logic and mathematics
Happy Bloomsday everyone,
http://www.rejoycedublin2004.com/
I am starting to reorganize the IFF stack (core hierarchy) somewhat along
the conceptual lines expressed in the book "Sets for Mathematics" by Lawvere
& Rosebrugh, and thought the following comment from Appendix A might be of
some interest to you all.
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Historical Note:
It is sometimes objected that logic is allegedly not algebra since - for
example - nobody thinks in cylindric algebra. That is an unfortunate
misunderstanding: cylindric algebras (and polyadic algebras) were an
important initial attempt in the 1950s to make explicit the objective
algebra briefly described above [Appendix A]; however, they were excessively
influenced by the styles of subjective *presentation* (of logical content)
that had become traditional since the 1930s under the name of "first order
predicate calculus" or the like. Those styles of presentation involved
various arbitrary choices (such as the specification of a single infinite
list of variables) that proved to be quite bizarre when confronted with
ordinary mathematical practice; surely the logic of mathematics is not
cylindric algebra. For about a hundred years now, mathematical scientists
have indeed had an intuitive distrust of attempts by some logicians to
interfere with mathematics. However, some explicit recognition of the role
of logical algebra is helpful and even necessary for the smooth and correct
learning, development, and use of mathematics, including even the high
school algebra of ordinary quantities.
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Robert E. Kent
rekent@ontologos.org