636 research outputs found
Leibnizian, Robinsonian, and Boolean Valued Monads
This is an overview of the present-day versions of monadology with some
applications to vector lattices and linear inequalities.Comment: This is a talk prepared for the 20th St. Petersburg Summer Meeting in
Mathematical Analysis, June 24-29, 201
A non-standard analysis of a cultural icon: The case of Paul Halmos
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil
worship, logic, and Robinson's infinitesimals. Halmos' scepticism about
category theory derives from his philosophical position of naive set-theoretic
realism. In the words of an MAA biography, Halmos thought that mathematics is
"certainty" and "architecture" yet 20th century logic teaches us is that
mathematics is full of uncertainty or more precisely incompleteness. If the
term architecture meant to imply that mathematics is one great solid castle,
then modern logic tends to teach us the opposite lession, namely that the
castle is floating in midair. Halmos' realism tends to color his judgment of
purely scientific aspects of logic and the way it is practiced and applied. He
often expressed distaste for nonstandard models, and made a sustained effort to
eliminate first-order logic, the logicians' concept of interpretation, and the
syntactic vs semantic distinction. He felt that these were vague, and sought to
replace them all by his polyadic algebra. Halmos claimed that Robinson's
framework is "unnecessary" but Henson and Keisler argue that Robinson's
framework allows one to dig deeper into set-theoretic resources than is common
in Archimedean mathematics. This can potentially prove theorems not accessible
by standard methods, undermining Halmos' criticisms.
Keywords: Archimedean axiom; bridge between discrete and continuous
mathematics; hyperreals; incomparable quantities; indispensability; infinity;
mathematical realism; Robinson.Comment: 15 pages, to appear in Logica Universali
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