Thought Experiments in Mathematics: Gabriel's Horn • Hacker News Board

Re: Thought Experiments in Mathematics: Gabriel's Horn

#41

Post by blippityBlop » Sat, Oct 08, 2016, 9:54 AM UTC

drdeca wrote: post #27

blippityBlop wrote: post #24

Okay, tra la la. But can we all take note that the volume is PI, and that PI's decimal places are also infinite. In this moment, I take issue with the idea that PI represents a "finite" volume. PI, is more accurately a "known-volume-other-than-infinity." Meanwhile, angels dance upon the head of a pin somewhere.

You can just re-scale it to have volume 1 if you want. Also, uh, you realize that pi is less than 5, and more than 0, right? If you draw a circle with radius 1, the area would be pi just as much as the volume of this at some scaling. Would you say that such a circle has infinite area? That seems like it would be refusing to use a term as it is generally used. You can object by saying that circles do not exist, I supp…

So, if I was going to argue about things not existing, the concept of pi, the irrational number, would be my first target. It's just an artifact of bean counting that people chase their tails over.

To say that pi is a non-repeating fraction with infinite decimal places, is to say that we cannot ever precisely know the boundary and definition of a circle, forever zooming in on a point never met.

That we find circles, and capitulate to what must be their true or approximate location is a cheat, and mere luck at the integrity of atoms or planck space or what have you.

But still, it's garbage in the same way that the divide-by-zero rule is garbage. Granted, dividing something into nothing isn't a practical operation. But, I can divide zero into as many parts as I like, and it's still zero.

So, if people want to keep stacking infinitesimally small tenths, hundreths, thounsandths, millionths and so on, to the end of any particular number, forever, well, forever sounds like an infinite quantity to me, and I'll argue to point that out.

Unroll the circumference of a circle. Where does it end? Approximately near X? Approximately???

Tell me the area of a circle. Approximately within this fuzzy boundary? Approximately???

I am dubious.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#42

Post by blippityBlop » Sat, Oct 08, 2016, 9:57 AM UTC

tedunangst wrote: post #25

blippityBlop wrote: post #24

Okay, tra la la. But can we all take note that the volume is PI, and that PI's decimal places are also infinite. In this moment, I take issue with the idea that PI represents a "finite" volume. PI, is more accurately a "known-volume-other-than-infinity." Meanwhile, angels dance upon the head of a pin somewhere.

uhm, 2/7 has infinite decimals. it's not infinite.

It's 22/7, and zeno's paradox aside, pi has an infinite component to it, if you look at it the wrong way.

That the horn's tail (or mouth piece, as it were) extends into an infinitely small value approaching zero, is as fair as the never-ending irrational fraction pi represents.

Otherwise, the horn's volume is undefined. Where does the tip of the horn cross zero?

It doesn't??? Hmmm, that sounds like a non-finite value to me.

Discussions like this often wax political more than analytical. There are four lights.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#43

Post by drdeca » Sat, Oct 08, 2016, 9:48 PM UTC

blippityBlop wrote: post #41

drdeca wrote: post #27

Earlier quoted context omitted.

You can just re-scale it to have volume 1 if you want. Also, uh, you realize that pi is less than 5, and more than 0, right? If you draw a circle with radius 1, the area would be pi just as much as the volume of this at some scaling. Would you say that such a circle has infinite area? That seems like it would be refusing to use a term as it is generally used. You can object by saying that circles do not exist, I supp…

So, if I was going to argue about things not existing, the concept of pi, the irrational number, would be my first target. It's just an artifact of bean counting that people chase their tails over. To say that pi is a non-repeating fraction with infinite decimal places, is to say that we cannot ever precisely know the boundary and definition of a circle, forever zooming in on a point never met. That we find circles,…

Are you objecting that infinite precision or irrational numbers do not physically exist?

That may be. Or it might not. Whether (physical) space is infinitely divisible or not is, I think, an open question. I think it is even plausible that the answer might be that it can be described equally well both ways? Not sure.

I don't mind thinking about abstract objects which do not physically exist. (Though, I generally consider abstract objects to exist in some sense, though you may disagree, and that's fine.)

If you don't care to think about them, that's fine also.

I believe that thinking about them is often useful. Treating pi as existing allows for reasoning about things more efficiently, even if it turns out that it results in inaccuracies in predictions about the physical world which are far smaller than our ability to measure/notice. (If we can't notice the results of the error, it doesn't seem to be much of a problem.)

That doesn't mean you have to accept pi as actually existing. I think it would be to your benefit to at least accept its use though. I do consider it to actually exist.