The paradox is easily resolved though by pointing to time intervals that get smaller if smaller space intervals are chosen in the formulation of the paradox. Like that there is no ground to make motion impossible. — Prishon

That doesn't resolve Zeno's paradox. There is more to Zeno's paradox than the oft stated claim that it would take an infinite amount of time to traverse an infinite number of points.

Consider the notion of counting every ${1}\over{2^n}$ between 0 and 1 in ascending order. Simply saying that if it takes $n$ seconds to count from $0$ to $0.5$, ${n}\over{2}$ seconds to count from $0$ to $0.25$, etc. and using the convergent series to show that the sum is finite doesn't show that it's possible to perform such a count.

There's the far more practical problem of where such a count starts. There is no first ${1}\over{2^n}$ to count after 0, and so you can't even start counting. There is no first ${1\over{2^n}}m$ point to move to, and so you can't even start moving.

A solution is that motion isn't continuous; it's discrete. There is some smallest unit of movement, e.g. the Planck length, and that such movement doesn't involve passing through some halfway point.

Does Zeno's paradox prove that this has to be the case? — Prishon

Zeno had four paradoxes and he needed all of them for the reason you suggest. He assumed that time is either discrete or continuous and time likewise. He considered all combinations - four possibilities in all. Those pre-Socratics didn't have calculus or anything but speculative theory of matter: but they were no slouches with logic.

Infinitesimals are funny things. What about velocity, dx/dt (is there mathJax here?)? You think its a real physical quantity? — Prishon

I have no idea; all I know is infinitesimals are like near death experiences: deadish but not quite dead, if you know what I mean.

I have no idea; all I know is infinitesimals are like near death experiences: deadish but not quite dead, if you know what I mean. — TheMadFool

Yes. Newton and consortes did indeed introduce near-death or even death experiences, introducing them! Those unlucky kids at high school having to absorb them. Good if they wanna commit suicide. Thats the sunny side maybe.

discrete space — Metaphysician Undercover

An oxymoron for a classicist

How are they treated in NSA? — Prishon

Here's a good intro to the subject: Nonstandard Analysis

What is required is a proper analysis which separates space from time — Metaphysician Undercover

I shudder when I say this, but there might be something to this idea. Just a feeling, since the two are so different.

Can you word your feeling? You feel it is right? Are not space and time separated but living together?

Space is a container for matter. It's necessary for matter. Time is an effect that happens from motion. Physicists speak of "spacetime" because it's easier to do the math when space and time are seen as one entity. Scientists know time is not a substance but in practice you have to go with what is easiest to understand

Spacetime is space as substance. Time describes the affect of relativity

Consider the notion of counting every 1/2^n between 0 and 1 in ascending order. — Michael

Why? This leads nowhere, nor does it prove anything. It does, however, resemble something I looked into a few months ago concerning convergence of infinite compositions in the complex plane.

I think Raymond Tallis gave a good answer to this question. Zeno's paradox arises when we mistake mathematical space for manifest reality. In our daily lives, there are no infinite halfway points between things. When we enter into specific intellectual domains, things are different.

But I don't think Zeno's paradox should go beyond the problems it may cause to some of our intuitions about space, i.e. real life affairs.

Why? This leads nowhere, nor does it prove anything. — jgill

Because it's comparable to passing every ${1\over{2^n}}m$ point between $0m$ and $1m$ in order. For the same reason that the count is impossible, so too is the movement. The impossibility has nothing to do with the length of time it would take and so isn't solved by referencing a convergent series of time intervals.

Well, if space is not continuous, arent there gaps to stop the motio? — Prishon

Gaps do not necessarily stop motion. that would only be the case if motion is continuous. Doesn't quantum mechanics demonstrate that it is probably the case that the motion of fundamental particles in not continuous. And if the motion of fundamental particles is not continuous, why not consider that the motion of any body is not continuous. That motion is continuous was simply an assumption of convenience. Then the required mathematics was produced to support that assumption.

I shudder when I say this, but there might be something to this idea. Just a feeling, since the two are so different. — jgill

It wouldn't be the first time we agreed on something, even though the two of us are so different.

Space is a container for matter. It's necessary for matter. Time is an effect that happens from motion. — Gregory

Space is a concept, developed from studying the properties of bodies. It is not a container, but has been deemed as a necessary condition for motion, as a body needs a place, space, to move to. Time is not the effect of motion, but it is also a required condition for motion. Traditionally, space was conceived of as static, as an object and its properties were something static. But Aristotle demonstrated the need to allow for change, and motion if our conceptions are to be real representations. This produced the need to integrate the two distinct conceptions, space, and time, as the two necessary conditions for motion.

Gaps do not necessarily stop motion. — Metaphysician Undercover

Only if they're infinite. Math does need a better explanation of this imo. Saying infinite steps has a finite sumation doesn't answer the paradox

Then the required mathematics was produced to support that assumption. — Metaphysician Undercover

Which mathematics demonstrate space can be discrete? Isn't this contrary to the very definition of space? As I said a loop of some kind is a better idea

Aristotle demonstrated the need to allow for change, and motion if our conceptions are to be real representations. — Metaphysician Undercover

Aristotle didn't believe in space or time, just forms. Space is a physical container and humans use the concept of time to understand how relativity works within space. Aristotle was right actually in that space and time are both phantoms but modern physics doesn't work with these absolute ideas anymore

I don't know, but I steadfastly refuse to believe time comes in literal bite-sized intervals... That doesn't make sense on so many levels.

