You don't have any empirical proof that movement is either discrete or continuous; however, mathematically, calculus can show net change and rate of speed across differences on a continuous line. Your supertask, on the other hand, is nothing but a subjective classification of ill fitting paradoxes.

And I am sure that was your only take away.

Maybe this one is harder for people to sink their teeth into.

Gabriel's horn is an object that exist in math which has finite volume, but infinite surface area, that is a conundrum if I have ever seen one. Such objects should not exist, but mathematically we can show that the volume converges to a finite point, while the surface area diverges to infinity.

You can say, well it is not in the real world, and while it may be true I can't find a horn and point to it; however, it does exist in mathematics, and this is the math section of these forums and the title of this thread is "Mathematical Conundrum or Not?". The horn absolutely deserves its spot here, even if grasping it is not as intuitive as the other two I posted.

I don't really see you as an authority on what is and what is not a paradox. I mean all you have here is an assertion and a false one at that. On the other-hand academically Gabriel's horn is widely viewed as paradoxical. So you don't think is a paradox, OK fine, I don't really care.

Also volume is the amount of space an object takes up, paint or no paint.

Well this is not my paradox, I didn't invent it. It is a well known paradox, and widely recognized as such. Also the mathematical proof is posted in the OP. Saying there is no mathematical basis for this just tells me you can't read the math, as it is posted right there for you to review.

Claiming it is abstract does't prove that Gabriel's horn is not a mathematical conundrum.

Abstract things exist in reality, as reality is a very very very broad term.

Gabriel's horn exist in reality, the math was posted in the OP.

Any container or solid object that has an endless surface area, but a finite volume is paradoxical, abstractly or otherwise.

Volume is the amount of space it takes up, so if it has endless surface area it should have endless volume. However, Gabriel's horn does't, and this is why it is widely recognized as a paradox.

Like I said. You are confusing abstract and physical properties that happen to have the same name. — tom

If you recall I never said or agreed to any such notion in the last thread. I avoid that line of thought for a reason. There is nothing which says we can't think about this in more practical terms.

In what way is this in need of 'resolution'? You haven't stated a problem with this scenario. Is there some law somewhere being broken, like infinite surfaces must enclose infinite space? There is obviously no such law, as demonstrated by this example. — noAxioms

The paradox, seems clear to me, we have a container that stretches on forever, yet it has a finite volume.

It is only paradoxical if the same thing both converges and diverges. — noAxioms

The horn both converges and diverges, so it fits your personal take on what is needed for a paradox.

Clearly the paint would not run out, as it hasn't in your example. It covers the entire surface, and doesn't even need to be spread out to do so, since it has finite thickness (all the way to the center line) at any point being painted. — noAxioms

So you are suggesting a finite amount of paint that goes on forever. So in your suggestion the volume of the paint both converges and diverges? Well, mathematically we can prove the volume of the paint converges, that means there is a limited amount of it, but if you want to claim it is a endless bucket of paint go for it. The math just does not back you up.

So you are suggesting if it was filled with paint, you could use a finite amount of paint to paint an endless surface.


It seems to me, that you'd run out of paint, and even if you could stretch the paint infinitely thinner, that still does not resolve the paradox. As abstractly what you have is a cone with a converging volume and a diverging surface area.

This one is a bit trickier and as far as I know it has not been resolved. So if we can't get it off the ground, I have others waiting.

Any thoughts on the horn?

Paradox or not, I think this one passed the mathematical conundrum standard, and it generated far more discussion than I expected.

I am not sure what you are referring to , perhaps you are thinking about one of the test for convergence. I would have to review them to be sure.

I don't know how to input math notation into here so you'll have to bear with my lack of proper notation but by definition the sum of an infinite series is the limit S= lim_n→∞ S_n, if the limit exist then the infinite series converges to the sum S. If the limit does not exist the infinite series diverges.

You are describing infinite partial sums and if your series converges then you have net change, aka movement.

You have math degrees, but you never saw any of this in class?

It explains how we can have a finite net change from a to b with infinite partial sums, which is what the paradox was.

