Hi! > On Sat, Mar 14, 2009 at 3:38 AM, William Stein wrote: > > teragon:papers wstein$ sage -python > > Python 2.5.2 (r252:60911, Mar 12 2009, 23:58:30) > > [GCC 4.0.1 (Apple Inc. build 5488)] on darwin > > Type "help", "copyright", "credits" or "license" for more information. > >>>> a = 10**22; b = 10**22+1; c = complex(a) > >>>> a == c > > True > >>>> b == c > > True > >>>> a == b > > False
Impressive! Since the topic now changed into "is Sage implementing Mathematics": IMHO it is frankly impossible for *any* CAS to implement a mathematically meaningful notion of == that is both useful and rigorous. If you simply say "a == b should return True only of a and b are the same objects (i.e. same address in memory)" then you'd use a *non- mathematical* notion of ==. Namely, this would depend on the implementation, and without Sage's concept of caching Parents, you would then even have "ZZ[x] != ZZ[x]" (without uniqueness of parent structures, the two ZZ[x] would be two different objects). Side note: You can alway use "A is B" instead of "A==B" if you really need A and B to be identical objects. A mathematically meaningful and useful notion of == must involve some concept of mathematical equivalence (which is not identity!) between objects. However, there are lots of mathematically natural equivalences that are undecidable (equality of finitely presented groups, for starters); hence, this can not be implemented in a rigorous way. Moreover, the 'obvious' equivalence relation will depend on the nature of objects under consideration. So, the best one can hope for is to have a reasonable heuristics, where "reasonable" is in the eye of the beholder. E.g.: Take an element A of some algebraic structure (such as the integers) and an element B of a "numerical something" (e.g. RR, which is even not a ring AFAIK). What is a reasonable comparison of A and B? 1. From the viewpoint of A, one must refuse the idea that A and B can be equal, since B is only defined up to some error margin, that would be inacceptable for A 2. From the viewpoint of B, one must take into account the error margin subject to which B is defined. If A is within this margin, then A and B are "sufficiently equal". And I think you will certainly find both supporters of 1. and of 2., and you will find both situations in which 1. or 2. is the natural way of thinking. Sure, this can be a reason for bugs that are hard to detect. But I don't think there is a natural and rigorous way to avoid bugs of that kind. 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 -~----------~----~----~----~------~----~------~--~---
