Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118539] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: Richard Fateman <fateman at cs.berkeley.edu>
- Date: Mon, 2 May 2011 06:51:30 -0400 (EDT)
- References: <ip6834$bmt$1@smc.vnet.net> <4DB8C302.3060402@cs.berkeley.edu> <ipgm39$8p1$1@smc.vnet.net>
On 4/30/2011 2:51 AM, Noqsi wrote: > 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. I doubt that it is a huge set. However, the further one departs from mathematical correctness in the 'basics', the larger that set becomes. 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. The nice thing about much of mathematics is that more advanced notions (like other kinds of 'integrals') are generally an extension of the more elementary notions, and so someone who delves in the elementary world need not know that there are these extensions. Sometimes there are problems -- e.g. for someone who knows only about real numbers will be perplexed by complex numbers coming out of Solve, say. It would be a huge burden to do so, even for > the small minority who really understand these issues. Quite the contrary. Getting the basics wrong is a burden to people who understand or don't understand --- sometimes. You could, for example, create a version of Mathematica that did not know about complex numbers. Do you think it would be worthwhile? > > As for mathematical correctness, well, there exist computer proof > verification systems. You miss the point. You can't prove things in Mathematica correct because they are (in ways I've pointed out), mathematically wrong. End of story. A proof system starts with axioms and proof rules and such, and sometimes is able to generate a verification of correctness of a proof or perhaps of a kind of program. If the procedures in the programming language do not correctly reflect mathematical properties, then the chance that you will be able to prove that manipulations are correct, is dramatically reduced. They are much more limited in scope, and much > harder to use than Mathematica. Mathematica is more focused on > applications. I think you are misreading the literature on proof systems. They can prove (or attempt to ...) anything formal. 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. No, but it should say what the program does. What are the valid inputs to Limit ? Is ComplexInfinity not allowed? What else? What are the valid outputs from Limit? How is it that two equivalent expressions can have different limits? (e.g. when limit returns an Interval...) what does this mean for mathematics, where limits presumably do not have this property? 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. > This is, I think, an insult to applied mathematics educators. RJF > > >