Wikipedia:Featured article candidates/Euclidean algorithm/archive1: Difference between revisions

Revision as of 20:25, 27 April 2009

Revision as of 20:40, 27 April 2009

Line 18:

:::I've read the quote in the box. Is this "only" Knuth's opinion or a statement on whom most specialists agree ? --[[User:El Caro|El Caro]] ([[User talk:El Caro|talk]]) 19:39, 27 April 2009 (UTC)

::::If "algorithm" is defined only as a set of numerical rules (without requiring proof that the rules always work or understanding why they work), then surely there were algorithms for thousands of years before Euclid. I'll review the literature and list the opinions of other people besides Knuth. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 20:25, 27 April 2009 (UTC)

:::::I corrected this in the text before, but that is indeed what "[[algorithm]]" means. I have never seen anyone define algorithm to mean it must come with understanding of the person using it or a proof that it works. So I can't imagine Knuth would use some nonstandard definition of "algorithm", as you suggest, especially since he is a computer scientist and certainly computer scientists do not require algorithms come with proofs. That is probably why he says "nontrivial" as I mentioned above. --[[User:C S|C S]] ([[User talk:C S|talk]]) 20:40, 27 April 2009 (UTC)

"Euclid's algorithm can be applied to real numbers, as described by Euclid in Book 10 of his Elements" looks like an anachronism. Did Euclid know real numbers ? --[[User:El Caro|El Caro]] ([[User talk:El Caro|talk]]) 18:17, 27 April 2009 (UTC)

:Well, now we're getting into metaphysical matters that I think are tangential. For example, when Euclid considered arithmetical operations on whole numbers, it's not the same in a sense as what we consider such arithmetic, nor is probably what the ancient Greeks considered whole numbers the same as what we do now. So strictly speaking Euclid did not know real numbers, but he didn't know whole numbers, addition, or subtraction either. So that makes your point kind of moot. --[[User:C S|C S]] ([[User talk:C S|talk]]) 19:21, 27 April 2009 (UTC)

::The ancient Greek certainly knew about irrational numbers, at least as [[nth root|surds]], even if they hadn't defined [[Dedekind cut]]s. Book X of Euclid's ''Elements'' is devoted entirely to questions of [[Commensurability (mathematics)|incommensurability]], and this real-number version of Euclid's algorithm begins that exposition. Knuth states elsewhere that the Greeks treated real numbers by infinite continued fractions, but he doesn't explain his remark further; I took him to be referring to this version of Euclid's algorithm. One could argue, I suppose, that the modern concept of real numbers embraces more than just "the set of all rational and irrational numbers", but that seems beyond the level of this article. [[User:Proteins|Proteins]] ([[User talk:Proteins|talk]]) 20:18, 27 April 2009 (UTC)

:::Ancient Greeks considered numerical concepts mainly in terms of geometric constructions ala ruler and compass and that is how Euclid's treatment of commensurability goes. This is certainly not the way modern mathematicians think of them. Sure the ancient Greeks knew of some irrational numbers, but they certainly didn't know "e" or many other irrational numbers that are not constructible. So their concept of irrational number was far more limited than our modern understanding, even when one limits the concept of real number to mean "set of rational and irrational numbers". Even on the math where modern and ancient understanding would seem to overlap, it's clear the ancient Greeks just had a different way of thinking about it, so in a metaphysical sense, you could argue that the objects are really different. --[[User:C S|C S]] ([[User talk:C S|talk]]) 20:40, 27 April 2009 (UTC)