Anything moving from point A to pointB must first travel half of that distance. Before that it has to travel half of half of that distance and so on. It is possible to iterate this to infinity. Therefore, motion is impossible. — TheMadFool

Anything moving from point A to pointB must first travel half of that distance, which will take half the time of the whole trip. Before that it has to travel half of half of that distance, which will take a quarter of the time. Each step takes half the time of the previous step. There are an infinite number of steps, but they do not take forever.

So you left out a third possibility:
3. Zeno was wrong.

Each step takes half the time of the previous step. There are an infinite number of steps, but they do not take forever. — Banno

You brought time into the picture. However that is really not the issue. If it is it is secondary. The key premise is the infinite divisibility of space. If space is indeed infinitely divivisble then motion would be truly impossible. Only after this is established can time enter the picture.

3. Zeno was wrong — Banno

I did conclude that Zeno started off with the wrong premise - that space is infinitely divisible.

You're overthinking it.

½ + ¼ + ⅛...=1

So you believe space is infinitely divisible?

How do you know that?

Don't set off up the garden path so quickly.

:) ok

Can you see that the series ½ + ¼ + ⅛... is infinity long, and yet adds to 1?

Space is infinitely divisible in mathematics under the definition of continuous space. Unfortunately, we do not really know whether this is applicable to our world. However, scientifically, it works better if it is.

We don't know if reality is wrong, but before we can actually argue about that, we should be concerned about the logical fallacy made by Zeno.


The fact that "one must first travel half way before getting to point A", and that this applies infinitely, is true. That is, however, irrelevant to whether one can move or not. This is because the argument that one must first travel half way is simply a requirement. It has nothing to do with whether one can actually travel that distance or not.

Yes.

So let's suppose it takes the runner one minute to run 1 km. That OK?

ok

So after ½ a minute, she's traveled half the distance.

After ¾ minute she's traveled ¾ a kilometre.

And so on.

And, although there are in infinite number of steps in the process, it only takes a minute.

No paradox.

So let's suppose it takes the runner one minute to run 1 km. That OK?

...

So after ½ a minute, she's traveled half the distance.

After ¾ minute she's traveled ¾ a kilometre.

And so on.

And, although there are in infinite number of steps in the process, it only takes a minute.

No paradox. — Banno

I'm sure this argument begs the question. Zeno's tries to show that motion is impossible. You can't refute it argumentatively¹ by setting up an argument in which motion being possible is a premise.

¹ Although obviously we refute the conclusion experimentally every day.

I don't get why this is a topic.

How do folk get through high school without being exposed to calculus?

I don't get why this is a topic.

How do folk get through high school without being exposed to calculus? — Banno

There's no necessary reason to think that the mathematics of limits addresses the (meta)physical problem. Plenty of philosophers think it's a mistaken solution.

For example, there's Why mathematical solutions of Zeno's paradoxes miss the point: Zeno's one and many relation and Parmentides' prohibition.

Have a look at a Koch Curve:



How long is the circumference of the curve? But what is its area?

There's no necessary reason to think that the mathematics of limits addresses the metaphysical problem. — Michael

SO, what is the metaphysical problem? My suspicion is that there is none, once the mathematics is understood.

If I am wrong, then set out the paradox for us, clearly.

The mathematics shows that despite there being an infinite number of steps, the result is finite. So the conclusion, that motion is impossible, is a non sequitur.

SO, what is the metaphysical problem? My suspicion is that there is none, once the mathematics is understood.

If I am wrong, then set out the paradox for us, clearly. — Banno

The paradox is that if distance is infinitely divisible and if instantaneous travel between one point and the next is impossible (reasonable premises, at least at the time of Zeno) then any distance travelled would take an infinite amount of time, and so motion is impossible – but motion is possible.

The mathematics shows that despite there being an infinite number of steps, the result is finite. So the conclusion, that motion is impossible, is a non sequitur.

The mathematics only shows that one can calculate the sum of an infinite series. There's no prima facie reason that this says anything about the physics of distance and time. And the above seems to equivocate on "number of steps". The "number of steps" when it comes to motion is more akin to counting the elements of an infinite series, not summing them.

It is not a paradox, it is a question of rules, lack of adequate real-world parameters, and Infinity:

Rule: OVER-STEPPING: If you are allowed to OVER-STEP point B, then, once beyond it, it can be said that you 'reached it' at some point in time, however infinitesimally short in duration.

THE ROLE OF INFINITY: Since the 'line' between Point A and Point B is 3D (which it has to be in order to physically exist, and we ARE talking about physical reality), then you may have missed it along any dimensional axis, if your measurements get small enough. If you allow a tolerance of say 3ft, then the determination can be made.

LACK OF DEFINITIONS: First 'you' hasn't been defined, so the 'paradox' is rendered 'silly'. If the point is minuscule, say just a fraction of a millimeter, and 'you' are a circumference of two feet, then it can be easily determined if 'any of you' is 'on' the point - say if an outside observer cannot see the point through your shoes. Note that this assumes that you are allowed to 'OVER-STEP' the point (meaning part of your shoe has 'gone beyond' the point.

