Q&A: Infinity
Infinity
Question
Hello Rabbi,
I read your remarks about the cosmological proof, and about the distinction between potential infinity and concrete infinity.
The concept of infinity still bothers me quite a bit.
My question is basic: how can it be that at the time of writing this question, my finger passed through infinitely many parts? (I realize this is one of Zeno’s paradoxes.)
Even a potential reference to infinity — one that would say I did not pass through infinitely many parts, but rather that the distance I traveled can be divided into infinitely many parts — does not seem to me to solve the problem.
After all, there is still infinity here in the end, even if it is only potential. And why is it easier to understand that I passed through a potential infinity?
I cannot grasp the fact that infinitely many parts of a convergent series add up to a finite number. It is hard for me to grasp intellectually.
The more I think about it, the more I remain perplexed with the answer that it is indeed a great wonder. And without a real answer.
Do you have a satisfactory answer? Is it clear to you how infinity “works”?
Thank you,
Nathan
Answer
A simple grasp of infinity is of course impossible. But perhaps try thinking about it from the opposite direction. Take 1/2. Now add to it half the distance from it to 1 (that is, another 1/4), and you get 3/4. Now add to that again half the distance to 1 (that is, another 1/8), and you get 7/8. Now add again half the distance from it to 1, and so on and so on. You understand that even if you do this infinitely many times you will not reach 1, because each time you are only doing half of the remaining distance to 1. That means that this infinite sum converges to 1 even though infinitely many quantities are being added here.
Discussion on Answer
Nathan,
why not go the other way: think about a square from which you cut off half, and then cut the second half in half, and the next in half, and so on — you understand that in the end you are dealing with one whole.
https://steemit.com/steemstem/@nitesh9/ramanujan-s-sum-1-2-3-4-infinity-1-12-really
You will not get to one because you never finish infinitely many steps. But the limit is defined as something potential (you do not need to reach it in practice. Put simply, the limit describes what would happen if I kept going to infinity).
Okay, so the paradox is “solved” (to the extent that one can understand continuing infinitely many steps).
I am still left with a big puzzle: how can one traverse a distance that can potentially be divided into infinitely many parts?
But it may be that I am looking for an answer that does not exist.
Maybe understanding infinity is not really possible (as you already wrote in the first answer), except in the very basic sense of no-end.
(And maybe that is why Descartes attached such importance to infinity in the Meditations, since it is such an unusual concept.)
You are thinking about it incorrectly. One does not traverse infinitely many steps. The concept of a limit does not assume that. What it says is that as you increase the number of steps (= the number of steps is as large as you like), you get closer to 1 and never pass it. That is, there is no finite number of steps that will bring you to 1, but for every number less than 1 that you choose, there is a finite number of steps sufficient to go beyond it.
Irrational numbers too are represented by the limit of an infinite series. And still, both the square root of 2 and pi have finite length. And there it is even more pronounced and significant than a length that can be divided into infinitely many parts.
I explained what you asked. You can see that I defined the concept of a limit without resorting to the concept of infinity. Everything you are writing here is not relevant to our discussion. And the rest — go and learn.
I understand that I will never reach 1, and on the other hand I understand that the two halves (one of which I kept dividing and dividing) really do add up to one finite, definite whole.
How can those two understandings coexist? How can this paradox be solved?