·i·fy·ing, re·i·fies
To regard or treat (an abstraction) as if it had concrete or material existence.
Soo… you think that we’re not really passing through time, then? That it’s ‘only a model’? (‘Let’s not go there; it is a silly place.’) That explains a few things.
I admit if you keep this situation completely in the abstract, it’s easier to gloss over using the terminology for trends and potential as terminology for actual things. But unless you are arguing for a completely memory-inclusive solipsistic time-free existence, we have had a past, with time, and that time was as reified as it needed to be to work with nice solid objects and people in the rest of reality.
The argument becomes that you just described a backwards-growing tape, which I have no problem with except causality.
Oh, wait, did you want it to be endless, backwards, forever, infini-oh, sorry, you’re demanding that we don’t know mathematical definitions and don’t have an ability to comprehend the mathematical impossibility of infin-endless timespans.
Well, gee, george, if I’m not allowed to know math, I guess I just have to not be able to argue against such silly things, am I?
Well, except for the second law of thermodynamics. That puts a pretty big knife into an infinite past, too. (Oh, wait, I know - as long as we’re pulling infinities out of our backsides, we’ll say the universe has infinite energy in it too! We can tell that that’s what’s happening because we’re all instantly burned into a crisp. Since, of course, infinite energy cannot possibly have reduced or failed to spread or ceased to fill all space with infinitely powerful heat all the time.) This doesn’t have anything to do with my mathematical/definitional argument, of course. 
I must have missed your mainsteam definitions of infinite, unless they were in that quote glurge, which I really don’t think we want to get back into. I suppose you could pick the one you like best and lay it out briefly in your own words, the definition which allows an infinite ‘entity’. (Oh, hints of god-fantasy. The infities hit the fan real fast if there’s an observer their counting time.) Anyway, I doubt we’d agree on what any definition you propose means, though - and/or where the term/definition may legally be applied.
And math doesn’t preclude square roots of negative numbers, either. Point out one in reality, please. A quantity that size of something would be fine.
(My definition of ‘the past’ is just ordinary time with the added property of immutability. This clashes rather harshly with the ‘backwards growing’ model that nobody’s arguing for, but would ostensibly have no problems with a constantly-infinite past, as it is imagined to be. There are other problems with that, of course.)
Here’s how I use it in my arguments - it’s impossible by definition for an infinitely long interval of time to have passed.
You keep finding the word ‘beginning’ in that. I don’t know how. Perhaps it’s convenient for you to see it there to make it easier to argue against me?
And seriously - do you really argue against the idea that time passes? To me that seems to be what it does. That’s it’s thing. I can watch any clock, and observe the evidence that time is passing all the, uh, time. If you really think that the idea that time passes implies it has a beginning how can you possibly argue against that? (With a straight face, I mean.)
Yeah, but we’re not talking about how things actually are. We’re not talking about the Big Bang. We know time does not extend infinitely into the past. The question is: could it, logically?
“The amount of time that has passed” Since what? You can’t really talk about time passing if you don’t have a point you’re passing from. Asking how much time has passed without a beginning is nonsensical. Now, maybe that’s a proof by contradiction, or maybe it’s just nonsense, like asking how much volume it fills up. I don’t know.
They’re all around you, particularly if you don’t take time and space to be quantized; aside from the extremely obvious (the number of integers, the number of texts expressible in the Latin alphabet), the obvious (the number of pure sine waves contained in a triangle wave, the number of points on a ruler), and the not-so-obvious (the length of the coastline of Britain, the surface area of an orange), there are the hidden ones. E.g., hold up an index card, which we shall think of as occupying the coordinates [0, 1] x [0, 1]. Consider the piecewise linear curve whose vertices are given by the sequence (1, 0), (1, 1), (1/2, 0), (1/2, 1), (1/4, 0), (1/4, 1), (1/8, 0), (1/8, 1), … . This curve has infinite length. Similarly, extruding this, we see that there is a figure of infinite area contained in every finite cube. Etc., etc. Infinite quantities abound.
