> Martin Rubey wrote: >> 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? >
On Sat, Mar 14, 2009 at 1:25 PM, Simon King wrote: > ... 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). No, that is usually not the case, and as far as I understand it is not the case in Sage. What one can do is convert ZZ -> RR in a canonical way (aka. coercion), and then you are just comparing RR(1) to RR(1). One does not compare ZZ(1) to RR(1) directly. > 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. > There is no need to compare apples to oranges. Sometimes we can say in what way an apple is like an orange (or vice versa) and then we can compare oranges to oranges (or apples to apples). >> 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. That is not true. But Godel did show that one must choose either consistency or completeness. The more reasonable choice for any logical or compational (algebraic) system is consistency. As a consequence there are classes and some things are undecidable. > 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. > Indeed. > 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. > Although at best this might be viewed as a conflict of competing mathematical philosophies (and at worse a matter of personal philosophies), I think the fact that you (and perhaps others) might be "offended" by the implication of the subject of this thread shows a rather profound misunderstanding of the relationship between computer programming and mathematics. "Implementing mathematics" should not be confused with the (wrong) idea that it is somehow possible to "mechanize" all of mathematics! Regards, Bill Page. --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