THE LIMITING RULE and another LACK OF DEFINITION: If, on the other hand, you are NOT allowed to over-step the point, then the statement is true (and not a paradox), given infinite regression (that you will never 'reach' Point B). If an adequate parameter is given, say a 'zone' of adequacy (for instance, if you are 'close enough' for all practical purposes), then a determination can be made. So the real problem (in the real world) is in the lack of parameters. As for the math, you are using...

THE WRONG MATHEMATICAL TOOL: It is true that by halving the distance ad infinitum you will never reach Point B (nor will any part of you ever 'cross' it), so that presents no paradox in itself.

CONCLUSION
The 'paradox' is a good example of posing a real-world question (that which is beyond pure theoretical math) with a lack of real-world parameters.

That article that Michael posted is actually quite helpful.

The paradox is that if distance is infinitely divisible and if instantaneous travel between one point and the next is impossible... — Michael

It seems to me that what you have written here is misguided - an inaccurate picture of the number line.

Since a number line is infinitely divisible, there is no "one point and the next"; between any two points there are an infinite number of points.

It follows that the notion of instantaneous travel between one point and the next is muddled.

I didn't say this earlier, but I did kind of think that there is a smallest point, and things do actually just teleport kinda, from one point to the next. That's a thing, right?

It seems to me that what you have written here is misguided - an inaccurate picture of the number line. — Banno

I'm not talking about a number line. I'm talking about actual space.

Since a number line is infinitely divisible, there is no "one point and the next"; between any two points there are an infinite number of points.

If there are an infinite number of points then there are two points, so this doesn't contradict what I said.

It follows that the notion of instantaneous travel between one point and the next is muddled.

I don't know what you mean by muddled. If you just mean impossible, I'll agree. If I set up a starting point here and an ending point 10 metres away, I can't teleport directly there. It takes a non-zero amount of time to travel the distance in between. And this is true however close the points are. That's the paradox. It takes a non-zero amount of time to travel any distance, and so given the infinite divisibility of space it should take an infinite amount of time to travel from any arbitrary point to the other.

I didn't say this earlier, but I did kind of think that there is a smallest point, and things do actually just teleport kinda, from one point to the next. That's a thing, right? — Wosret

That would be a solution, but I believe the best theories suggest that space is continuous, not discrete, and so there isn't a smallest point. Given that this seems to entail that motion is impossible, it seems to be that continuous space and motion are inconsistent, and the paradox arises because we're unwilling to reject one of them.

Although I think that if continuous space and motion really are inconsistent, and assuming that motion is more obvious than continuous space, I'd take this as proof that, contrary to our best theories, space really is discrete – even if the scale is below the measurable, and so unable to be empirically determined (unless you count the evidence of motion and the reasoning that leads to the paradox as empirical proof that space is discrete).

I'm not talking about a number line. I'm talking about actual space. — Michael

However the paradox is framed precisely as a purported paradox of the infinite divisibility of a line. I agree with Banno; the realization that the finite time taken to traverse the line is infinitely divisible (in principle) just as the finite line is, dissolves the supposed paradox.

However the paradox is framed precisely as a purported paradox of the infinite divisibility of a line — John

A line, not a number line.

I agree with Banno; the realization that the finite time taken to traverse the line is infinitely divisible (in principle) just as the finite line is, dissolves the supposed paradox.

The conclusion of the paradox is that it doesn't take a finite time to traverse an infinitely divisible line. It takes an infinite amount of time, and so any such traversal is impossible.

Even if we consider the infinite series of ever-shrinking fractions that Banno provided we can see the problem. How long would it take for you to sequentially add the fractions that make up the series? An infinite amount of time, and so it can't be done. The maths that allow us to determine the sum without sequential addition is analogous to teleporting to the finishing line, which is something we claim isn't possible.

It takes a non-zero amount of time to travel any distance, and so given the infinite divisibility of space it should take an infinite amount of time to travel from any arbitrary point to the other. — Michael

No; and that's were the limits fit into the argument. There are an infinite number of steps between 0 and 1; it takes a specific time to travel between any two points; but the sum of those specific times is not infinite. Just as the sum of ½, ¼, ⅛... is not infinite.

Here's the rub: you must agree that the sum of an infinite series is not necessarily infinite. So it does not follow that "it take an infinite amount of time to travel from any arbitrary point to the other."

No; and that's were the limits fit into the argument. There are an infinite number of steps between 0 and 1; it takes a specific time to travel between any two points; but the sum of those specific times is not infinite. Just as the sum of ½, ¼, ⅛... is not infinite. — Banno

As I said before, this reasoning begs the question by assuming from the start that it takes a finite time amount of time to travel a certain distance, and then considers the infinite divisibility of that finite time. In using an infinite series of fractions you've assumed your conclusion.

The article (i'm sure you know already, having linked it, but just so you know that I know) suggested that the whole thing wasn't about denying movement in any case, but criticizing a suggested movement from the one to the many. More of a critic of metaphysical reductionism of the universe to concrete multiplicity rather than a single substance, as he believed on account of Parmenides. That aside...

This is what wikipedia says about it: "the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that."

And "In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart." - https://en.wikipedia.org/wiki/Planck_length

So, taken together, and if right (maybe not), then there is a smallest unit, and no movement within that unit can be determined, as the whole deal counts as a single location.