And, yes, as Strinka said, and perhaps this is the whole misunderstanding between us, we are not arguing about how the universe is. We are arguing about what it is logically capable of being like; at least, that was what the discussion seemed to be about, given how your arguments were presented as being so generally applicable, having so little in the way of background assumptions. But, if some of your arguments are actually grounded in enthymematic empirical premises, there is nothing wrong with that; but, it would be best if you stated those premises explicitly.
You’re asking me to? Weren’t you the one who stated “sqrt(-1)… describes a specific fixed value relating to quantity”? Weren’t you the one who defended sqrt(-1) as more legitimately a number than any infinite value?
Not that I couldn’t easily point to applications of complex numbers in modelling physical reality (e.g., the ring structure of complex numbers corresponds to that of the angle-preserving linear transformations of two-dimensional space, with sqrt(-1) corresponding to a 90 degree rotation; similarly, we can encode the phase information of a wave as part of its amplitude using complex numbers. Though, I wouldn’t be particularly inclined to call these things “quantities”, for what it’s worth. Indeed, I, and I would think most others, would be much more readily inclined to refer to certain infinite values as “quantities”.) I just don’t see what the purpose of this challenge is.
No, but because it’s mathematically possible, it’s not explicitly impossible in the real world, and at least a plausible way for things to be. Whether or not it actually is that way is a matter for observation and empiricism to decide.
Your definition, then, imposes finite constraints on reality without logical necessity. You’re basically saying ‘infinity doesn’t exist because it can’t exist’.
But to the contrary, there’s numerous physical theories that include infinities in one way or another, in the form of singularities of some kind – the most famous being the gravitational singularities of General Relativity, i.e. black holes, where matter has a volume of zero and infinite density. And while it’s true that it’s thought those singularities indicate that something’s missing in the theory, and it’s hoped that an eventual GUT will eliminate them, that’s not a logical necessity.
We should probably abandon the concept of a ‘growing timeline’, it doesn’t make all that much sense. Generally, the proper time of anything moving through space-time depends on its path; take, for example, the famous twin paradox: one twin takes off, comes back, and finds the other twin has aged faster than he has. There’s an allowed path for every time difference, hence every arbitrarily far ‘future’ can be reached in this manner. This implies that the timeline is not ‘growing’, but that in fact there exist points in space-time arbitrarily far in our causal future. Thus, the future is infinite, nobody’s adding cells or laying down new tiles to walk on at the ‘front end’. So your argument for why the past cannot be infinite should apply perfectly symmetrically to the future, yet you yourself contend that that’s not so.
No, but you yourself derive the contradiction in the original form of the argument from essentially saying:
However, point 2) is equivalently stated as ‘there exists a point y in time such that x - y = infinity’, and as such assumes the existence of a beginning point; if it didn’t (i.e. if there were no point y since when an infinite amount of time has passed), the conclusion wouldn’t follow, since there would be no contradiction.
I don’t think that the idea that time passes implies it has a beginning, I think that the idea that a certain amount of time has passed implies a point since when it has passed – otherwise, there would be no meaning to the idea of ‘an amount of time has passed’.
Also, I don’t necessarily agree that time passes, at all – it can be equivalently said that it is us that move through time which causes our experience of the passage of time (so that, even if your argument did hold, it would at best be our movement that had to have a beginning).
Instead of getting caught up on the term “infinity”, why not side step the issue?
Hypothesis: For any point in time, it is possible to find an earlier point.
begbert2, is this statement logically consistent? You are not allowed to invoke the term “infinite” in any form. I believe, absent of evidence for the Big Bang, that there is no logical reason to reject this hypothesis.
I get the gist of what you want to say here, but I believe it’d be better stated as ‘for any point in time, it is possible to find another point a second (or some other arbitrary distance) earlier’, or else your statement doesn’t actually say anything about whether or not time is infinite – it’s possible to find an earlier point for any point between 12:00 and 12:01 today (if time isn’t quantized).
Albert Einstein and the Fabric of Time The past ,present and future all exist simultaneously according to Einstein.
I can’t speak for him, but my complaint about your question of “how much time has passed?” implies that there was a beginning point in time to measure from. The fact that you’re asking for a measurement of the time difference, implies that you have two endpoints to measure between. Therefore your question presumes a starting point. I’m really surprised that you haven’t been able to grasp this concept.
Good point, Zeno!
