This is true. But Zeno would still counter:
Infinitely many time intervals, however short, cannot have passed after finite time.
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True.
Then I would counter with:
We traveled 1/2 of the way, so we still have 1/2 to travel.
Since we travel at constant speed the 2nd half of the entire way cannot take longer to travel than the 1st half of the entire way.
So each subdivision can NOT ADD more time than the previous 1/2 interval.
So we have an upper bound on time, since time can never be larger.
So while we do have an infinite amount of time intervals, the sum will never grow, it's upwards limited.
But since each "parent interval" is upwards limited, it's irrelevant to how many "child" intervals you subdivide it as the parent time interval is always upwards limited.
So no matter how many times you subdivide, the TOTAL TIME never grows, thus time is not infinite.
I like this explanation. Every other argument I've seen against Zeno's paradox either takes as a given that an infinite series can converge to a finite sum (which I don't think is a self evident truth) or relies on theorems about infinite sums that weren't rigorously proven until well over 1000 years later. This is the only counterargument I've seen that seems like it would have held up in Zeno's time.
I like this point, but I'm not sure exactly how well concepts such as velocity and time, etc. were understood. They were certainly thinking about it, but even the idea of no action meaning constant motion (without other forces) was about 1500+ years away.
I think Zeno might have invented calculus if it weren't for the fact that math wasn't even nearly sophisticated[1] enough to admit any sensible formalization of the ideas.
[1] Perhaps it was even just because the power of symbolic algebra hadn't been fully realized. Geometry and logic only takes you so far.
Infinitesimals exist precisely to create an reciprocal quantity to infinities.
Infinitely many finitely short time intervals cannot have passed after finite time, but infinitely many infinitesimally short time intervals can. In the same way that one way of conceiving infinity is to imagine it as an arbitrarily large value at any given expansion, but infinity in the limit, an infinitesimal is an arbitrarily small value at any given expansion, and zero in the limit.
That's the whole point. The atomic distance of a subdivided step goes to infinitesimal at the same rate that the time associated with the distance does, so we're fine!
Infinitesimals are cool but I think this is not really about infinitesimals - it is about ordinary real numbers.
(so I think this is not true: "Infinitely many finitely short time intervals cannot have passed after finite time")
(I agree here: "infinitely many infinitesimally" is finite) (an infinitesimal is smaller than any real number, especially smaller than 1/n for every natural number)
https://en.wikipedia.org/wiki/Infinitesimal
"Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632."
My eight year old son (who watches a lot of youtube videos about physics stuff) countered with the comment that there can’t be infinitely many time intervals because eventually they get down to Planck time and time doesn’t really make sense past that point.
> Infinitely many time intervals, however short, cannot have passed after finite time.
This isn't true, because we know from math that infinite series can converge to finite sums. In particular, 1/2 + 1/4 + 1/8 + 1/2^n does, in fact, converge to 1.
No one seems to be bold enough to actually go this far, so I’ll make the claim:
Zeno’s paradox is wrong because its key premise is wrong: there aren’t infinitely many time intervals or distances, or arbitrarily small time intervals or distances.
That just raises a new paradox: how can you jump across those discrete distances without passing through the space between?
If space is discrete, then there is no space in between. If you shift a bit to the right or left, it does not pass through some in between space, it disappears in one position and appears in one next to it.
Similar to Richard Feynman’s famous answer to the question “why do I feel the force of a magnet from several inches away?” What counts as a satisfying answer depends on the person, but the general answer is “that’s how the physical world works.” This one doesn’t particularly feel like a “paradox.”
With my feet, usually.
Solvitur ambulando, as St. Augustine and/or Lewis Carroll would say.
I would say it is the other way around, it is totally feasible to traverse an infinite number of intervals in a fixed time. As I also responded to a different comment, just assume space is E³. Would you want to argue that you can not move or reach the goal under that assumption?
But if it's an infinite number of spatial intervals, then it's also an infinite number of temporal intervals, so you can't argue they cancel without quantifying how fast they grow.
Sure, but we know from the setup and I think it is fair to just assume a constant velocity which gives us this countably infinite sequence of distances and times, both getting shorter geometrically. And their partial sums have a finite limit. In essence you seem to be stuck because you have to complete an infinite number of tasks, but that is just the wrong measure, i.e. you must measure the time or distance, not the number of tasks in order to determine if you can complete all the tasks. Just walk from start to goal, there is no task in Xeno's infinite task list where he could say that you did not complete it. The problem is that Xeno can not give you the complete list of tasks if he tries to enumerate them one by one, not that you can not complete them all. Instead of enumerating them, he should just tell you to visit all places 2^-n for all natural n, you walk one meter and are done.
>> Infinitely many time intervals, however short, cannot have passed after finite time.
If they are finite time intervals that's true. But not true for infinitessimally short time intervals. If Zeno doesn't like this, we can say the time intervals are just as large as his distances. He can't argue infinitessimally short distances and not accept the same for time intervals. Playing dumb here backfires.
Yes they can. Zeno is wrong. There are an infinite amount of points between 0 and 1.
The way I had the 0.999…=1 concept explained to me (evidently late, I was out of high school when I encountered it) is that there are infinities which are larger or smaller than other ones.
I now know this more familiarly as sets: the infinite set of half distances in Zeno’s paradox is a subset of another infinite set in the same paradox, where the distance to travel is greater. If, for instance, your destination is the chemist down the road, that’s a smaller[1] infinite set of half distances than traversing all of space.
Because we are living beings who move beyond our initial destinations, with compounding goals we reach the smaller infinite set because it’s a subset of a larger one. And because we're living beings who are mortal, we eventually cease movement presumably at some increment less than 1/1 of some particular destination or goal.
There, not a paradox! We are simultaneously able to move because there is no singular infinite subset of distances to traverse, and unable to move at a distinct point in time because we die before we exhaust the infinite superset of other distances to traverse, but we’ve already traversed some subset of it by that point.
1: peanuts/1, and this is the first time I’ve got to reference the same joke on HN twice in totally different contexts just a few days apart.