Mathematical Conundrum or Not? Number Three
Think of an ice cream cone where it is possible to eat all the ice cream but not the cone, because even though the ice cream fills the cone, it is finite, but the cone goes on forever. This paradox is commonly known as Gabriel's Horn.
When you graph the function y=1/x on [1, inf) and then rotate about the x-axis you get Gabriel's Horn, an object that has a finite volume but infinite surface area.
Here is a visual:
https://ngthuhoa.files.wordpress.com/2011/09/gabriels_horn.gif
If you want to review the math, it is explained here, it is not too heavy but you need some calculus (starting on page 2).
http://www.math.montana.edu/pernarow/m172/resources/Gabriels_Horn_new.pdf
So how it is possible this horn can have limited volume but endless surface area?
When you graph the function y=1/x on [1, inf) and then rotate about the x-axis you get Gabriel's Horn, an object that has a finite volume but infinite surface area.
Here is a visual:
https://ngthuhoa.files.wordpress.com/2011/09/gabriels_horn.gif
If you want to review the math, it is explained here, it is not too heavy but you need some calculus (starting on page 2).
http://www.math.montana.edu/pernarow/m172/resources/Gabriels_Horn_new.pdf
So how it is possible this horn can have limited volume but endless surface area?
Comments (52)
Wise advice.
Fortunately, it is impossible to sit on it, because it has no tip. The pointy bit just recedes endlessly, never culminating in a spike. The ultimate in child-safety mathematical structures.
As for getting it off the ground, that would be impossible because, even though it has finite volume, and hence finite mass (if we assume constant density), its moment of inertia would be infinite because of its being infinitely long. So it would require an infinite torque to rotate it to an erect position.
Short version - funny things happen with infinity. (one reason why maths is so much fun)
Unlike the horn, my post had a real point. :) That was informative and fun to read though. :up:
In mathematics, any volume can be divided in such a way to cover any surface.
This is just like Zeno's paradox. The paradox arises from the confusion of abstract properties with real ones of the same name.
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.
It is trivial to divide any volume to cover an infinite surface. There are plenty of convergent infinite series that will divide the volume for you.
Quoting Jeremiah
Like I said. You are confusing abstract and physical properties that happen to have the same name.
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.
Quoting JeremiahClearly 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.
I see no paradox in need of resolution. The volume converges and something different (the area) does not. It is only paradoxical if the same thing both converges and diverges.
The paradox, seems clear to me, we have a container that stretches on forever, yet it has a finite volume.
Quoting noAxioms
The horn both converges and diverges, so it fits your personal take on what is needed for a paradox.
Quoting 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.
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.
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.
No such object can exist in Reality, so it cannot be "abstractly or otherwise".
Gabriel's horn exist in reality, the math was posted in the OP.
that paints an infinite surface. 'goes on forever' is not what I said, and seems a sort of undefined wording.
The alternative is that there is some points along your surface that do not enclose volume and are thus not painted.
No, the volume is finite. You said that. There is finite (convergent as you put it) volume of ice cream, which could be 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.
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.
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.
Fair enough. The relevant definition of paradox that pops up says this:
The funny cone seems to fall under definition 'a' since it seems opposed to common sense to many people. So yes, it makes sense to 'resolve' such paradoxes by showing that the seeming contradiction is something that is actually the case. The mathematics (a computation of the area and volume) is linked in the OP, but not sure what part of that is a 'proof' of something.
'b' seems to be the opposite of 'a': something that seems true at first but false on closer inspection.
I guess my idea of a paradox falls under 'c', the most basic example being "This statement is false". Any truth value assigned to that seems to be incorrect. I've seen it resolved in law-of-form using an imaginary truth value (square root of false) just like imaginary numbers solve square root of -1. There is application for such logic in quantum computing.
I never contested the mathematics, which simply shows that the object indeed has infinite area but finite volume. I can think of more trivial objects that are finite in one way but infinite in another, and your cone did not strike me as a connundrum. But I retract my assertion that it is not a paradox. The definition above speaks.
Quoting JeremiahIndeed, it is only a mathematical object. A real one could not be implemented, growing too thin to insert ice cream particles after a while.
Interestingly, a liter of physical paint contains insufficient paint to actually cover a square meter of surface. There is a finite quantity of fundamental particles making up the volume of paint, and no fundamental particle has ever been found that occupies actual volume. So the paint is all empty space with effectively dimensionless objects which are incapable of being arranged to cover a given area without gaps. Instead, paint atoms work by deflecting light and water and such using its EM properties, not by actually covering a surface without gaps. Point is that this particular mathematical object has little relevance to even a hypothetical physical object.
She wasn't working with infinite amounts, but I wonder if there's a way to figure you could encompass an infinite area with a finite mass?
That is a feeling. The 18th century British invaders of Australia had a similar feeling when they first saw a platypus. When they found that the object in question was undeniably there in front of them, their 'should not exist' transformed to 'well, I am very surprised'.
Is the aim of this thread then to muse over the nature of the emotion we call Surprise?
Let me know if you figure it out.
I'll give you a hint, it has nothing to do with a platypus.
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?
What AndrewK may be getting to is that you are the topic. You just want the attention.
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.
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 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?
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.
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.
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.
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?