[FOM] Mathematical conceptualism
Nik Weaver
nweaver at dax.wustl.edu
Mon Sep 19 18:10:59 EDT 2005
I had expected that my previous message, together with the papers
that I posted to the ArXiv, would spark some discussion on this
list. I would be disappointed to think that because I am unknown
in foundational circles (my specialty is functional analysis) people
in this area consider these papers not worth reading. Perhaps
everyone already has a favorite foundational stance and is not
interested in looking at work that supports an opposing view? If
so, this would make my readership very small, as I represent a view
that I gather is almost universally rejected. (Jeremy Avigad says
that "it is an awkward fact that there seems to be no strong case
that predicativity is a notion worthy of our attention".)
To the contrary, I think there is a strong case for predicativism
given the natural numbers, or as I prefer to call it, mathematical
conceptualism.
The case for predicativism: (1) the idea that there exists an abstract,
metaphysical, Platonic world of sets is nonsensical and is discredited
by the set-theoretic paradoxes; (2) once this is accepted, any domain in
which set-theoretic reasoning is to take place must be in some sense
constructed; (3) it would be absurd to allow such a construction to
be circular.
The case for "given the natural numbers": (1) the scope of mathematical
reasoning is the realm of logical possiblity; (2) once this is accepted,
a necessary and sufficient condition that a putative construction be
considered legitimate is that we be able to concretely imagine how
it could be carried out; (3) we have such a concrete picture for
constructions of length omega, but not for constructions carried out
along larger cardinals.
These arguments are made in greater detail in my paper "Mathematical
conceptualism". (I also give reasons there for the terminological
change.)
It is my impression that predicativism is rejected not because of any
logical defects in the arguments summarized above, but rather because
one has been told that it fails to support important mainstream results
(e.g., Kruskal's theorem) and that it is limited by a rather small
ordinal, Gamma_0. However, neither of these statments is correct, as
I demonstrate at some length in my paper "Predicativity beyond Gamma_0".
As the latter assertion has been conventional wisdom for the past forty
years, and I am claiming to have decisively refuted it, I should think
that this paper would be of interest to a number of people on this list.
I have also written a paper, "Analysis in J_2", in which I show how
core mathematics can be straightforwardly developed in a predicatively
valid system which is faithful to classical intuition and avoids some
of the coding machinery involved in subsystems of second order
arithmetic. All three papers are available at
http://www.math.wustl.edu/~nweaver/conceptualism.html
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
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