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Re: Replacement Rule with Sqrt in denominator. Also Bug in Series

On 12/10/10 at 2:28 AM, wpthurston at (Bill Thurston) wrote:

>I'd like to chime in that I appreciate Richard Fateman's analysis of
>the behavior of replacement rules.  (See his paper
>  I've been
>using mathematica since version 1 (or even before, when Steve
>Wolfram was still involved with SMP), and I've often been annoyed at
>the difficulties he's discussed. My interest in Mathematica is as a
>peripheral tool; I don't want to have to know too much about its
>quirks and its inner workings. I'm a mathematician, I don't want to
>be a mathematica-ologist.  Usually if I start looking at
>FullForm[expression], I feel like I'm wading in more deeply than I
>want to or should.  It quickly becomes very frustrating, since as
>Richard noted, FullForm doesn't tell the whole story, the whole
>story is not public and not documented, and the undocumented
>internal behavior can and does change from version to version --- so
>I'd prefer to have to think about it as little as possible.

With respect to replacement rules or any other behavior of
Mathematica you have a pretty limited choice for solving problems.

You can:

1) use different functions in Mathematica and ignore the
behavior you don't care for or don't want to take the time to
understand more thoroughly.

2) you can dig into the operation of the function until you do
understand how it behaves and can use it effectively

3) you can use something other than Mathematica to solve the problem.

There really isn't any other useful choice in terms of getting a
problem solved.

With respect to 2) above and replacement rules, it is not
correct to say the behavior is undocumented. Most of the posters
here are not WRI employees and have no access to the source
code. All they have access to is the same documentation you have
access to. And since it is clear some users do understand the
behavior of replacement rules it cannot be true the behavior is undocumente=

>Because of problems like those cited, I've often ended up just
>giving up on mathematica finding formal solutions even when I know
>in principle it can find them, and instead I attack them from other
>directions or by alternate tools. Mathematica has done many things
>well for me, but I've been tripped up miserably on others.

Which is basically choice 3). For myself, I generally choose
option 2). The payoff for option 2) is as I increase my
understanding of how Mathematica behaves in some specific case,
I find I have more confidence in the results I get and I am
better able to predict behavior for other problems. In this
respect, Mathematica is like any other complex tool. The better
you understand a tool, the more effectively you can apply that
tool to a new problem.

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