You might as well be asking Star Trek fans why they talk about Star Trek.

That's the fourth dodge.

I imagine there are plenty of Star Trek discussion boards and that on those boards, each thread has a point, that is generally posed as a question, eg:

- Do you think we will ever have teleporters?
- Do you think Spock has emotions but just doesn't show them?
- Who do you think would win in a fight between a Klingon and a Sontaran?
- Who is your favourite commander of the Enterprise?

or sometimes they might be propositions put out as challenges, and seeking opinions for or against, eg:

- I think Captain Kirk is really evil, and here's why
- I think it's unrealistic that nearly all aliens are bipeds with only superficial differences from humans

What is the proposition you'd like to put out for challenge, or the question you want to ask, in relation to the mathematical construct in the OP?

I am sorry, but I answered your question, the fact that it went over your head is not something I care about. Now you are dragging this thread off topic, so do you have anything to say about the horn? Can you resolve the paradox?

If you could, would that resolve the paradox?

you are dragging this thread off topic — Jeremiah

What AndrewK may be getting to is that you are the topic. You just want the attention.

Not interested, stay on topic.

Gabriel's Horn is a mathematical paradox, and, like I said before, this is the math section for discussion of math topics.

Any mathematician of any sort, and any philosopher with an interests in mathematics, generally finds that interesting. It is definitely worthy of contemplation and discussion. If you can't wrap your head around that notion, then maybe, just maybe you don't belong here.

Star Trek fans talk about Star Trek, as mathematicians and mathematically incline philosophers talk about math. There is no greater reason beyond that simple fact. Those that want to make it out as if there is some other hidden agenda here are drama seekers.

I plan on posting more of these paradoxes, these are exactly the type of content mathematically incline philosophers should be turning their skills towards. Some will generate a good amount of discussion, while others may not. I can't help that; however, Gabriel's Horn is one of the major paradoxes, it should be in these posts.

What paradox? You have described a mathematical structure. If you think there's a paradox in it that needs to be resolved, you need to explain what it is that is paradoxical about the structure and what you would regard as a resolution.

Paradoxes are either (1) logical contradictions, or (2) logically consistent but surprising to some.

Contradictions require resolution, but there is nothing contradictory about the structure, so it is not the first type. So if you see the structure as a paradox it must be the second type - surprising to some. A surprise does not need resolution.

So there's your answer - there is nothing to resolve.

We went over this already, when you read the whole thread let me know.

I plan on posting more of these paradoxes, these are exactly the type of content mathematically incline philosophers should be turning their skills towards. Some will generate a good amount of discussion, while others may not. I can't help that; however, Gabriel's Horn is one of the major paradoxes, it should be in these posts. — Jeremiah

I have a question for you.

Gabriel's horn is a paradox of Riemann integration, accessible to students of freshman calculus. As others have noted it's a paradox in the sense of being counterintuitive, not a paradox in the sense of being a logical contradiction.

Now, why aren't you bothered by the following more basic counterintuitive paradox of Riemann integration? Let's say we integrate 1 over the unit interval. That is, we compute the integral ∫dx between the limits of integration 0 and 1. Any calculus student will tell you the answer is 1.

But if you think about it, how can this be? We are literally adding up infinitely many zeros to get the number 1. And if we were to change the limits of integration to go between 0 and 2, we would be adding up infinitely many zeros to get an answer of 2. And the number of zeros, or dimensionless points, in the interval between 0 and 1 has the exact same cardinality as the interval between 0 and 2. You can see this by noting that the map f(x) = 2x is a bijection between [0,1] and [0,2].

How can Riemann integration make sense? How can we add up infinitely many dimensionless points to get 1; and then add up the same infinite number of dimensionless points to get 2? One answer is that it's mathematically true. But by your own argument, that's not very satisfying. We have a formalism that works out integrals. But what kind of sense does it really make to add up infinitely many dimensionless points and end up with a nonzero answer? And not only that, but by rearranging the points, we can get any answer we want.

Why don't you consider this an incomprehensible paradox? After all, once you believe that you can add up infinitely many zeros to get 1, and then add up infinitely many zeros to get 2; why should you be surprised that Riemann integration leads to other counterintuitive results?

Gabriel's horn rests on Riemann integration. If you object to Gabriel's horn, why don't you object to the more fundamental mystery of Riemann integration in the first place?

Put more simply: How does a collection of dimensionless points, each of size zero, add up to any volume we care to name? Isn't that a puzzler deeper than the mere rotation trick of Gabriel's horn?

0+0 is not a change in x.

0+0 is not a change in x. — Jeremiah

Which has what to do with anything I wrote?

Let me tl;dr this for you. Why are you so focussed on a particular paradox of Riemann integration, when it's Riemann integration itself that is philosophically murky?

The FTC is the total change F(b) - F(a) equal to the sum of small changes F(x of i) - F(x of i -1) and that is equal to the sum of the areas of rectangles in a Riemann sum approximation for f(x).

0+0 is not a change in x. You forgot the area slice.

The FTC is the total change F(b) - F(a) equal to the sum of small changes F(x of i) - F(x of i -1) and that is equal to the sum of the areas of rectangles in a Riemann sum approximation for f(x). — Jeremiah

Yes, that is the mathematical formalism.

So in this case you fall back on the mathematical formalism to ignore the philosophical paradox; but in the case of Gabriel's horn, you dismiss the mathematical formalism and focus on the philosophical paradox. Why is that?

Because your understanding of calculus is very poor and incorrect.

Because your understanding of calculus is very poor and incorrect. — Jeremiah

Even if that were true, it wouldn't answer my question.

Yes it did, you are trying to do calculus without a delta x. You are doing it wrong, that is not a paradox just an error.

Yes it did, you are trying to do calculus without a delta x. You are doing it wrong, that is not a paradox just an error. — Jeremiah

LOL.

I ask again: Why is it that in one case, you invoke the standard mathematical formalism to explain or ignore the underlying philosophical issues; and in the other case, you reject the standard mathematical formalism and insist that there's a paradox that must somehow be explained?

Did you even follow my meaning when I said delta x?

Well I gave it the old college try. At least @apokrisis didn't show up to hurl gratuitous insults. I'm outta here again.

You have no clue what I am even talking about.