Dear Martin, On 14 Mrz., 16:08, Martin Rubey <martin.ru...@math.uni-hannover.de> wrote: > > IMHO it is frankly impossible for *any* CAS to implement a > > mathematically meaningful notion of == that is both useful and > > rigorous. > > This is certainly not true. The point is that you have to make equality > mean different things for different domains.
Well, see the rest of my above post...: As I indicated, it is indeed necessary (it you want something useful) to have different notions of == for different classes, and .. > I must admit that I do not understand (yet) how Sage works here, but > I thought it would define equality separately for every "parent". > Doesn't it, or did I miss something? ... yes, Sage does. But I wanted to point out that in order to have a *useful* notion of ==, you should be able to compare, say, ZZ(1) and RR(1). And once you start to compare apples and oranges, sooner or later you can't avoid to find cases that can't be treated in a mathematically rigorous way. > For many many many domains you can have a "mathematical equality". Agreed. And for many many useful domains (the reals, for example, or the category of finitely presented groups) you can't. Let me try to put it differently: * "A is B" is certainly useful and (computationally) rigorous, but it is not a mathematical notion. * Any implementation of a useful (!), thorough (!!) and consistent mathematical notion of "==" will sooner or later show inconsistencies. Of course, there is a very consistent mathematical notion of ==: "A==B for all A and B" -- but this is not useful. And of course there is a consistent and useful notion of == restricted to the integers. But mathematics is more than the integers. The simple reason for my post was that I felt offended by the subject of this thread: "Is Sage implementing mathematics" -- well, of course it does not. I think it is a theorem that no CAS implements mathematics (once it reaches a certain level). But this is no show stopper, since still CASs are useful for doing mathematics. Cheers, Simon --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
