New
Directions in the Philosophy of Mathematics by Thomas
Tymoczko (Editor)
The traditional debate among philosophers of
mathematics is whether there is an external mathematical reality,
something out there to be discovered, or whether mathematics is the
product of the human mind. This provocative book, now available in a
revised and expanded paperback edition, goes beyond foundationalist
questions to offer what has been called a "postmodern"
assessment of the philosophy of mathematics one that addresses issues of
theoretical importance in terms of mathematical experience. By bringing
together essays of leading philosophers, mathematicians, logicians, and
computer scientists, Thomas Tymoczko reveals an evolving effort to
account for the nature of mathematics in relation to other human
activities. These accounts include such topics as the history of
mathematics as a field of study, predictions about how computers will
influence the future organization of mathematics, and what processes a
proof undergoes before it reaches publishable form. This expanded
edition now contains essays by Penelope Maddy, Michael D. Resnik, and
William P. Thurston that address the nature of mathematical proofs. The
editor has provided a new afterword and a supplemental bibliography of
recent work.
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Steven Cullinane
Richard J. Trudeau's book The NonEuclidean Revolution
opposes his Story Theory of truth to the more traditional Diamond Theory
of truth. This site defends the Diamond Theory with various
geometric examples, beginning with the diamond in Plato's Meno.

by David S. Ross, Ph.D.
Excerpt:
The philosophy of mathematics is the philosophical
study of the concepts and methods of mathematics. It is concerned with
the nature of numbers, geometric objects, and other mathematical
concepts; it is concerned with their cognitive origins and with their
application to reality. It addresses the validation of methods of
mathematical inference. In particular, it deals with the logical
problems associated with mathematical infinitude...

Web resources; Regional mathematics; Subjects; Books
and other resources; Chronology; Timelines. The Canadian Society for the
History and Philosophy of Mathematics, founded in 1974 and more.

Stanford
Encyclopedia of Philosophy
Excerpt:
One of the most intriguing features of mathematics is its
applicability to empirical science. Every branch of science draws upon
large and often diverse portions of mathematics, from the use of Hilbert
spaces in quantum mechanics to the use of differential geometry in
general relativity. It’s not just the physical sciences that avail
themselves of the services of mathematics either. Biology, for instance,
makes extensive use of difference equations and statistics. The roles
mathematics plays in these theories is also varied. Not only does
mathematics help with empirical predictions, it allows elegant and
economical statement of many theories. Indeed, so important is the
language of mathematics to science, that it is hard to imagine how
theories such as quantum mechanics and general relativity could even be
stated without employing a substantial amount of mathematics...

by Kurt Gödel, 1961
Excerpt:
I would like to attempt here to describe, in terms of
philosophical concepts, the development of foundational research in
mathematics since around the turn of the century, and to fit it into a
general schema of possible philosophical worldviews [Weltanschauungen].
For this, it is necessary first of all to become clear about the schema
itself. I believe that the most fruitful principle for gaining an
overall view of the possible worldviews will be to divide them up
according to the degree and the manner of their affinity to or,
respectively, turning away from metaphysics (or religion). In this way
we immediately obtain a division into two groups: skepticism,
materialism and positivism stand on one side, spiritualism, idealism and
theology on the other. We also at once see degrees of difference in this
sequence, in that skepticism stands even farther away from theology than
does materialism, while on the other hand idealism, e.g., in its
pantheistic form, is a weakened form of theology in the proper sense...

The papers indexed below were given at the Twentieth
World Congress of Philosophy, in Boston, Massachusetts from August
1015, 1998. Additional papers may be added to this web site as
electronic versions are acquired and formatted for the archive.

The increasing levels of experimentalization in
mathematics and other traditionally theoretical fields has brought focus
to bear on the issues of human perception and thought, intuition,
perception and the nature of knowledge. These issues are embodied in the
problem of scientific visualization and its relationship to mathematics,
philosophy and traditional scientific methodologies.
The projects described here represent ongoing work
being done at the CECM with respect to the experimentalization of
mathematics and related processes.

Paul Ernest
This site lists The Philosophy of Mathematics
Education Journal and Philosophy of Mathematics Education
Newsletters. Site also includes essays by Paul Ernest on the
Philosophy of Mathematics.

RBJ Mathematics Philosophy Foundation
Writings on the Philosophy of Mathematics mainly concerned with the foundations of mathematics, but also touching upon other problems in the Philosophy of Mathematics and providing some historical background.

