Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118502] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: Noqsi <noqsiaerospace at gmail.com>
- Date: Sat, 30 Apr 2011 05:51:23 -0400 (EDT)
- References: <ip6834$bmt$1@smc.vnet.net> <4DB8C302.3060402@cs.berkeley.edu>
On Apr 29, 5:29 am, Richard Fateman <fate... at eecs.berkeley.edu> wrote: > On 4/28/2011 1:30 AM, Andrzej Kozlowski wrote: > > In fact I once suggested that an options should be available for the user to decide which compatification he wants to use when taking limits etc, > > Sounds plausible to me. >> but now I think that this additional functionality would almost never be used and thus is not worth the effort. > to require the user > to check something that the system could check is to say, in effect, we > could make Mathematica do mathematics correctly, but we will settle for > it doing mathematics "pretty well" and in particular, sometimes wrong. One difficulty here is that few Mathematica users would understand how to navigate the maze, if that was the general approach. The issue you've identified is only one in a huge set of potential "mathematical context" issues that one might address. Like essentially everyone who applies mathematics, you yourself throw around terms like "integral", "function", "neighborhood", "(in)finite", "continuous", and "differentiable" without stating the precise mathematical context in which you are speaking. It would be a huge burden to do so, even for the small minority who really understand these issues. As for mathematical correctness, well, there exist computer proof verification systems. They are much more limited in scope, and much harder to use than Mathematica. Mathematica is more focused on applications. Usually, the "application context" is stricter discipline anyway. There is no guarantee that a well-formed mathematical expression or a formally correct mathematical result makes physical sense. Nor, in this case, have you identified a result that is actually incorrect: you merely found a case where you got different answers to different questions. > Or perhaps we really don't know how to do this correctly, and we should > just say (as I've suggested a day or two ago) that it be documented. The documentation is already impenetrable, and it should not assume the additional burden of providing a graduate-level education in mathematics to Mathematica users. Mathematica relates well to the common practice of applied mathematics education, where this kind of mathematical context issue is not covered in any great detail. It would be too distracting.