## Zeno's paradoxes in the modern era

• 3.2k
Yes.
• 5.7k
OK. End of problem. Which paradox is next?
• 3.2k
:cool:
• 8k
I don't follow what you are claiming here.

I'm claiming that it's impossible to count in order the rational numbers between 0 and 1 and that for the exact same reason it's impossible to pass through in order the rational-numbered distances between 0m and 1m.

May I ask, have you studied differential calculus and limits?

Up to A-Level, yes. And calculus can't help in this case. That the sum of a geometric series is finite can't show that it's possible to count in order the rational numbers between 0 and 1 and for the exact same reason can't show that it's possible to pass through in order the rational-numbered distances between 0m and 1m.

Returning to my previous point, Wolfram's argument is that if it takes n seconds to move from 0m to 0.5m and n/2 seconds to move from 0.5m to 0.75m, and so on, then it will take 2n seconds to move from 0m to 1m. But this is akin to arguing that if it takes n seconds to count in order the rational numbers between 0 and 0.5 and n/2 seconds to count in order the rational numbers between 0.5 and 0.75, and so on, then it will take 2n seconds to count in order the rational numbers between 0 and 1. Such an argument begs the question because Zeno's argument is that it's impossible to count in order the rational numbers between 0 and 1 (or move from 0m to 1m) tout court – i.e. its impossibility has nothing to do with it allegedly taking an infinite amount of time.

That there's a finite sum to a geometric series of time intervals is a red herring.
• 5.7k
I'm claiming that it's impossible to count in order the rational numbers between 0 and 1 and that for the exact same reason it's impossible to pass through in order the rational-numbered distances between 0m and 1m.

Tough. It is possible to travel a metre. I've even done it a few times. Bet you have, too. SO there is something wrong with your account.

Now, why do you feel the need to replace the geometric sequence in Zeno with the rational numbers?
• 8k
Tough. It is possible to travel a metre. I've even done it a few times. Bet you have, too. SO there is something wrong with your account.

If it's impossible to pass through in order the rational-numbered distances between 0m and 1m and if it's possible to travel from 0m to 1m then when we travel from 0m to 1m we don't pass through in order the rational-numbered distances between 0m and 1m.

Now, why do you feel the need to replace the geometric sequence in Zeno with the rational numbers?

It doesn't need to be every rational number. It can be every 1/(2n). Or Zeno's paradox can consider every rational numbered-distance rather than just every 1/(2n) metre. It makes no difference, so yours is a strange response. Zeno isn't saying that you have to pass through the 1/2- and 1/4-way points but not the 1/5-way point. :brow:
• 3.9k
Zeno's paradox is a mismatch between a priori and a posteriori knowledge. I think it's wonderful. Do we trust logic and math or do we trust our eyes? Both logic and observation are standard ways of acquiring knowledge. It's a difficult choice.

However, if I find any fault in Zeno it's that he made a choice, a choice in favor of math and logic and declared motion as an illusion. Since Zeno actually doesn't say why he made that choice, I think he was, and is still, fooling around with us.

We can, by that reason or to be truthful, lack of reason, declare observation trumps logic and math and we'd still be on an equal footing as the great Zeno himself.

The better way out of the paradox is to realize that both [logic/math] and observation are true but that there's an explanation as to why they don't agree.

I think calculus does the trick.
• 3.9k
That there's a finite sum to a geometric series of time intervals is a red herring.
2h

I don't think it has do with time but the distance too is a geometric series, no? What do you get if you add all the terms in the series (1/2 + 1/4 + 1/8 + 1/16 +...}? 1 no? If the math shows anything it's that adding smaller and smaller quantities to a number doesn't actually result in an unmanageable infinity (Zeno would've loved that). Rather the sum tends to a finite limit - exactly what we need to resolve the paradox.
• 533
A unit of distance can be infinitely halved, but these mathematical "tasks" are not required for motion. Otherwise, only mathematicians could move.
• 6k
That question can only be fairly answered by an Aristotelean. I am not one, but I think there are plenty on this board. IIRC Metaphysician Undercover is one (apologies in advance if I have misread your position MU).

I believe I've read Aristotle's work quite well, and thanks for the reference.

Does anyone philosopher still think that they prove that change is impossible?

Aristotle demonstrates that change (becoming) is fundamentally incompatible with being (represented as a describable state. If change is described in states of being, there would be one state followed by a different state. To account for the change between them we'd have to posit another state as intermediary. But this would just introduce another different state, so we'd need to posit more states to account for the change, resulting in an infinite regress of described states, without any real change. So he concludes that a state of being is fundamentally distinct from becoming, change.

Accordingly he divides reality into two distinct aspects, form and matter. Form is described as actual, active, while matter is described as potency, or potential. All reality is composed of these two aspects, and the separation is theoretical only. The difficult part to understand is the distinction between "forms" in the physical world, and "forms" in the human mind. In the physical world, forms are what have actual existence, and are actively changing. In the human mind forms have actual existence, as what is real to the mind, but they exist as formulae which are described, or defined. states of being. So there is an incompatibility between what is "actual" within the human mind, and what is "actual" in the physical world. This is what creates paradoxes like Zeno's. In the physical world, the forms of existence are actively changing and this is fundamentally incompatible with the forms by which the human mind describes physical existence, as states of being.

