The Logical Methods in the Humanities Workshop will be having its first meeting on Friday, October 10th, in 90-92Q, at 3PM. This will be a casual introductory meeting, where several faculty members will introduce themselves, talk about their research interests, and how they began their research during their early careers. Everyone who is interested in the workshop is encouraged to attend.
The Platonist answer to the question, "What is mathematical
language about?", is that it is about abstract individuals (such as
zero,
the null set, omega, etc.) and abstract relations (successor,
membership,
group addition, etc.). One way to make this answer precise is to provide
a
formal, background theory of abstract individuals and abstract
relations.
I review one such formal theory and explain the special way in which the
language and theorems of arbitrary mathematical theories can be
interpreted in this formalism. (A full analysis is developed in my paper
"Neologicism? An Ontological Reduction of Mathematics to Metaphysics",
Erkenntnis, vol. 53, nos. 1-2 (2000), 219-265.)
However, it turns out that the background formalism for abstracta itself
is subject to interpretation. The Platonistic interpretation is just one
of (at least) four ways of interpreting the theory. I'll explain how one
can develop fictionalist, structuralist, and inferentialist
interpretations of the formalism. Since each interpretation offers us a
clear, but different, answer to our initial question, the resulting
analysis not only offers a way to make these philosophies of mathematics
more precise, but also unifies them in a new and unsuspected way. (It
also
has the consequence that no matter how the mathematicians decide to
extend
mathematics with new axioms or mathematical foundations, the philosopher
will have something to say about the mathematical language used in the
extension.)
We discuss the search for philosophical principles that appear more fundamental than set theoretic principles, and which are sufficiently powerful to allow for the interpretation of set theory, including ZFC and large cardinal axioms. The principle examples take on the character of principles of "reflection".