From: David Eppstein <eppstein@ics.uci.edu>

Date: Wed, 12 Apr 2000 10:33:47 -0700

To: www-math@w3.org

cc: roconnor@uwaterloo.ca, jsdevitt@radicalflow.com

Message-ID: <12958114.3164524427@cx344290-c.irvn1.occa.home.com>

Date: Wed, 12 Apr 2000 10:33:47 -0700

To: www-math@w3.org

cc: roconnor@uwaterloo.ca, jsdevitt@radicalflow.com

Message-ID: <12958114.3164524427@cx344290-c.irvn1.occa.home.com>

"Russell Steven Shawn O'Connor" <roconnor@uwaterloo.ca> writes: > but note that in both your case and in my case we hand to translate the > statement into something else. I rearranged stuff and used set > inclusion. Your rearranged a bit less, but added two quantifiers. > Either way the point is that if the statement is going to get used by a > computer, it is going to have to get translated into a first-order > statement. In that sense (1) is meaningless because it is not a > first-order statement. Bogus argument. The whole point of any notation is as a shorthand for something else. Plenty of other notations need translation to become first order e.g. lim_{x->0} (sin x / x) = 1 "really means" Exists(L) all(epsilon>0) exists(delta>0) all(x:|x|<=delta) |(sin x / x) - L| <= epsilon, and furthermore L=1. People don't usually do this sort of explicit translation whenever they manipulate limits or O's, why do you think a computer should have to? For instance, in Mathematica or other symbolic symbols, it would be perfectly straightforward to add a simplification rule that O(x^i)+p(x) simplifies to O(x^i) whenever p is a polynomial with degree at most i. Such a rule doesn't require any deeper understanding of what O "really means". -- David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/Received on Wednesday, 12 April 2000 13:33:53 UTC

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