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Time Puzzle

Here's a paradox I've been thinking about for a very long time, but haven't yet been able to fully articulate. Hopefully this comes close:

We can speak of existence, or experience, with regard to three modes of time: past, present, and future. The past can only be inferred by the existence of the present, along with tools and capacities such as memory and other symbolic records. The future is also a posited existence. Of the three, only the present exists in a true sense; while we may delve into our memories for the past, or daydream about the future, it is only the present "now" that we can subjectively experience, and even scientific "objective" observations can only be made in (or from) the present. 

Yet the present itself, continually ocurring as a moment, has no duration, for if it did it would admit itself to division, and the present would then entail a past and a future, for of any present moment that admitted duration one could speak of that portion of the present which "had happened," "is happening," and "is yet to happen." This is clearly self-contradictory. If the present is something other than that which has a past and a future, then the present moment cannot be said to have duration, but must exist as a point travelling from the past and continuously slipping over the horizon of the future. 

Again, a point in time, by definition, cannot contain within it any duration. And as a point in time, and without duration, no series of points could ever be said to have duration, or exist sequentially to "add up" to any duration. Just as a point in space contains absolutely no space (absolutely no volume), a point in time cannot itself contain any duration, for if it did it would be divisible, and thus not a point in time but a measurable unit of time.

If time, then, were to exist in a linear fashion, how does a continuum of present "nows," an infinite series of points, add up to the processes we observe and experience? For even an infinite set of points can never be said to have duration, for a point in time, by definition, has a duration of zero. Again, if the present moment, as a point in time, had a non-zero duration, the present would contain a past and a future, which is absurd. But if the present moment does not contain any duration, and the present is the only point in time which truly exists, how does a series of present moments ever become the sum of past events?
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Eric said…
I would argue that a point in time does not actually have a duration of zero, but instead that it has an infinitely-short duration. Therefore...


As I understand the mathematics of infinity, it's not true that "For even an infinite set of points can never be said to have duration, for a point in time, by definition, has a duration of zero." I think an infinite collection of infinitely-small objects can have a collective size of any number in R+ (that is, zero, infinity, or anything in between), depending on how quickly the two infinitys grow relative to each other. Consider the limits as x->infinity of x^2/x, 2x/x, and x/x^2 (and ignore the obvious algebraic simplifications).

The point of all that was to say that time must therefore consist of an infinite number of infinitely-small points, but where the number of points is a "larger infinity" than the (inverse-)duration of the points. This would give us an actual aggregate duration from a set of infinitely-small points, allowing the passage of time. And since we can't distinguish between each of these infinitely-small points in time, they appear to be a coherent/continuous "flow", just like a video does to the human eye, even though it may only be 30 or 60 images/second.
January 22, 2009 at 11:55 PM
Glen said…
This reminds me of the "painter's paradox": say you completely fill a paint can with paint. Clearly, the paint inside the can must wet the entire interior surface of the paint can. In other words, you obviously have enough paint to paint the surface of the can. But Gabriel's horn is a "paint can" that has infinite surface area but a finite volume. In other words, the amount of paint inside the can is finite, so there is no way it can wet the surface of the can that contains it, because that surface is infinite! (see http://en.wikipedia.org/wiki/Gabriel%27s_Horn)
January 23, 2009 at 11:00 PM
Glen said…
(Maybe this will help Marie get it?) With regard to the time paradox, one might start by thinking of the “present time” as being of some short time interval, say one second. Then to describe the past minute one would simply stream together the most recent sixty seconds, that is, the sixty most recent “presents.” But, as Dirk and Eric argue, the “present” isn’t one millisecond, it is something much shorter than that. So say it is only one nanosecond. Then to describe the past minute one would simple stream together the most recent sixty-giga “presents”. But, as Dirk and Eric argue, the “present” isn’t one nanosecond, it is something much shorter than that. So say it is only one yoctosecond. Then to describe the past minute one would simple stream together the most recent yotta “presents.” And we keep going and going with this type of argument, making the “present” shorter and shorter, until the present is described by some infinitesimally small time interval and the last minute described by some infinitely large number of the most recent “presents.”
January 25, 2009 at 1:27 AM
Glen said…
Oops, I guess I left out one step. Let's try again: One might start by thinking of the “present” as being of some short time interval, say one second. Then to describe the past minute one would simply stream together the most recent sixty seconds, that is, the sixty most recent “presents.” But, as Dirk and Eric argue, the “present” isn’t one second, it is something much shorter than that. So say it is only one millisecond. Then to describe the past minute one would simply stream together the most recent sixty thousand “presents.” But, as Dirk and Eric argue, the “present” isn’t one millisecond, it is something much shorter than that. So say it is only one nanosecond. Then to describe the past minute one would simple stream together the most recent sixty-giga “presents”. But, as Dirk and Eric argue, the “present” isn’t one nanosecond, it is something much shorter than that. So say it is only one yoctosecond. Then to describe the past minute one would simple stream together the most recent yotta “presents.” And we keep going and going with this type of argument, making the “present” shorter and shorter, until the present is described by some infinitesimally small time interval and the last minute described by some infinitely large number of the most recent “presents.”
January 25, 2009 at 1:36 AM
brian roosjen said…
This comment has been removed by the author.
January 28, 2009 at 2:58 PM
brian roosjen said…
0 and infinity are incomprehensible concepts except when one is asleep. so let's all go to sleep. it's late.
January 28, 2009 at 3:00 PM
Unknown said…
mista - deep sleep is the only time in which, other than death (or enlightened states), our most subtle mind is active and unobscured by ignorance (or so they say)...