(Note: these responses are in no particular order, by which I mean, “mostly reversed order”)
The only way “how much time has passed?” implies that there was a beginning point in time to measure from, is if you, not me, are a priori assuming that there is no such thing as an infinite quantity. If not, the answer “an infinite amount of time has passed” answers the question without implying a beginning.
I, of course, am quite explicitly not assuming that there isn’t such a thing as an infinite amount - I’m referring to the definition and usages of infinite/infinity and arguing forward from that to demonstrate that it is not a term that applies to quantities. (Which is very likely the reason for this new tactic of challenging me to argue my point from a mathematical knowledge of no greater than junior high - presumably they’d ask me to argue from the perspective of a caveman if they thought they could sell the idea.)
So, no, there is nothing in asking how much time has passed that assumes a conclusion - you lot who think it does are making extra assumptions on your own, which would make it circular, and then attributing them to me. Problem is, I don’t make any such assumptions a priori, which is why I haven’t been able to grasp the notion that I have.
The question you ask is identical in meaning to the question under discussion. And, you’re forbidding me from citing the information that forms my argument. So in other words, you’re demanding that I assert the truth of my conclusion without making my argument.
Gee, that’s a reasonable demand to make of a person. Sidestepping the issue indeed!
What did you think of my reference to the alternate thermodynamic argument, by the way? Or are am I not allowed to refer to any science at all, having instead to prove the beginning of the universe using basic symbolic logic only?
“hence every arbitrarily far ‘future’ can be reached in this manner” - this manner being, to wait. The fact that you experience time as moving more slowly does not mean that you are just teleporting into the future - your ship still exists the entire time, just moving very very fast through space to outside observers. So, this adds nothing at all that we didn’t have already with people just passing through time at normal speed, and does not prove anything new, does not prove a pre-existent infinite quantity of future time, does not prove that the timeline is not growing - it doesn’t add or prove anything. (Not even if you assert that it has done so.)
Nice try, though.
I concede that there’s a possibility that einsteinein relativity can maybe be used to make an argument that all time exists simultaneously - of course, that doesn’t contradict anything about my position, because a timeline can be finite in both directions and all exist simultaneously, the same way that an entire strip of film exists simultaneously even if the frame that is being projected is constantly advanced.
I don’t even recall making this argument - though I suppose I might have made various ones similar to it in various ways, but differently. Perhaps if you quoted what I actually said, rather than just telling me what I “essentially” said, then we might be able to get somewhere without tripping over strawmen.
As you say, that’s equivalent. Though it’s suddenly becoming popular in this thread, I’m not impressed by the practice of swapping out words and phrases with synonymous ones and then pretending that things which are true with the original words, aren’t true with the new ones.
The coastline of britian isn’t infinite - you’re misunderstanding the argument that refers to. (And/or grasping at straws). I didn’t manage to google up a reference to your orange but I’d bet it’s similar. So, all you have to offer me is “abstract things that we can think of and imagine, but which don’t have an actualized existence in reality”. This notably includes mostly things which you assert as being infinite but describe in terms of trends, like the number line.
I assumed we were talking about logically capable of being possible in reality. Not in an abstract model. If the model is incompatible with reality, I would consider it irrelevent. You appear to have no such reservations, which may be where we differ. (Heck, in an abstract model entropy doesn’t need to apply either so the thermodynamics problem doesn’t exist either.) I also was avoiding reference to science since, let’s face it, if we were using science properly we’d be back to the big bang, which takes all the fun out of it.
But, why not, let’s introduce an entirely new argument, designed to be more logical and requiring less of an understanding of the fact that math depends on context. (Okay, I’ve hinted around this argument at least once before, but never with this formulation, with its explicit P1.)
Premise 1: (observed) there is a ‘now point’, that moves along the timeline in a ‘forward’ direction at a constant speed. (Depending on their velocities different observers may experience time at a slower ‘sampling rate’, but they can still be observed externally as existing continually at the ‘now’ point.) The ‘now point’ is currently at some specific point in time N.
Premise 2 (speculated for sake of argument): instead of a ‘starting point’, where the timeline and the ‘now point’ began, there is an ‘infinite quantity’ of time prior to N.
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’.
