[coldtea]

>It took 2040 years before science/math could answer Zeno's paradoxes.

And that's only if we assume calculus answered them. Which is not the absolute consensus of logicians/mathematicians.

Many seem to believe they are simply about limits and that's all, but they may also point to a more fundamental aspect of reality. E.g. from Wiki:

A suggested problem with using calculus to try to solve Zeno's paradoxes is that this only addresses the geometry of the situation, and not its dynamics. It has been argued that the core of Zeno's paradoxes is the idea that one cannot finish the act of sequentially going through an infinite sequence, and while calculus shows that the sum of an infinite number of terms can be finite, calculus does not explain how one is able to finish going through an infinite number of points, if one has to go through these points one by one. Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself, independent of how much time such an act would require.

Another way of putting this is as follows: If Zeno's paradox would say that "adding an infinite number of time intervals together would amount to an infinite amount of time", then the calculus-solution is perfectly correct in pointing out that adding an infinite number of intervals can add up to a finite amount of time. However, any descriptions of Zeno's paradox that talk about time make the paradox into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler inherent paradox that does not at all consider any quantifications of time. Rather, this much simpler paradox simply states that: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible". The calculus-based solution offers no insight into this much simpler, much more stinging, paradox.

A thought experiment used against the calculus-based solution is as follows. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle in finite time, the counting process will be complete, and we could ask Achilles what the greatest number he counted to was. Here we encounter another paradox: while there is no "largest" number in the sequence, as for every finite number the turtle is still ahead of Achilles, there must be such a number because Achilles did stop counting.

[im2w1l]

>Imagine that Achilles notes the position ... and so on

If in logic you assume/imagine something, and reach a contradiction, then your reaction should be to reject that assumption.

In this case we reach the conclusion that Achilles cannot note all the positions.

[coldtea]

That's not adding anything new. The very idea of the thought experiment is that we reach a contradiction and that we therefore should reject the assumption.

The problem, and the whole point of the thought experiment, is that the assumption is tied to Achilles being able to make the distance, so we should ditch that too.

[hodwik]

To be honest, I don't actually think calculus solves Zeno's paradox either, I was just using it for the sake of the argument.

I'm of the opinion that Achilles and the Tortoise is solved by there being a smallest distance, a pixel (Not smallest measurable. Smallest period.)

I make amends with Achilles' Arrow by believing the universal clock is independent of the moments that it measures -- like the internal clock of a computer's being independent from a pixel arrow moving across a pixel screen.

[chrisdone]

But the point of Zeno's paradox is to show that our model of reality is purely definitional, rather than objective in any absolute sense.

[choosername]

doesn't quantum mechanics mean everything has a lowest resolution of discrete steps, ie. Plank-time and Plank-length. In that sense, calculus over finite terms is enough.

The paradox is illogical anyhow, it fails to equate simple v*t=s.

[coldtea]

>doesn't quantum mechanics mean everything has a lowest resolution of discrete steps, ie. Plank-time and Plank-length. In that sense, calculus over finite terms is enough.

That's one solution yes, but whether the universe is continuous or discreet is not a "solved" problem in physics in general. The lowest resolution could just be a "sampling restriction" (or quantum laws etc) and not the actual reality. There have been some experiments to discern between the two cases, but nothing conclusive IIRC.

>The paradox is illogical anyhow, it fails to equate simple vt=s.*

That's not being illogical. vt=s is not logic, is just a formula that we deduced and which matches what we see in everyday life (and in fact it's not absolute, e.g. when it comes close to the speed of light etc).

The paradox just checks whether vt=s is compatible with certain descriptions of the chain of events that look intuitive at first.