What does calculus prove about Zeno?

[thinkandmull]

Careful now, I haven’t done advanced math in quite awhile, so if you could walk me through this I’d be thankful. I have for a long time believed Zeno’s paradox to be unsolvable. How can something be both infinite (infinite points) and finite (beginning and end). Some say that calculus shows how something infinite can have a finite solution. I can’t remember how the equations go, but to my mind these days its sounds to me that calculus is bunk too, if that is what it says. Finite and infinite are opposed. Does calculus attempt to rigorously prove that the infinite can be finite, and if so, why is this not simply a new paradox?

 

[STT]

Careful now, I haven’t done advanced math in quite awhile, so if you could walk me through this I’d be thankful. I have for a long time believed Zeno’s paradox to be unsolvable. How can something be both infinite (infinite points) and finite (beginning and end). Some say that calculus shows how something infinite can have a finite solution. I can’t remember how the equations go, but to my mind these days its sounds to me that calculus is bunk too, if that is what it says. Finite and infinite are opposed. Does calculus attempt to rigorously prove that the infinite can be finite, and if so, why is this not simply a new paradox?

What calculus says is that the infinite sum of element of a series containing finite numbers could be finite and indeed the Zeno’s series’s sum is finite.

 

[Beryllos]

Does calculus attempt to rigorously prove that the infinite can be finite, and if so, why is this not simply a new paradox?

I can’t explain it, but I think Cauchy worked it out and published it in 1821.

 

[Peter_J]

It might help a little if you specify which of Zeno’s paradoxes you have in mind. Or all of them?

 

[thinkandmull]

What calculus says is that the infinite sum of element of a series containing finite numbers could be finite and indeed the Zeno’s series’s sum is finite.

That is what I believe is faulty. The distance from point A to point B requires infinite halfsteps because if there isn’t another half step you are already there. The process goes on forever. Its not potentially infinite as Aristotle said. Is it, I ask him, potentially one yard, or potentially 3 feet only, or 36 inches? The process goes on forever. I have never considered what I know about calculus to answer the question. Instead it compounds it, adding for confusion. I believe our minds aren’t equipped to understand the infinite

 

[PJH_74]

As far as I can remember part of the proof for an Integral. There are plenty of uses for an integral, but finding the area under the curve is one of them. For finite areas we’d use a delimited integral of the equation that defines the are. I’ll leave it at there since it’s going on 25 years since I look a calculus class.

 

[Peter_J]

This isn’t an answer to the question, but one thing to keep in mind is that often times the use of “infinity” in mathematics is a matter of convenience/convention.

 

[Ridgerunner]

That is what I believe is faulty. The distance from point A to point B requires infinite halfsteps because if there isn’t another half step you are already there. The process goes on forever. Its not potentially infinite as Aristotle said. Is it, I ask him, potentially one yard, or potentially 3 feet only, or 36 inches? The process goes on forever. I have never considered what I know about calculus to answer the question. Instead it compounds it, adding for confusion. I believe our minds aren’t equipped to understand the infinite

As I recall, he posited Achilleus doing the stepping. But as we know, Achilleus was killed by Paris, so Achilleus can’t do any more steps. However many he did before he died…that’s it.

 

[STT]

That is what I believe is faulty. The distance from point A to point B requires infinite halfsteps because if there isn’t another half step you are already there. The process goes on forever. Its not potentially infinite as Aristotle said. Is it, I ask him, potentially one yard, or potentially 3 feet only, or 36 inches? The process goes on forever. I have never considered what I know about calculus to answer the question. Instead it compounds it, adding for confusion. I believe our minds aren’t equipped to understand the infinite.

You are thinking about time when you say “the process goes on forever”. That is true that the number of steps is infinite but we need this fact into consideration that the time needed for each step tends to zero too as we go from the beginning of series to the end. That resolve the problem. In fact infinite events in one instance is also possible, what God does.

 

[thinkandmull]

You are thinking about time when you say “the process goes on forever”. That is true that the number of steps is infinite but we need this fact into consideration that the time needed for each step tends to zero too as we go from the beginning of series to the end. That resolve the problem. In fact infinite events in one instance is also possible, what God does.

I don’t feel like that resolves it because the time would be infinite as well. Infinite and finite are two concepts that are mutually exclusive but we are thrown into a world where they appear to merge. That is why I find it a paradox, one that we can’t get a hold of