[FOM] Mathematical conceptualism
Nik Weaver
nweaver at dax.wustl.edu
Mon Sep 12 17:54:53 EDT 2005
I have posted three papers to the Mathematics ArXiv that may be
of interest to members of this list. They are also available at
http://www.math.wustl.edu/~nweaver/conceptualism.html
All three deal with a foundational stance I call "mathematical
conceptualism", which is in effect equivalent to predicativism
given the natural numbers.
The first paper, "Mathematical conceptualism", is an explanation
and defense of the position for a general mathematical and
philosophical audience. The main arguments (against platonism
and impredicative definitions) are not new but the paper contains
several novelties, including a conceptualist discussion of the
distinction between sets and classes and an example from functional
analysis illustrating the exact fit of mathematical conceptualism
with ordinary mathematical practice.
The second paper, "Analysis in J_2", shows how core mathematics
can be developed within the second set in Jensen's constructible
hierarchy. The advantage this approach has over other developments
of core mathematics in minimalist systems is its retention of the
standard language of set theory, which permits a more faithful
expression of the classical material. For example, measures are
defined as functions from sigma-algebras into [0,infty], not as
linear functionals on C(X) as one sees in other approaches.
The third paper, "Predicativity beyond Gamma_0", refutes the claim
that predicative reasoning (given the natural numbers) is limited
by the Feferman-Schutte ordinal Gamma_0. This may be a provocative
assertion, but I think the flaws that I identify in the arguments
for this claim are so severe that a serious question is raised as
to why the foundations community accepted it so uncritically, for
so long. I also develop a general method for producing predicative
well-ordering proofs and in particular establish that the "small"
Veblen ordinal \phi_{\Omega^\omega}(0) is predicatively provable.
However, I expect that much larger ordinals should be accessible
and this opens up an interesting avenue for future research.
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
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