eric - hurray! i love it. since i don't understand infinite mathematics, i have a few questions. (1) are these infinitely small time-points divisible? (2) where are they (that is, is there one giant time moment for the universe, or an infinite set of them for the infinite set of points in space? if the latter, do we admit different infinitesmly small durations for different points in space, or are they all the same infinitely small size?)

Dad - yes, although 100 attoseconds is the smallest duration measured so far, so maybe the yoctosecond is still theoretical?

http://news.bbc.co.uk/2/hi/science/nature/3486160.stm

"Einstein showed that people travelling at different speeds, whilst agreeing on cause and effect, will measure different time separations between events and can even observe different chronological orderings between non-causally related events."
January 28, 2009 at 9:07 PM
Eric said…
The short answer is I don't know, but here are my guesses:
1. No, they're not. How do you divide something that's infinitely small? There's no such thing as more-infinitely small.
2. If we consider it as a timeline (and I'm not sure how this fits in with the time-space theories of physics), I would imagine it has an infinite set of infinitely small points along the line (just like a graph of y=x). And I don't think you can have different sizes of infinitely-small.

To address a potential next question/confusion: In my first comment, I mentioned that things can approach infinity at different rates. This is true. But I don't think that this affects the eventual "size" of the infinity reached - just how quickly you get there.

On the other hand, maybe moments in time aren't actually "infinitely-small", but rather "infinitesimal" - http://en.wikipedia.org/wiki/Infinitesimal (See also http://en.wikipedia.org/wiki/Infinity)

As a third option, it's also possible that moments in time do have a true duration, and we just can't innately sense it (and haven't figure out how to measure it yet), just like our eyes cannot perceive changes faster than ~120 frames/sec (exact number unclear, but probably in the high 10s or low 100s of frames/sec). Maybe moments in time are really just a yoctosecond long.

I'm sure I didn't actually clarify anything here...
January 29, 2009 at 2:39 AM
Glen said…
I guess I had assumed not, but maybe you are suggesting there is in fact an "absolute zero" for time like there is for temperature? From http://en.wikipedia.org/wiki/Planck_time : "According to quantum theory, 1 Planck time should be the smallest unit of time physics can reason about in a meaningful way. But according to news reports, analyses of Hubble Space Telescope Deep Field images in 2003 raised a possible discrepancy. Very distant images that should have been blurry were not, contradicting the notion that Planck time is indeed the smallest measurable unit of time."
January 31, 2009 at 6:41 AM
Glen said…
How much of life is there that remains incomprehensible to the human mind? We are presumably just a bit more evolved than say a chimp, who presumably is not able to understand many things that we understand. Is it possible that evolution could progress infinitely further, and thus we are only capable of understanding an infinitesimal amount of "truth" (scientific or otherwise)?
January 31, 2009 at 6:46 AM
Unknown said…
Eric - Continuing your video metaphor, would we describe each infinitely small point in time as a "still frame", wherein change and causation are suspended until the next point in time? That is, does the change that occurs over time occur within the moment of time, or somehow in between the moments? And since we have an infinite field of them, and we must admit different durations for different observers (ie, spacial locations) (if we accept relativity), I'm wondering what governs causation, and what binds sequences of events, ie, what is the causative mechanism? I think this all just came out as nonsense, but I take solace in the thought that, according to our friends at wikipedia, "defining time in a non-controversial manner applicable to all fields of study has consistently eluded the greatest scholars."

Dad - Maybe if we have a large enough network of quantum computers (an infinite one?), at some point they will creatively solve the mysteries of the universe for us to the point where they can communicate them to us in a simplified manner... We'll just have to hope they don't deceive us for their own purposes. Robots!
January 31, 2009 at 11:28 AM
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