I'm claiming that it's impossible to count in order the rational numbers between 0 and 1 and that for the exact same reason it's impossible to pass through in order the rational-numbered distances between 0m and 1m.

That's right, there can be no definite order to the rational numbers, because any "first" number is arbitrary and randomly chosen. The better question is whether the principles which assert zero as a rational number are truly consistent, and this questions the validity of negative integers. But that's off topic of the thread.
• 8k
Rather the sum tends to a finite limit - exactly what we need to resolve the paradox.

It doesn't resolve the dichotomy paradox.
• 12.6k
Zeno's paradoxes are a good example of theory-worship--you take the theory to trump reality, and when the theory results in something absurd, you conclude that we must have reality wrong rather than realizing that we're f---ing up theoretically somehow.
• 8k
Zeno's paradoxes are a good example of theory-worship--you take the theory to trump reality, and when the theory results in something absurd, you conclude that we must have reality wrong rather than realizing that we're f---ing up theoretically somehow.

Or the paradox shows that reality isn't as we think it is, e.g. space isn't infinitely divisible and/or motion isn't continuous.
• 12.6k
Or the paradox shows that reality isn't as we think it is, e.g. space isn't infinitely divisible

"Space is infinitely divisible" is theory. So, right, when that theory leads you to conclude something obviously absurd, you don't go with the absurdity. You realize you screwed up somewhere.
• 8k
A unit of distance can be infinitely halved, but these mathematical "tasks" are not required for motion.Luke

If motion is continuous (and space infinitely divisible) then I must pass through each 1/(2n) unit of distance (even if I don't stop at them), and surely that counts as a "task".

"Space is infinitely divisible" is theory. So, right, when that theory leads you to conclude something obviously absurd, you don't go with the absurdity. You realize you screwed up somewhere.

Ah, I see. Misunderstood you.
• 533
If motion is continuous (and space infinitely divisible) then I must pass through each 1/(2n) unit of distance (even if I don't stop at them), and surely that counts as a "task".

I suppose. Is there a problem?
• 8k
I suppose. Is there a problem?Luke

The problem is that it is no more possible to pass through each 1/(2n) unit of distance than it is to count each 1/(2n). It's a task that can't even start.
• 533
What makes it impossible?
• 8k
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.
• 533
You assume that mathematical tasks must be completed before motion can begin. I reject this idea. Motion does not require the completion of any mathematical tasks.
• 8k
You assume that mathematical tasks must be completed before motion can begin. I reject this idea. Motion does not require the completion of any mathematical tasks.Luke

I don't know what you mean by a mathematical task. Walking 1m requires physically passing through the 0.5m mark, and before that physically passing through the 0.25m mark, and so on. If space is infinitely divisible and motion continuous then each 1/(2n)m mark physically exists and must be physically passed through.
• 533
So why do I need to figure out "the first distance I pass through" before I can pass through any?
• 8k
So why do I need to figure out "the first distance I pass through" before I can pass through any?Luke

You don't need to figure it out. You need to actually do it, whatever's going on in your head. Just as you there's no first number to count to, making counting them impossible, there's no first distance to move to, making moving through them impossible.
• 533
Why equate the impossibility of counting with the impossibility of motion?
• 8k
Why equate the impossibility of counting with the impossibility of motion?Luke

Because the principle behind counting each 1/(2n) number is the same as passing each 1/(2n)m mark.
• 533
What does "there's no first distance to move to" mean (in practice)? Again, you seem to be saying that I need to figure out the first distance before I can pass through it. I still don't see why this mathematical step is necessary.
• 8k
I can't explain it in any simpler way. You might as well ask me what "there's no first number to count to" means in practice. It means precisely that.
• 533
Fair enough, and I have no issue that there might be a mathematical problem. I just don't see that there's any related problem of motion. I'm suggesting that the assumed relation between the maths and the motion, and that one somehow prevents the other, may be the only problem here.
• 8k
Again, you seem to be saying that I need to figure out the first distance before I can pass through it. I still don't see why this mathematical step is necessary.Luke

I'm not saying you have to figure it out. I'm saying that, assuming the infinite divisibility of space and continuous motion, each 1/(2n)m mark must be physically passed in ascending order, but that because there's no first 1/(2n)m mark, movement cannot start, just as because there's no first 1/(2n) number one cannot start to count each 1/(2n) number in ascending order.
• 533
I'm not saying you have to figure it out. I'm saying that, assuming the infinite divisibility of space and continuous motion, each 1/(2n)m mark must be physically passed in ascending order, but that because there's no first 1/(2n)m mark, movement cannot start, just as because there's no first 1/(2n) number one cannot start to count each 1/(2n) number in ascending order.

I don't see why I should accept that the mathematical problem of infinite divisibility should prevent movement from starting.
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