Accuracy and error analysis (was Re: [Maxima] primes)

--- Albert Reiner  wrote:
> [C Y , Fri, 13 May 2005 07:54:10 -0700 (PDT)]:
> > For physical measurements the price of calculating error
> > propagation MUST be paid, at least at the macroscopic error 
> > level - it is integral to getting a meaningful answer.
> 
> Of course, but this should be the result of an error analysis.  What
> you get with significance arithmetic as in Mma is, however, largely
> meaningless and, by being suggestive and ``looking reasonable'' in
> many cases, actually harmful.

Hmm.  OK.

> As long as the error terms of your calculation or measurement are 
> not understood, some work remains to be done.  Automated and 
> unreliable estimates, OTOH, are likely to induce
> people to skip that part of the work.

True.  If an answer is given at all, it needs to be reliable.  But I
still fail to understand why the correct procedure (whatever it may be)
cannot be an automated one.  It might involve defining error procedures
for every function in Maxima, but I just can't believe it's impossible
unless proper error analysis is also impossible.  If a human being can
do it, surely a computer can.  At an absolute minimum, it must be able
to verify something I propose as correct.  I suppose people have solved
mathematical problems no currently available software could solve on
its own, since things like integration are difficult, but I tend to
view human approaches to solving such things as a combination of a huge
number of heuristics and quasi random guessing with bounds defined by
those heuristics, which (in my experience) is as unreliable as it is
occasionally lucky/powerful.  And since taking derivatives IS well
defined, the computer can verify the correctness of this solution.  I
have the same opinion of mathematical proofs - namely, if a computer
cannot verify its correctness I don't believe a human can either - but
that's getting more into philosophy than math.  I'll concede, because
of the difference in approach and ability to apply background
information, humans can CREATE a proof a computer couldn't, but I'll
believe it more when a computer verifies it's valid and correct.

CY

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