Statement 5 (from P2 and S4): Point N is defined as being after an ‘infinite quantity’ of time. Thus, if the ‘now point’ was ever anywhere within that ‘infinite quantity’, it would never advance to N.
Statement 6 (from P1): Because the ‘now point’ is constantly advancing, we know that the ‘now point’ was at some point at all points of time prior to N and after a beginning-of-time-if-any (which is presumed not to exist by P2).
Statement 7 (from S5 and S6): we know that some of the points prior to N are in the ‘infinite quantity’ of time, and that the ‘now point’ has been in those point. Thus, we know that the ‘now point’ has never advanced to N.
Statement 8 (from S7 and P1): We have proven that the ‘now point’ has never reached N, and from P1, we know that it has reached P1. This is a contradiction; logically, one of the premises must be false.
Reviewing the premises, P3 is by definition and thus unassailable. Thus the only options are P1 and P2. Note that equivocation about whether time passes or we are passing through time is irrelevent to P1; if you accept any variation on the theme of time passing at all, you probably have to accept P1. (It actually holds even if thought there were multiple ‘now’ points, which I say in case relativity is weirder than my vague notion of
P2 is not similarly solid. In fact, assuming you accept P1 and P3, it is disproven, via proof by contradiction.
I think this is both free of “enthymematic” premises and avoids equivoaction (by any side) about the definition of ‘infinite’. So: Invalid, Unsound, (or both,) or both Sound and Valid?
(Hopefully adding in the explicit reference to our perception of unidirectional time flow will underscore the unreversibility of this - we don’t perceive time as operating in a way that is equivalent to its reversal; so we can’t argue that it should work the same if reversed. If in the forward direction it is “heading towards infinity and never getting there”, then in the backward direction it is “heading back from infinity and never getting here”.)
Those should be:
Statement 7 (from S5 and S6): we know that some of the points prior to N are in the ‘infinite quantity’ of time, and that the ‘now point’ has been at those points. Thus, we know that the ‘now point’ has never advanced to N.
Statement 8 (from S7 and P1): We have proven that the ‘now point’ has never reached N, and from P1, we know that it has reached N. This is a contradiction; logically, one of the premises must be false.
Sorry about that.
Hm… some things you’ve said now challenge what I previously thought your argument was.
Note that I don’t hold with my brethren that “X amount of time has passed until now” implies a starting point at the other end of that interval being measured. Like you, I think the answer “an infinite amount of time has passed” would be the right way to describe the situation where years 3000 BC, 4000 BC, 5000 BC, 6000 BC, …, all lie in the past.
This I don’t understand. I thought your argument was “There is no such thing as an infinite amount”. If it is something more complex, what exactly is it?
Incidentally, the definition of “infinite” I’d consider most appropriate to this context would be “not finite”, where “finite” means “having a magnitude upper-bounded by a natural number”, where the “natural numbers” are “those numbers inductively generated from the constant 0 and the operation of adding 1”.
Again, I agree with you on this much.
Well, I thought that was the idea… we are talking about what is logically possible. If we begin to invoke the science built up from evidence of how our world happens, contingently, to be, then we might as well simply demonstrate the Big Bang Theory.
As in:
I consider every abstract model to be something logically capable of being possible in reality; those facts which constrain reality but not abstract models are not logically necessary facts (e.g., the law of gravity, or electrical attraction, or conservation of momentum, or the fact that the universe is apparently spatially 3-dimensional rather than some other number).
Ok, I’m all for it.
All good so far [though I’d rather say “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’.”]
This does not follow from P3. For example, suppose the “infinite quantity” of time were the negative real numbers; it’s conceivable that our now point is currently at 0 (and thus has advanced outside of that quantity), but was previously within that infinite quantity (specifically, yesterday, it was at -1, the day before, it was at -2, the day before that, it was at -3).
This is all fine to me, except that it relies on Statement 4, which, as mentioned above, is not supported by your premises.
I’d say it’s invalid, on the grounds that Statement 4 does not actually actually follow from Premise 3 as you assert it does.
But I do appreciate your writing the argument out in this form; it makes it much easier to follow.
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 is some notion of “interval of time” and the “magnitude” of such things (it would not be difficult to formalize these in terms of other concepts, but I do not think it would be productive for the purposes here either). 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 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.