[FOM] The Role of Formalization
henriknordmark@mac.com
henriknordmark at mac.com
Mon Aug 2 07:10:24 EDT 2004
Recently, there has been a fair amount of interesting discussions about
the scope of mathematics and about mathematical rigor. This reminded me
of an anecdote of a student of axiomatic set theory who insisted in
writing all of his proofs for homework exercises in a extremely
rigorous fashion. All of his proofs were symbolic and with the
appropriate software these proofs could probably have been checked by a
computer. The grader was a bit confused as to what to do with these
proofs since they were correct and very rigorous but it was not really
what was expected from him. After a while, the professor and the grader
recognized the fact that it was in some sense admirable that he wrote
his proofs so formally, but requested him to start writing his proofs
at a higher level of abstraction, which was more appropriate for
human-readability.
I think this points out the interesting fact that although we strive
for all of our Mathematics to be formalizable for reasons of rigor,
most mathematicians are not really that interested in having formal
symbolic proofs of their theorems.
There is of course some interest in formal symbolic proofs of our
current mathematical knowledge. The QED project and the Mizar project
attest to this.
http://www-unix.mcs.anl.gov/qed/
However, this type of work does not seem to interest too many
mathematicians and perhaps for good reasons.
Is there a need for a repertoire of formal symbolic proofs? Is it in
some sense mathematically interesting to provide symbolic proofs of our
current mathematical knowledge? Is it worth investing time and money
for such an endeavor? Would we be able to gain any new insights by
doing this? Will future generations in any way profit from having a
repertoire of symbolic proofs at their disposal?
Henrik Nordmark
Institute for Logic, Language and Computation
Universiteit van Amsterdam
www.homepage.mac.com/henriknordmark
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