On a related point, is there any worked out finitist arithmetic in which, for example, (10^500 + 1)=10^500? Brent Meeker On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick wrote:
Hi all,
I'm wondering if anyone knows when and why it was agreed that 0.999...=1.
That is, the expression meant its limit of 1 and not something that got closer and closer to 1.
I've asked several people and have gotten some big pieces but not quite the whole story: Keith Devlin (last week): before Cantor mathematicians considered "0.999..." to mean a *growing sequence* of 9's after the decimal. After Cantor it was decided that the expression meant the *limit* of this sequence: an infinite number of 9's after the decimal all at once. Steven Strogatz (this afternoon) suggested talking to John Stillwell John Stillwell (a couple hours ago) seemed to indicate the shift in perspective happened gradually, with Zeno arguing for the growing sequence (which would never get to 1) and everyone after the Axiom of Choice agreeing that the expression was referring to its limit.
-400 Zeno: 1/2 + 1/4 + 1/8 etc will never get to 1 -350 Aristotle: 1/2 + 1/4 + 1/8 etc eventually gets to 1 -300 Euclid: 1/4 + 1/4^2 + 1/4^3 + etc eventually gets to 1/3 1585 Stevin: 0.999... = 1 1671 Newton: 0.999... = 1 1858 Dedekind cuts 1880? Cantor 1904 Zermelo: Axiom of Choice
Does anyone have more precise timing for this shift from thinking?
All the best,
Gary _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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