Prerequisite: 302 or 305; or at least two from 206, 214, 223, 225
Distribution: Mathematical Modeling. Majors can fulfill the major presentation requirement in this course in 2005-06.
What are the
foundations of mathematics? What is a logical starting point for
studying mathematics? If you ask different mathematicians, there is a
good chance you'll get very different answers to these questions.
Some mathematicians think that mathematics should be based on set
theory, so that set theory would be their starting point for
mathematics. Other mathematicians might consider the study of logic
itself to be the right starting point for mathematics. Still other
mathematicians believe that mathematical objects exist in our
imagination or are products of our intuition, so there doesn't need
to be an agreed-upon starting point.
In this course we study the mathematics that results from considering
one or more of these points of view, and we spend time developing
different tools that the mathematician routinely uses. The most
important of these tools are the basic number systems (such as
N, Z, Q, R), set theory, and the notion
of cardinal and ordinal number. Two sets have the same cardinal
number if they can be put in 1-1 correspondence, and two ordered sets
have the same ordinal number if there is a 1-1 correspondence between
them that respects their ordering. This course places these notions
within axiomatic set theory, invented to avoid contradictions that
arise when sets are considered from a purely intuitive point of
view.
This course will be taught using a modified Moore method. Students
will develop answers and proofs for a structured collection of
questions and theorems and present their results in class. Students
generally find they understand things better when they work through
examples and find proofs for themselves, and this course lends itself
well to that. Hints and/or outlines will be provided for the more
difficult proofs, students will work in pairs or small groups, and
there will be ample opportunity to confer with the instructor. In
many ways, then, this course will operate like a seminar. Majors can
fulfill the major presentation requirement in this course in
2005-06.
Prerequisite: 302 or 305; or at least two from 206, 214, 223,
225
Distribution: Mathematical Modeling. Majors can fulfill the major
presentation requirement in this course in 2005-06.
