[FOM] Advertising an approach to the philosophy of mathematics (with some logical technical work)
Feng Ye
yefeng at phil.pku.edu.cn
Fri Sep 7 23:04:20 EDT 2007
Dear FOMers,
First, let me apologize for advertising personal researches here. I really
like to see comments on this kind of approach to the philosophy of
mathematics by logicians. It differs from those well-known philosophies of
mathematics (from the old logicism, formalism, intuitionism Platonism and
so, to the recent fictionalism, naturalism, neo-logicism etc.), and it
contains some technical work as well, which might interest some logicians.
The goal for the entire research is to explore a completely scientific
account for human mathematical practices. The technical part of it is an
attempt to offer a logical explanation of the applicability of mathematics
to things in this strictly finite (!) universe (from the Planck scale up to
the cosmological scale). The idea is to show that all applications of
classical mathematics in realistic sciences are in principle reducible to
the applications of 'strict finitism', which is a fragment of the
quantifier-free Primitive Recursive Arithmetic (PRA), with recognized
functions restricted to elementary functions (the proper subclass of
primitive recursive functions). Technical work on this topic mostly consists
of developing ordinary mathematics within strict finitism. So far (in the
monograph cited below) it has contained some basics of calculus, metric
spaces, complex analysis, Lebesgue integration and linear operators on
Hilbert spaces. This technical work grew out of the early work in my
dissertation. (F. Ye, 'Toward a constructive theory of unbounded linear
operators on Hilbert spaces', Journal of Symbolic Logic, 65(2000), no. 1; F.
Ye, Strict Constructivism and the Philosophy of Mathematics, PhD
dissertation, Princeton University, 2000.) Besides that fact that some
impressive applied mathematics can be developed within strict finitism,
there are other intuitive reasons supporting the conjecture (!) that all
applications of classical mathematics in realistic sciences are in principle
reducible to the applications of strict finitism.
The philosophical part of the research might also interest those who are not
professional philosophers, because its goal is to show that we can replace
metaphysical speculations regarding mathematics (as in Platonism,
intuitionism and so on) by a completely scientific description of human
mathematical practices, including a scientific explanation of the
applicability of mathematics. It will view human mathematical practices as
human brains' cognitive activities and take a study of human mathematical
practices as a continuation of cognitive sciences, for the specific subject
matter of human mathematical cognitive activities. That is, its goal is to
reduce the traditional philosophical problems to ordinary scientific
problems, whose answers will then be pursued by ordinary scientific methods
(e.g. by postulating human cognitive models) and evaluated by ordinary
scientific standards. From that perspective, the research discusses
philosophical issues regarding mathematics, such as the nature of human
mathematical knowledge, objectivity in mathematics, apriority of logic and
arithmetic and so on.
A monograph 'Strict Finitism and the Logic of Mathematical Applications' and
several articles belonging to the research project are available online at
http://www.phil.pku.edu.cn/cllc/people/fengye/index_en.html
The monograph focuses on the logical explanation of applicability of
mathematics. It should be accessible to logicians who are not familiar with
the contemporary debates in philosophy of mathematics among philosophers.
Other articles are mostly on philosophical issues, including an introduction
to the research project, titled 'Introduction to a Naturalistic Philosophy
of Mathematics'. These belong to the context of contemporary debates in
philosophy of mathematics among philosophers in the analytic tradition.
All comments are greatly appreciated, and I am very happy to answer any
questions.
Sincerely,
Feng Ye (PhD, Princeton, 2000)
Associate Professor
Department of Philosophy
Peking University, China
yefeng at phil.pku.edu.cn
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