We know exactly how to carry on. — Banno

Thanks, Banno. I knew I would not get it quite right.

:wink:

My apologies for my curtness. I'v'e in mind heading off a divergence into discussions of rules.

Zeno's paradox is a convergent series, dude. It doesn't matter what order you sum it in. — frank

Yep.

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$

is

$\sum_{n=1}^{\infty} \frac{1}{2^n}$

that is, 1.

there are an infinite number of steps in this description of the distance between 0 and 1, but that simply does not stop it being traversed in a finite time.

Zeno mistook an infinite description of motion for an infinite obstacle to motion.

but that simply does not stop it being traversed in a finite time.

Zeno mistook an infinite description of motion for an infinite obstacle to motion. — Banno

Zeno saw himself as proving that all motion is an illusion. You're saying that he's wrong, but you aren't providing an argument. That's fine.

My apologies for my curtness. I'v'e in mind heading off a divergence into discussions of rules. — Banno

Of course. The ghost at the feast, perhaps.

Zeno saw himself as proving that all motion is an illusion. You're saying that he's wrong, but you aren't providing an argument. That's fine. — frank

Well, one could simply argue that the argument is not a proof, but a reductio of a certain approach to space, time and infinity.
We can compute when Achilles will achieve his goal as soon as we know how fast he is running and how large the distance is. That figure does not change as the race progresses. Unless Zeno can find a fault in that calculation, it proves that the issue is in the approach to the question, not in the situation as described.
In a fixed period of time, Achilles passes an increasing number of distances, culminating, no doubt, in his traversing infinitely many distances in an infinitely small amount of time. Zeno seems to think that he takes a non-infinitesimal amount of time to traverse an infinitesimally small distance.

Zeno pointed out the impossibility of enumerating, in order, the dense order of rationals.

E.g, starting from 0, what is the next rational number to count? since this doesn't have an answer (when counting in order), this means that the topology of the rational numbers cannot represent dynamics. Sure, the rationals can represent displacement, including an infinite sum of displacements, but not the process of displacement, namely motion.

To represent motion in a way that avoids the paradox, requires a smooth and differentiable continuous topology that doesn't contain points that are in need of traversal, but only open sets that can finitely intersect to create spots, but not infinitely often so as to create points. Yet on the other hand, to represent positions requires a discrete point-based topology of infinitely thin spikes that doesn't blur position information. Hence motion and position require incompatible topologies.

Fourier Transforms are the best way to visually understand the solution to zeno's paradox, but on the proviso that the FT is understood as creating an output topology (e.g motion) from a different input topology (e.g position).

It's not well enough known, to summaries?

see .

No need to overcomplicate things.

No need to overcomplicate things. — Banno

That's not an argument either. Some people are just emotionally averse to paying attention to things like Zeno's paradox. You can lead a horse to water, but you can't make him drink, that sort of thing.

And some can’t do the maths.

And some can’t do the maths. — Banno

What does math have to do with the structure of space and time? Read the SEP article on Zeno's paradox.

What does math have to do with the structure of space and time? — frank

:grin:

I just did. Did you? Which paradox would you like explained?

Maybe tomorrow.

What I need is for you to explain why you think calculus tells us something about space and time. It's in the article.

No one ever says either of those things. You're arguing with someone in your head who knows no more about mathematics than you do. — Srap Tasmaner

Huh? Someone in my head knows more about mathematics than I do? Isn't that contradictory?

...there are an infinite number of steps in this description of the distance between 0 and 1, but that simply does not stop it being traversed in a finite time. — Banno

Obviously then, the description is wrong.

Unless Zeno can find a fault in that calculation, it proves that the issue is in the approach to the question, not in the situation as described. — Ludwig V

The approach to the situation, is logically prior to the description. What appears to be the case, is that the description, and consequently the approach, are both wrong.

To represent motion in a way that avoids the paradox, requires a smooth and differentiable continuous topology that doesn't contain points that are in need of traversal, but only open sets that can finitely intersect to create spots, but not infinitely often so as to create points. Yet on the other hand, to represent positions requires a discrete point-based topology of infinitely thin spikes that doesn't blur position information. Hence motion and position require incompatible topologies. — sime

Or, we can represent motion as discontinuous, which is the way that quantum physics seems to demonstrate is the real way. The particle has a position, then it has another position, without traversing the intermediary. I believe, that what happens in between cannot completely be represented as "a smooth and differentiable continuous topology". Issues with the wavefunction demonstrate that this is not quite right. So what happens in between ought to be represented as truly unknown, though it is actually represented in a not very accurate way, as a continuous topology of superpositions.

No need to overcomplicate things. — Banno

Reality is complex, this is philosophy, and the common mistake is to oversimplify. Sometimes Ockham's blade just doesn't cut it.

