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Re: Limit question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg31561] Re: Limit question
*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>
*Date*: Sun, 11 Nov 2001 00:34:53 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Allan
I think this argument has moved somewhat beyond Mathematica so I will
this aspect first. I don't think on this point there is any real
disagreement involved. I wrote in my message: "Of course one could
formulate it in different but equivalent terms...". Thus the
epsilon-delta approach is less general than talking about topology since
it uses metric space structures, but apart from that is equivalent. It
is not important whether one starts with the notion of continuity of a
function or convergence, but topology is in any case used, even if only
implicitly, since it underlies any metric space structure. One need not
to consider infinities to be objects which you "add" , but even if one
does not one ends up describing with mathematically equivalent concept
of "converging to infinity", expressed in a more metaphorical and, in
my opinion less intuitive, language.
However, let's come back to Mathematica. This thread started with the
following observation:
In[1]:=
Limit[Tan[x],x->Pi/2]
Out[1]=
-Infinity
I agree with the original poster that this is clearly wrong. It just
does not make sense to give this answer from any mathematical view
point. To explain how Mathematica arrives at this sort of thing (by
making clear out it's hidden defaults) does not, in my opinion, address
the original question: what is the mathematical justification for this
answer? I know there are a number of situations (particularly involving
integration) when Mathematica faced with an input which which from its
point of view is not sufficiently informative, will default to some
particular defaults. I think this is to some extent unavoidable, because
of the way Mathematica is constructed. Unlike more specialized programs
(like, for example, Macaulay) you do not start a Mathematica computation
by specifying rigorously the mathematical setting (e.g. domain and range
of your functions.). This results in wider applicability of Mathematica
constructions and much easier programming language, but at a cost of a
certain vagueness. Personally I am not bothered by these matters,
because in almost all all cases I can think of the "confusion" is not
really significant and one can adjust things to fit ones own intentions.
But still I think whenever possible Mathematica ought to give
mathematically sensible answers, and -Infinity is not really a sensible
answer in the example above.
With best regards
Andrzej
On Sunday, November 11, 2001, at 02:43 AM, Allan Hayes wrote:
> I agree that if we want Infinity etc. to be objects, rather than just
> words
> in a phrase, then we need to "add them" to the original spaces (or
> embed the
> original space suitably). But the language seems to me to come first,
> and to
> be more directly related to the epsilon-delta definitions and more
> amenable
> to computational treatment. Also it provides the motivation for the
> extension and an opportunity to illustrate the fascinating idea of
> creating
> structures to our needs --- Cauchy sequences ---> completions,
> ultrafilters --> compactifications, prime ideals --> .... and
> consistency --> model.
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
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