Except
for a huge poster of saxophonist John Coltrane, the walls of Prof. Kimmo
Rosenthal's office are nearly bare.
The jazz icon has been an inspiration for Rosenthal, he
says, because Trane was never satisfied with where he was musically,
shifting from bebop to the more obscure avant garde, sometimes to the
dismay of his once-loyal fans.
“At any point in his career, he could have said,
'That's it,'” says Rosenthal, a student of jazz and host of Dr.
Kimmo's Jazz Brunch Sundays at 9 a.m. on WRUC. “But Coltrane was
always interested in saying something new.”
So perhaps it should not be surprising that Rosenthal,
professor of mathematics, is looking to say something new in his own
field. Just as his idol did with his musical improvisations, Rosenthal is
interested in mathematics that pushes some of the boundaries.
Rosenthal will deliver a faculty colloquium titled
“Can We Do Mathematics Without Classical Logic?” on Thursday,
May 4, at 11:30 a.m. in the Olin Auditorium.
His talk will explore his field of category theory,
referred to by some mathematicians as “abstract nonsense” or not
suitable as a foundation for mathematics, in some sense equivalent to the
skepticism regarding abstract art or avant-garde music, Rosenthal says.
Category theorists study all fields of mathematics for
certain constructions that are universal across different fields. These
investigations have led to considering a rather unorthodox
“intuitionistic logic.”
Rosenthal was introduced to category theory and
intuitionistic logic through an undergraduate course in algebra taught by
William Lawvere at SUNY Buffalo, who eventually would supervise Rosenthal's
Ph.D. thesis.
Unlike classical logic, which relies on the law of an
“excluded middle” — every statement is true, or its negation
is true — intuitionistic logic allows for a middle ground, abandoning
the excluded middle rule that most practicing mathematicians would say has
to be there.
To illustrate these ideas, Rosenthal will teach a
freshman calculus proof the way it would have been taught by German
philosopher Gottfried Liebniz in the late 1600's using some
infinitesimally small numbers that would “not be legal if you insist
on staying in the realm of classical logic.” Then, using
intuitionistic logic, he will discuss how that calculus proof is valid.
“Just as some discount abstract art or avant-garde
music, intuitionistic logic doesn't fit into the preconception of how
mathematics works,” Rosenthal says. “I hope to provoke some
interest into the question of intuitionistic logic and the foundations of
mathematics, but I don't necessarily expect to win any converts.”
Rosenthal, at Union since 1979, holds a bachelor's,
master's and Ph.D. from SUNY Buffalo. He is the author of two books – Quantales
and the Applications (1990) and The Theory of Quantaloids
(1996) – and numerous articles. He has a range of college service to his
credit including acting director of AOP, Middle States Review Committee on
the Mathematics Major, department chair, director of General Education,
Faculty Review Board and FRB chair, and Student Affairs Council chair.