My intuition wants to say there is no real passage of time, and that this all occurs in the same space (or lack thereof, as it were) at once.

Ultimately, I think we can take bites out of truth and be led where we may, but ultimate knowledge is just out of our league.

My intuition wants to say there is no real passage of time, and that this all occurs in the same space (or lack thereof, as it were) at once. — theRiddler

That's my point as well. Time is a mystical concept that is helpful in physics but there is really no stuff called time. Physics deals with stuff. Time and space, understood in an absolute sense, are a kind of Platonic heaven, designed to help people see this world as a Platonic place. That's philosophy though, not empirical thought

For the same reason that the count is impossible, so too is the movement. The impossibility has nothing to do with the length of time it would take and so isn't solved by referencing a convergent series of time intervals. — Michael

Here's something I've looked into on numerous occasions that bears some resemblance to this topic. Instead of dealing with 0 to 1 or 1 to 0, this is a sequence that goes from n to 1 where n is unbounded.

${{F}_{n}}(z)={{F}_{n-1}}({{f}_{n}}(z)),\text{ }{{F}_{1}}(z)={{f}_{1}}(z)$, ${{f}_{n}}(z)\to f(z)$

The question is does ${{F}_{n}}({{z}_{0}})\to L$ as $n\to \infty$ ?

So it might seem that n being unbounded raises a similar issue of where to start since backward recursion is involved? But an example of where this appears in math literature is in the analytic theory of continued fractions, and it is quite solvable.

Agree completely. Probabilistic notions maybe especially...contrived formulas for Platonic, as you put it, "realities" that are neither here nor there.

It's all rigidly fixed academia, though.

I think humans can't understand the world at all unless something remains a mystery. Once I think I understand everything suddenly nothing makes sense. It's assumed we know what material existence is so we posit other realities. But Heidegger asked, "do we really know what 'to be' means?" At such a point one forgets about other realities and does science, but Platonic ideas always creep in nonetheless

Which mathematics demonstrate space can be discrete? Isn't this contrary to the very definition of space? As I said a loop of some kind is a better idea — Gregory

I said the mathematics supports the assumption of continuity. "That motion is continuous was simply an assumption of convenience. Then the required mathematics was produced to support that assumption.

Aristotle didn't believe in space or time, just forms. Space is a physical container and humans use the concept of time to understand how relativity works within space. Aristotle was right actually in that space and time are both phantoms but modern physics doesn't work with these absolute ideas anymore — Gregory

I conclude that you haven't read Aristotle's "Physics".

I have read the Physics. There is no middle ground between absolute time and space on one hand and relational theory. Aristotle rejected the former, calling it a void, and so falls in the other camp

And youre not being clear about continuity and discreteness. Space can't be discrete. Space necessarily has parts. You say mathematics backed up motion being continuous and yet this was exactly Zeno's point.

You said "Aristotle didn't believe in space or time", though Bk.4 of his "Physics" indicates that he believed in both "place" and "time". Though he rejected the prevailing conception of "void", this does not mean that he did not believe in "space", because he replaced "void" with the more comprehensive and practical "place". And, he stated that "time" has two distinct senses, primarily it is a measurement, and secondarily it is the thing measured. In modern usage this separation is not maintained and equivocation is the result. When pressed for an explanation, most people simply deny the second, 'there is no such thing as time', as something which is being measured. You can see this in Einstein's famous quote where he states that time is a persistent illusion.

And youre not being clear about continuity and discreteness. Space can't be discrete. Space necessarily has parts. You say mathematics backed up motion being continuous and yet this was exactly Zeno's point. — Gregory

I don't see what you're objecting to. If space necessarily has parts, then we must conclude that it is discrete, as each part is a distinct and therefore discrete entity. If space were continuous, then it would have no parts, as being partitioned means that it is divided, therefore necessarily not continuous.

You said "Aristotle didn't believe in space or time", though Bk.4 of his "Physics" indicates that he believed in both "place" and "time". Though he rejected the prevailing conception of "void", this does not mean that he did not believe in "space", because he replaced "void" with the more comprehensive and practical "place". And, he stated that "time" has two distinct senses, primarily it is a measurement, and secondarily it is the thing measured. In modern usage this separation is not maintained and equivocation is the result. When pressed for an explanation, most people simply deny the second, 'there is no such thing as time', as something which is being measured. You can see this in Einstein's famous quote where he states that time is a persistent illusion. — Metaphysician Undercover

Einstein did not deny that place and time exist in the Aristotelian sense. People who believe in the universe believe in this, but it is a relational theory and in Aristotle it was the quintessence instead of spacetime that provided the means for parts to talk to each other in the language of space and time. Your attempt to find a middle ground between absolute plenum and relational theory doesnt work as I already pointed out. Aristotle believed in relationship theory but God(s) held the relations together through the 5th element

If space necessarily has parts, then we must conclude that it is discrete, as each part is a distinct and therefore discrete entity. — Metaphysician Undercover

No because each part of space has parts which have parts which have parts which have parts... to infinity.

If space were continuous, then it would have no parts, as being partitioned means that it is divided, therefore necessarily not continuous. — Metaphysician Undercover

You have discrete and continuous mixed up. Discrete is pointsize. Continuous is infinitely divisible, which even Aristotle said was the case. Discrete space doesn't exist. The question is how to understand infinite divisibility because it leads to problems as Zeno showed