Miguel A. Lerma
mlerma at math.northwestern.edu
Mon Jan 8 15:16:52 EST 2001
Alexander R. Pruss said:
> Another way to highlight the problem is this. I said errata are
> rare. However, as we sadly know, errors in math papers are perhaps not
> all that rare. But when a paper has been published after the usual
> refereeing process, then any errors are likely to be confined to small
> details of the proof, details that the reader can fix, and such errors do
> not merit an erratum. Why is it that small errors of detail in proofs
> usually do not vitiate the main theorems?
Because the published proof is just the tip of the iceberg
in the process by which we become convinced that our results
are right. That proof is just a summary of our work that we
present to the mathematical community in order for our results
to be accepted as part of the mathematical corpus. By the time
we submit a work for publication we have probably worked on it
under different angles, rewritten and simplified the proofs, and
discussed the matter with other colleagues. The results must still
go through the refereeing process, and after publication resist
the proof of time. The longer a result remains uncontested the
more likely it is that it does not contain non-fixable errors.
> After all, speaking completely
> abstractly, as someone has noted, a mathematical "proof" with an error in
> it is like a person who traced his genealogy to William the Conqueror with
> only two gaps in the proof.
A mathematical proof published in a refereed journal with an error
is more like someone who studied his genealogy, traced it in various
ways each time finding that it leads to William the Conqueror,
showed it to other colleagues and experts, and then, when putting
it in writing for publication made a few trivial mistakes (such
as misspelling a name) or even errors (such as changing the order
of two of the elements in one of the ascending lines) that even
the referees missed but are easily fixable.
A related matter that interests me is the status of the mathematical
(and in general scientific) results known to the NSA (National
Science Agency) but unknown to the rest of the mathematical
(or scientific) community because they are considered "classified
material" by the NSA. Can the NSA be considered a part of the
mathematical community and their results considered part of the
mathematical corpus, or should we call "mathematical community"
only the set of mathematicians willing to share their results
with the rest of the world? We should note also that since the
NSA contains only a small fraction of the mathematicians in the
world, the validation of theirs results through time and scrutiny
is weaker than that of the results of their exterior world colleagues.
Miguel A. Lerma
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