## Zeno's paradoxes in the modern era

• 5.7k
I'm saying no more than that your description does not work.
• 8k
I'm saying no more than that your description does not work.

Why doesn't it work? If space really is infinitely divisible and if motion really is continuous then movement requires passing though each 1/(2n)m position, which is a sequential series with no start - and a series with no start by definition can't be started.
• 5.7k
And yet it moves.
• 8k
Which means that space is not infinitely divisible and/or motion is not continuous.
• 533
We can say that the space between two points is infinitely divisible but that it is impossible to move through them all.

Sounds odd. It's a bit like saying that since there is no highest number to count to, I can't ride a bike. How does one have any effect on the other?
• 8k
Sounds odd. It's a bit like saying that since there is no highest number to count to, I can't ride a bike. How does one have any effect on the other?Luke

I'm not saying that. I'm saying that because there's no first 1/(2n)m position to pass through - a physical fact about space (assuming infinite divisibility) - then movement cannot start.
• 533
But there is a first position to pass through, regardless of whether you can calculate it.
• 8k
But there is a first position to pass through, regardless of whether you can calculate it.Luke

There isn't. Just as there's no first rational number after 0. This isn't just some epistemic problem where we're unable to calculate the first rational number - there just isn't a first rational number. And there just isn't a first 1/n metre position to pass through.

Unless movement is discrete.
• 533
I still don't see how it prevents motion. I reject the assumption that motion requires counting or requires determining a first position. You (and Zeno) derive the impossibility of motion via the infinite divisibility of space, but space is not motion; position is not momentum. So how does one affect the other?
• 8k
I reject the assumption that motion requires counting or requires determining a first position.Luke

It doesn't require counting or determining a first position. It requires there being a first position. But if motion is continuous then there isn't a first position.
• 8k
You (and Zeno) derive the impossibility of motion via the infinite divisibility of space, but space is not motion; position is not momentum.Luke

I'm not saying that motion is impossible because space is infinitely divisible. I'm saying that continuous motion is impossible because continuous motion entails having started a sequential series with no start, which is contradictory.

So because motion is possible it isn't continuous (even if space is infinitely divisible).
• 533
I'm saying that continuous motion is impossible because continuous motion entails having started a sequential series with no start, which is contradictory.

I thought the problem was in determining the first step after the start of the sequence (or after 0)?
• 199
I think there is an even greater problem with Zeno's (and Parmenides) argument against change.

The arguments that they give are supposed to be deductive arguments that demonstrate the impossibility of change, but pure reason alone cannot tell us what the actual state of affairs is. Pure reason alone can only tell us whether something is logically possible or impossible.

The question then is this: is there any logical contradiction in proposing that change occurs?

If there is no logical impossibility in proposing that change occurs, then no argument from pure reason alone could ever demonstrate the impossibility of change and Zeno and Parmenides have pursued a fool's errand.

Thus, the question of whether change occurs or not can only be proven empirically.
• 8k
I thought the problem was in determining the first step after the start of the sequence (or after 0)?Luke

It’s not about determining the first step but about there (not) being one.
• 533
is there no start to the sequence or no first step after the start of the sequence? It makes a difference.
• 2.1k
If I want to count in order the 1/(2n) numbers between 0 and 1, which is the first number I count? If I want to move from 0m to 1m which is the first 1/(2n)m distance I pass through? There isn't one, and if there isn't a first step the task cannot start.
Thinking about this leads me to what I think is a fairly precise mathematical statement of the controversial assumption that Zeno's argument makes. It is this:

(A) An object O cannot move from location A to a different location B on a path Y unless, for any countable subset S of points in Y, there exists an order-preserving map from the natural numbers onto S.

In Zeno's example, S is the set of points at proportions 2^-k along the track Y, for natural number k. But there are infinitely many different types of S that present the same problem. For any real number x>1, the set of numbers x^-k for natural number k does the same thing. The sequences 1/k, 1/k^2, 1/k^3 and so on do it too. In fact, take any monotonic-decreasing function f:R+ -> [0,1]. Then the set S = f(N) provides the required "blockage".

There are maps from N onto any countable set S in Y, but they do not preserve order. For example, the map f: k |-> 2^-k has 1/2 = f(1) < f(0) = 1, reversing the order that 0<1.

The question is, why should we accept assumption (A)?

As (I think) I said earlier, we don't need calculus to dissolve Zeno's problem. All we need do is identify the questionable assumption on which it relies.
• 8k
The question is, why should we accept assumption (A)?

Because if space is infinitely divisible then there exists such a subset and if motion is continuous then it must pass through each member sequentially? I don’t see how that can be avoided.
• 199

"Another response—given by Aristotle himself—is to point out that as we divide the distances run, we should also divide the total time taken: there is 1/2 the time for the final 1/2, a 1/4 of the time for the previous 1/4, an 1/8 of the time for the 1/8 of the run and so on. Thus each fractional distance has just the right fraction of the finite total time for Atalanta to complete it, and thus the distance can be completed in a finite time. Aristotle felt that this reply should satisfy Zeno, however he also realized (Physics, 263a15) that it could not be the end of the matter. For now we are saying that the time Atalanta takes to reach the bus stop is composed of an infinite number of finite pieces—…, 1/8, 1/4, and 1/2 of the total time—and isn’t that an infinite time?

Of course, one could again claim that some infinite sums have finite totals, and in particular that the sum of these pieces is 1× the total time, which is of course finite (and again a complete solution would demand a rigorous account of infinite summation, like Cauchy’s)."

Does this part of the article answer your objection?
• 8k
Does this part of the article answer your objection?