Or, we can represent motion as discontinuous, which is the way that quantum physics seems to demonstrate is the real way. The particle has a position, then it has another position, without traversing the intermediary. I believe, that what happens in between cannot completely be represented as "a smooth and differentiable continuous topology". Issues with the wavefunction demonstrate that this is not quite right. So what happens in between ought to be represented as truly unknown, though it is actually represented in a not very accurate way, as a continuous topology of superpositions. — Metaphysician Undercover

Yes, there is no traversing anything unless a particle is in a smooth motion-state as a result of applying a motion operator to it, which cannot be the case if the particle is in a spiky position-state as a resulting of applying a position operator to it. The question is, at what level of explanation should this incompatibility be situated? at the physical level, as physics usually assumes, or at the level of the rules of mathematics?

I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. Hence the mathematical definition of differentiation that we inherited from them and use today, isn't defined as a resource-transforming operation that takes a mutable function and mutates it into its derivative; rather our classical differentiation is merely defined as a mapping between two stateless and immutable functions.

But if Zeno's paradox is to be exorcised from calculus, such that calculus has a dynamical model, then I can't see an alternative than to treat abstract functions like pieces of plasticine, that can be sliced into bits or rolled into a smooth curve, but not at the same time.

Zeno mistook an infinite description of motion for an infinite obstacle to motion. — Banno

Or it was a critique of Plato and other mainstream philosopher's idea of the potential infinite.

We should remember that we unfortunately have lost Plato's original book, where likely the Eleatic school would have made their own viewpoint. Now we have just the texts of those who were against the Eleatic school, the "mainstream" Socratic-Platonic school.

Boy, would that book be nice to resurface. It's said that Zeno had even more paradoxes. Loved to have known what they were.

I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. — sime

No, but the issue in the core of Zeno's paradoxes. And we should note that calculus had problems with the infinitesimals, like the famous critique from bishop Berkeley.

And basically logism and the set theoretic approach hoped to find some rigorous ground for calculus, but the paradox resisted to die with Russell's paradox. And with Cantor's hierarchial system, there's still questions...

I really think that there's more to it than we know now. Math is just so beautiful and so awesome.

Wittgenstein would say it worked, so, so what?

What I need is for you to explain why you think calculus tells us something about space and time. It's in the article. — frank

I'm not claiming calculus tells us what space and time are; I'm denying that this is a coherent question.

l

I'm not claiming calculus tells us what space and time are; I'm denying that this is a coherent question. — Banno

If we want calculus to solve Zeno's paradox, we have to assume that the math is telling us something about space and time.

If we want calculus to solve Zeno's paradox, we have to assume that the math is telling us something about space and time. — frank

It doesn't solve, it dissolves.

The paradoxes only appear to work because they slide between the mathematical and ontological games.

Pick one, and set it out, and we can see how this happens in the detail.

For starters, I think we can agree on what space is. What is time and how it relates to space is another question.

One can argue that calculus doesn't solve Zeno's paradoxes as we don't have yet a clear understanding of infinity.

The point is that the paradox isn't fundamentally a math problem. It's a series of questions that point to a contradiction.

Applying the Mathematical Continuum to Physical Space and Time: As noted in §1.2, the ‘received view’ of Zeno (developed in the latter part of the Twentieth century by philosophers developing the ideas of Grünbaum 1967) aimed at showing how modern mathematics resolves the paradoxes. However, central to this project was the recognition that a purely mathematical solution is not sufficient: the paradoxes not only question abstract mathematics, but also the nature of physical reality. So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space. It would not answer Zeno’s paradoxes if the mathematical framework we invoked was not a good description of actual space, time, and motion — SEP

One can argue that calculus doesn't solve Zeno's paradoxes as we don't have yet a clear understanding of infinity. — ssu

I guess you could put it that way.

So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space. — SEP

See how explicit the admixture of two differing language games is here?

One can argue that calculus doesn't solve Zeno's paradoxes as we don't have yet a clear understanding of infinity. — ssu

What we have are ways of talking, language games, a grammar, or a paradigm - whatever you want to call it. Infinity is a mathematical notion that we can use to calculate physical results. It is not an ontology.

See how explicit the admixture of two differing language games is here? — Banno

I suppose so, yes.

So the paradox involves confusing a way of talking, the maths, with a description of how things are, the ontology. We can be pretty confident that space is not infinitely divisible and yet still use calculus to plot satellite orbits.

(And all of this makes sense only if we agree that there is a whole number between one and three.)

So the paradox involves confusing a way of talking, the maths, with a description of how things are, the ontology. We can be pretty confident that space is not infinitely divisible and yet still use calculus to plot satellite orbits. — Banno

Zeno wasn't arguing that we can't plot satellite orbits with acceptable precision.

I do admire your devotion to the practical. Detaching yourself from it and purely following the contours of the mind will set you out in front of contradictions.

Zeno wasn't arguing that we can't plot satellite orbits with acceptable precision. — frank

Well, he was, from what we know, arguing that motion was not real.

Paradoxes occur when we say things incorrectly. The world cannot be wrong, but what we say about it can be.

Well, he was, from what we know, arguing that motion was not real. — Banno

Yes, but we can get along just fine in an illusion. Contradictions are just little sign posts that things aren't exactly as we're imagining them. They can't be.

Paradoxes occur when we say things incorrectly. The world cannot be wrong, but what we say about it can be. — Banno

Thank you, John Locke. Your faith is commendable.

Not faith so much as care and attention.

Have you some alternative? :wink: