Andy Latto - very helpful. First of all, I did mean the series 1+1/2+1/4... Second - the odometer wasn't a very good analogy. Good point. Third - the bit about the line wasn't well stated. I'll try again below. Anyway, your note is great. Perhaps the big distinction historically is whether or not the sum of an infinite series was a meaningful operation. And I do have a couple questions. Regarding "0.999...", by "anyone who thought it was a meaningful operation felt that this series converged to 1" you're saying that within this group it's always been thought (notation issues aside) that the expressions "0.999..." and "1/2+1/4+1/8..." and "1" all referred to the same number? And are they interchangeable terms? If so, you could say 1 converges to 0.999... Someone might argue that 1 can't converge. But neither can 0.999... the ellipses, if I understand it correctly, mean an infinite number of nines, so the expression already refers to the number 1. Similarly - Euler: "Nobody doubts that the geometric series 1 + 1/2 + 1/4 + 1/8 +... converges to 2." Does Euler mean "1 + 1/2 + 1/4 + 1/8 +..." and "2" are referring to the same number? There is a bit of a difference between "converged" and "converges," and am wondering if the distinction makes a difference. Related to convergence -- let's take the the hyperbola xy=1. Would mathematicians consider this hyperbola to touch the x axis at x=infinity? If not, how is this convergence different from the convergence above? If so, has this always been considered to be the case among mathematicians when defining hyperbolas? On Thu, Nov 15, 2012 at 1:53 PM, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Nov 15, 2012 at 12:08 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category.
Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions.
*1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2
I'll assume you meant either "y = 1 + 1/2 + 1/4 + 1/8 ..." or "Y = 1"; otherwise the answer is "of course not".
I think Zeno and Aristotle would have answered "I'm unfamiliar with your notation; can you explain it to me?" Zeno and Aristotle would not have been familiar with the = sign, much less the "...". The answer would no doubt depend on details of the explanation you gave of what ... means. If you gave the the modern definition of "...", incorporating the modern epsilon-delta definition of limit, Im sure they would say that y = Y; they may not have had that notion of limit before, but they were bright guys, and would have figured it out if you had explained it to them. Cauchy and Cantor would certainly have agreed that y = Y.
I think they would have all been confused by the reference to lines meeting; frankly, so am I. If you mean these as equations of two vertical lines in the x-y plane (if so, why use y in one and Y in the other?) then I think they all would agree (after you explained Cartesian coordinates to Zeno and Aristotle) that these are the same line, so that they of course "meet". If you mean something else, I don't know what it is.
It's not so much that before Cauchy people had a different, precise, definition of the sum of an infinite series than we have now, so that you could ask the question "using the old definition, did .9 + .09 + .009 + .0009... sum to 1"? They had the idea that some infinite series had sums, but not a precise definition of the sum of an infinite series, or the idea that such a precise definition was important or even useful, though one way to read Zeno is as saying "If you don't have a better definition of the sum of an infinite series than we have now, it's going to get you in trouble; you're going to end up with paradoxes and contradictions".
People worked with infinite series and their sums before they had a rigorous definion. Euler in 1760 had the formula for summing a geometric series, so he believed .9 + .09 + .009 + .0009 ... was = 1. He also believed that 1 + 2 + 4 + 8 + ... = -1, and 1 - 1 + 1 - 1 + 1 ... = 1/2, and 1 - 2 + 3 - 4 + 5... = - 1/4. Though he also said
“Nobody doubts that the geometric series 1 + 1/2 + 1/4 + 1/8 +... converges to 2. As more terms are added, the sum approaches 2, and if 100 terms are added, the difference between the sum and 2 is a fraction with 30 digits in its denominator and a 1 in its numerator. The series 1 + 1 + 1 + 1 + 1 + etc. and 1 + 2 + 3 + 4 + 5 + 6 + etc. whose terms do not tend toward zero, will grow to infinity and are divergent.”
So he also had some notion of a divergent series. See
http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%2... for more about how Euler viewed divergent series and their sums.
I think the notion that people used to think .99999... was different from 1, and then learned that it was the same, is incorrect, so asking when the change happened is the wrong question. While some, including implicitly Zeno, certainly questioned whether the sum of an infinite series had a meaning at all, anyone who thought it was a meaningful operation felt that this series converged to 1. The idea that it converges to some number just slightly less than 1 is not an old notion of old incorrect mathematicians, but a new notion of people misled by notation and a confusion between numbers and numerals leading to an implicit belief that different numerals must always denote different numbers.
2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1?
Well, all of the above's answer would of course be "what's an odometer?" If you asked me, I'd ask you what you meant by a physical object "indicating" a number. If you gave a precise definition, and then asked me about two hypothetical odometers, I'd say they were indicating the same number if you gave any reasonable meaning to an odometer "indicating" a number. If instead of describing hypothetical odometers, you confronted me with two physical objects, one of them stretching out into the distance beyond my field of vision, I would assume that it was nontheless a finite object and answer "no". If you assured me that it was in fact infinite, and all its dials read 9, I wouldn't believe you, but I suppose that if I pretended to believe you, I would answer "yes".
- - - - I'm thinking the answers would be
First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how
this
happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y
Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing.
On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
unless I'm missing something, there would still be the following problems: Zeno's arrow will never actually hit the tree (eg 1/2 + 1/4 + 1/8 will never quite get to 1) a polygon with more and more sides will not quite ever become a circle 9/10 + 9/100 + 9/1000 etc will not quite ever equal 1
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