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Why is Zeilberger so willing to give up on absolute truths?
The most reasonable answer is that he is pursuing deeper truths.
In Identities in Search of Identities, Zeilberger
advocates an examination of identities for the sake of studying
identities. Still as Herb Wilf and others have pointed out it is
possible to produce an unlimited number of identities. It is
the context, the ability to use and manipulate these identities, that
make them interesting. Why then might we think that studying
identities for their own sake may lead us down the golden path
rather than the garden path?
We are now looking for what might be called meta-mathematical
structures. We remove the math from its original context and isolate
it, trying to detect new structures. When doing this it is impossible
to collect only the relevant information that will lead to the new
discovery. One collects objects (theorems, statistics, conjectures,
etc.) that have a reasonable degree of
similarity and familiarity and then attempts to eliminate the irrelevant or
the untrue (counter examples).
We are preparing for some form of eliminative induction.
There is a built in stage, where
objects are subject to censorship. In this context, it is not
unreasonable to introduce objects where one is not sure of their truth,
since all the objects, whether proved or not, will be subject to the
same degree of scrutiny. Moreover, if these probably true objects
fall into the class of reliable (i.e., they fit the new conjecture)
objects, it may be possible to
find a legitimate proof in the new context.
Recall that the fast WZ algorithms transform the problem of proving an
identity to one of solving a system of linear equations with symbolic
coefficients.
It is very time consuming to solve a system of
linear equations with symbolic coefficients. By plugging in specific
values for n and other parameters if present, one gets a system with
numerical coefficients, which is much faster to handle. Since it
is unlikely that a random system of inhomogeneous linear equations
with more equations than unknowns can be solved, the solvability of
the system for a number of special values of n and the other
parameters is a very good indication that the identity is indeed true.
([16] p. 980)
Suppose we can solve the system above for ten different assignments
for n and the other parameters but cannot solve the general system.
What do we do if we really need this identity?
We are in a peculiar position. We have reduced the problem
of proving identities involving sums and integrals of
proper-hypergeometric terms to the problem of solving a possibly
gigantic system of inhomogeneous linear equations with more equations
than unknowns. We have an appropriately strong belief that this
system has a solution but do not have the resources to uncover this
solution.
What can we do with our result?
If we agree with G. J. Chaitin,
we may want to introduce it as an
`axiom`.
I believe that elementary number theory and the rest of mathematics
should be pursued more in the spirit of experimental science, and that
you
should be willing to adopt new principles. I believe that Euclid's
statement
that an axiom is a self-evident truth is a big mistake
.
The Schrödinger equation certainly isn't a self-evident truth! And
the
Riemann hypothesis isn't self-evident either, but it's very useful.
A physicist would say that there is ample experimental evidence for
the Riemann hypothesis
and would go ahead and take it as a working assumption. ([5]
p. 24)
In this case, we have ample experimental evidence for the truth of our
identity and we may want to take it as something more than just a
working assumption. We may want to introduce it formally into our
mathematical system.
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