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. — Wosret

That it's the smallest measurable length is not that it's the smallest length.

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. — Michael

If you take that approach, your paradox also begs the question; it becomes "if it takes an infinite time to travel from one point to anther, then motion is impossible".

Indeed.

But measurable in principle, not just in practice, as it says that this isn't a problem of technology, or measuring ability.

If you take that approach, your paradox also begs the question; it becomes "if it takes an infinite time to travel from one point to anther, then motion is impossible".

Indeed. — Banno

That's not the argument. The argument is "if space is infinitely divisible and if no distance can be travelled instantaneously then it takes an infinite amount of time to travel from one point to another".

An infinite number of non-instantaneous movements to get from A to B makes it impossible to get from A to B. You just can't dissolve this problem by appealing to a mathematical formula that allows you to sum an infinite series of fractions. It's strange to even think that you can.

But measurable in principle, not just in practice, as it says that this isn't a problem of technology, or measuring ability. — Wosret

That it's the smallest measurable length in principle is not (necessarily) that it's the smallest length.

A line, not a number line. — Michael

The idea of the infinite divisibility of a line necessarily involves number; and the notion of traversing it necessarily involves time and the measurement of it; which is also number.

I do actually think that is precisely what it means. When something can't be done in principle, that means that it can't be done at all, not because of any failure to meet any conditions.

Why else would it be the smallest in principle, and why else would it be that it would be inconceivable for a better future technology to measure a smaller scale?

I won't labour it, I'm not that good at physics anyhow, but that's does seem to be the suggestion to me.

Maybe that's that concrete vagueness Apo's always on about!

The argument is "if space is infinitely divisible and if no distance can be travelled instantaneously then it takes an infinite amount of time to travel from one point to another". — Michael

The problem here is equating continuity with infinite divisibility, as if space and time consisted of infinitely many points and instants, respectively. The reality is that there are no actual points, just continuous space; and there are no actual instants, just continuous time. An infinitesimal distance can be traveled in an infinitesimal interval of time. A finite distance can be traveled in a finite interval of time.

An infinite number of non-instantaneous movements to get from A to B makes it impossible to get from A to B. — Michael

But this is not an argument; it is an assertion.

You will need to fill it out to turn it into an argument. It appears that the missing assumption is that the sum of an infinite series must itself be infinite; but we agree that this is not the case.

So the case rests with you: why is it impossible to get from A to B?

I put it to you that the demonstration that the sum of an infinite series can be finite kills the paradox.

Incidentally, the inevitable appeal to Plank's Length indicates movement down yet another garden path.

Better scenery.

Perhaps.

This is akin to saying that counting each half division of a segment (of an infinite series) takes half the time as counting the whole segment, and that the sum of those times adds up, not to an infinite magnitude, but to the total time it takes to count the whole series. — Michael

You are still not getting the difference. Counting each half-division of a segment takes the same time as it does to count any other. Not so with traversing them. This is because the time taken to count them is not dependent on the magnitude of the segments, whereas the time taken to traverse them is.

Who designed the garden and built the path?

Probably cobbler elves. You know... on vacation from cobbling...

Having a rest from making shoes for Planck? Must be tight work!

So let's take on the OP and search for the deeper point. The argument was set up to lead us into concluding either that our logic was wrong, or that reality is wrong.

The posited paradox was set up using a picture of the world, and a corresponding grammar, that were incapable of complex analysis of infinity. Later, with the introduction of the maths of limits, we had to hand a grammar capable of showing us some of the detail of how infinity works. The development of that grammar leads us to reject the picture of the world that leads to the paradox.

Logic cannot be wrong; but it can be inappropriate. Given a paradox, one ought look for a better grammar, a new logic; rejecting logic as "wrong" shows a profound misunderstanding. Logic is mere grammatical structure.

Hm. So wosret is walking the plank?

So the case rests with you: why is it impossible to get from A to B? — Banno

For the same reason that it's impossible to sequentially count the rational numbers from 1 to 2. Does your demonstration that the sum of an infinite series can be finite "kill" the claim that this would take an infinite amount of time? Of course not. It's a non sequitur.

Traversing each half division of a segment takes half the time as traversing the whole segment did, and the sum of those times adds up, not to an infinite magnitude, but to the total time it takes to traverse the whole line. — John

This is akin to saying that counting each half division of a segment (of an infinite series) takes half the time as counting the whole segment, and that the sum of those times adds up, not to an infinite magnitude, but to the total time it takes to count the whole series.

It just doesn't work that way. You can never count to the end of an infinite series of numbers. And so, by the same logic, you can never travel to the end of an infinite series of spatial divisions. As I said to Banno, this use of the sum of a geometric series is a non sequitur.

>:O

For the same reason that it's impossible to sequentially count the rational numbers from 1 to 2. — Michael

Is this an argument by analogy? Then it doesn't get you where you want to go.

If the length of time it took to count a number reduced as the size of the number, we would be able to count the rational numbers in a finite time.

You can never count to the end of an infinite series of numbers — Michael

But you can sum them. Which is what is asked in the supposed paradox.

But you can sum them. Which is what is asked in the supposed paradox. — Banno

No it isn't. The paradox is about moving from one point to another, which is analogous to counting from one number to another. It's got nothing to do with summing, which is why this use of limits is a non sequitur.

If the length of time it took to count a number reduced as the size of the number, we would be able to count the rational numbers in a finite time. — Banno

So what would be the first number we count after 1?

Nuh. The argument, on your own account, is that there are an infinite number of steps, each of finite length, and that therefore the total time taken must be infinite.

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

Summation is there at the start.

8-) Whatever you want. And the number of them would be denumerably infinite.

Whatever you want. — Banno

It's not whatever I want. The task is to count all the rational numbers between 1 and 2. I can't skip any.

And the number of them would be denumerably infinite. — Banno

The paradox is about moving from one point to another, which is analogous to counting from one number to another. — Michael

I can count from 1 to 2 in a finite time (see, I just did it); there is no need to count every rational number in between. Likewise, I can move from point A to point B in a finite time; there is no need to "touch" every point in between.

I can count from 1 to 2 in a finite time (see, I just did it); there is no need to count every rational number in between. — aletheist

But the task is to count every rational number between 1 and 2.

Likewise, I can move from point A to point B in a finite time; there is no need to "touch" every point in between. — aletheist

There is if motion is continuous, which is a premise of the argument that gives rise to the paradox. If motion is discrete then the paradox wouldn't arise.