How does Achilles achieve a net change with infinte small intermediate distances to cross. Zeno's paradox only existed because no one knew how to sum infinite parts to a finite amount. Now we do, so paradox resloved.

It's neat what I can do that with three little dots, right? Math is very cool.

I know they are a real thing, but then they are backed by real evidence.

That's not sufficient to dismiss any theory that suggests that it is discrete, hence why such theories are studied. And I would argue that the logic of Zeno's paradox and supertasks would suggest that such theories must be correct. — Michael

I am sorry, but this is where I lose respect for philosophy. Zeno's paradox certainly doesn't prove such theories "must be correct". That is a very bold claim off the back of a few conceptual paradoxes whose relation is a subjective classification called supertask. As far as I can tell, there is not even consensus that all supertask are impossible, or even relevant.

In mathematics Zeno's paradox does not prove partial sums, it lends to the notion yes, but those are backed by formal proofs that have been through the works. Deciding something must be correct just off the back of this paradox is sloppy and lazy. These paradoxes are guides, not proof.

Because in math ". . ." means the pattern repeats forever and that was easier to type. You know shorthand.

That's the flawed assumption of Zeno's Paradox, that is what sequence and series resolved, a finite being can converge an infinite partial sums.

Watch, I a finite creature will converge an infinite partial sums to a finite number.

.3 +.03 + .003 + .0003 + .00003 . . .

On forever. That is an infinite series, do you recgonize it?

I could rewire it this way. .33333....

Which is 1/3 a finite sum of the infinite parts.

Real numbers are all whole numbers, rational numbers and irrational numbers and they are absolutely on the number line. Distance is definitely mapped with real numbers.

Numbers not on the number line are imagery numbers and infinity, as infinity is not a number. If you mean something else I would suggest you don't use the phrase real numbers.

I said in that very post you qouted it converges to a finite number.

Infinitesimal infinite parts. There are more parts than I can count without approximating somehow or using calculus.

The paradox was solved by partial sums a long time ago.

It truly does not matter. Infinite is just another word for more than I can count. Which is why we use calculus.

I don't care about the supertask thing, I feel it is moot. We don't need it one way or another.
--

Now back on topic...

We already know that motion is possible and we know there are infinite parts between Achilles and the turtle. Whether you want to say those parts are continuous or discrete, that does not really matter, as you can't count them anyways, so they are effectively continuous and effectively infinite. We know Achilles can pass the turtle. This sounds like a job for calculus.

Actually the conclusion in Wikipedia makes more sense than yours.

Hey man, I am just citing the reference you gave me, maybe next time don't use Wikipedia.

Maybe you should go back and read your own links.

Form one of your own links on Zeno's paradox.

Zeno's argument to reach the conclusion that either motion is not a supertask or not all supertasks are impossible.

Ya, I read your post the linked pages.

And given that completing a supertask is demonstrably impossible, it must be that motion isn't a supertask — Michael

It must mean motion is not a supertask OR supertasks are not impossible. At anyrate it is moot.

Who cares if I can count an infinite number of points or not, I don't need to I have calculus.

Take the room crossing example, where I have to keep going to the half way point. If that is a supertask it should be impossible for me to cross the room, but I can prove empirically that I am able to cross the room, hence motion is not a supertask or supertasks are not impossible. Making the aspect, when considering motion, of being a supertask moot.

In Thomson's lamp the end point is unknowable, but we know our end point already and we know our start point in Zeno's Paradox. So maybe a supertask is not the best way to think about this.

Maybe the concept of a supertask is what is being misplaced here. 0 is your starting point, now all I need is the end point, let's pick 1. Net change on a straight line is 1.

Physics is ultimately in the realm of mathematics. The entire backbone of science is mathematics, as well as the rest of the skeleton. Basic human comprehension is rooted in mathematics, everything is in the realm of mathematics. That's the very reason I study it.

That is the different between a convergent series and a divergent series. We know the distance between Achilles and the turtle converges to a finite number, so you are kind of arguing a moot point.

Also it should be pointed out that infinity is just an amount too big to measure. It could be discrete; it does not necessarily have to be continuous; such as, there are an infinite number of stars in the heavens.