No, because the point I’m making has nothing to do with time.
• 2.1k
Because if space is infinitely divisible then there exists such a subset and if motion is continuous then it must pass through each member sequentially
I agree with the first point. For the second point, we need to be careful about what we mean by 'sequentially'. If we mean that we pass through x before y iff x<y then there's no problem. If we take a different meaning of 'sequentially' I suspect we are going to get another dubious, controversial assumption.
• 8k
If we mean that we pass through x before y iff x<y then there's no problem.

The problem is that there’s no smallest x after 0, so movement cannot start.
• 2.1k
The second part of that sentence is either an assumption (so the 'so' should be an 'and') or based on an unstated assumption, which will be something equivalent to (A). What I'm questioning is why anybody would accept that assumption.
• 8k
I don’t know how to explain it any clearer than by saying that if each member has to be passed from smallest to largest and if there is no smallest then the act of passing each member cannot start. The logic of that seems evident enough.

How exactly could one show that one can’t start counting each rational number between 0 and 1 from smallest to largest? The reasoning will be the same as that. Or are you saying that even this is a questionable assumption?
• 2.1k
How exactly could one show that one can’t start counting each rational number between 0 and 1 from smallest to largest?
Proving that there is no smallest will do it. Like this: Assume there is a smallest, call it x. Then x must equal 2^-M for some M. But 2^-(M+1) is less than that and is also in S, which contradicts our assumption that x was the smallest. Hence there can be no smallest.

But that proof doesn't do anything to support a belief that it is impossible to move from A to B.
if there is no smallest then the act of passing each member cannot start
The single act of passing all members of S is the single act of traversing track Y from A to B. In doing so, object O will pass each member, in order. The act starts at time 0 with object O at location A.

So we have an act that does everything we need, and which has a beginning at time 0. Observing that there is no smallest element of S and hence no 'first passing of a member of S' does nothing to obstruct that. It just demonstrates that the notion 'the first passing of a member of S' is empty, just as the notion 'the present king of France' or 'the beginning of this circle' is.
• 6k
When I imagine zeno's paradox, I tend to imagine an arrow travelling for a bit and then I stop it momentarily in my imagination and say to myself "This is now the arrow's position. Now how did it get here?". But of course I am not allowed to mentally stop the arrow from moving, for I would no longer thinking of a moving arrow.

Is it even possible to imagine a moving object that has a precise velocity and/or position? Personally I don't think so. I always find myself either fantasising that I have mentally stopped the arrow in order to measure it's position, or that I am entirely ignoring it's position when thinking about it's motion.
sime

Yes, I agree this is where the problem is right here. If the arrow is moving, there is no such thing as its position. That's why Aristotle sought to create a separation between these two, "becoming" and "being". The arrow paradox is clearly based in this problem. "Moving" implies that time is passing, "having a position", if we're not referring to something motionless, requires a moment at which time is not passing. So the arrow cannot be moving if it has a position. Simply put, there is an incompatibility between "having a position", and "moving", these two would be contradictory. So we cannot describe the same situation in these two contradictory ways.

But there is a first position to pass through, regardless of whether you can calculate it.Luke

The problem is that there is an incompatibility between moving, and having a position. The two are contradictory. So it doesn't make sense to describe the movement of a thing in terms of position. And if, or when we do, such as to say that a thing moves from position A to position B, then we are not saying anything about the movement itself, only that it was at position A and is now at position B. To describe the movement is to describe how it got from A to B. But to say it moved from A to B is not to describe the movement, which is how it got from A to B.
• 8k
The single act of passing all members of S is the single act of traversing track Y from A to B. In doing so, object O will pass each member, in order.

But this is just like saying that the single act of counting all members of S is the single act of counting the rationals from 0 to 1. It’s a nonsense assertion. You can’t just say that this task happens and so it doesn’t matter that there is no first rational number, and so you can’t just say that continuous movement happens and so it doesn’t matter that there is no first position to move through.
• 2.1k
But this is just like saying that the single act of counting all members of S is the single act of counting the rationals from 0 to 1.
I chose to use the word 'passing' rather than 'counting', with intent. There is a critical difference between 'counting' and 'passing'.

To me, 'counting, in order' [you didn't say 'in order' but it was implied] means identifying an order-preserving map from the natural numbers to the set S. That is impossible.

'Passing, in order', means identifying an order-preserving map f: [0,1] -> [0,1] such that the map obtained by restricting the domain of f to f^-1(S) is order-preserving. That is easily done.

It is the insistence that the points must not only be Passed In Order, but also Counted In Order that is equivalent to assumption (A), and which is unacceptable (and I would say also unintuitive, but intuition is in the eye of the beholder).
• 8k
It is the insistence that the points must not only be Passed In Order, but also Counted In Order

I’m not saying that. I’m saying that passing in order is a sequential series of events with no start and so cannot be started. My mention of counting is an analogy to show why one cannot start a sequential series of events with no start. We all agree that we can’t count the rational numbers in order but we don’t all agree that we can’t pass the rational divisions between two points in order, even though it’s no different in kind.
• 2.1k
I’m saying that passing in order is a sequential series of events with no start and so cannot be started.
'Sequential' is the problem word here. I would say that passing in order is not sequential, because the events are not sequential if we use the usual meaning of being in order-preserving bijection with the natural numbers.

We cannot 'start' a sequence that has no start because by definition a sequence has a start - the item that is the image of natural number 0 under the bijection. But the set S, with the natural order, is not a sequence, so there is no sequence to be started. To turn S into a sequence we need to change the order. But since order-preservation is required, that cannot be done, hence there is no sequence, and no problem.
• 3.9k
It doesn't resolve the dichotomy paradox.

What is this dichotomy paradox?
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal