Gary asked:

Are we ever likely to get to the truth about the nature of mathematics? I guess I mean; decide
between the competing views on the nature of the subject (I'm thinking Logicism, Formalism, and
Intuitionism, but then I studied back in the 70's and there must be some new 'isms' by now). Actually
I'm prompted to write after having read about Greg Chaitin and his view that math is empirical (I used
to laugh at John Stuart Mill for holding an empiricist view of the nature of mathematical truth but I'm
more inclined to take GC seriously).

So, to sum up:

What are the modern views on the nature of mathematics? (Can you recommend any good reference
works?) Will we ever be able to decide between them? Could mathematics be empirically based?

============

I like the empiricist viewpoint myself. Take a look at these:

Hacking, I. The Emergence of Probability: A Philosophical Study of Early Ideas About Probability,
Induction and Statistical Inference.Cambridge, England: Cambridge University Press, 1975.

Lakoff, G., and R.E. Nunez. Where Mathematics Comes From: How the Embodied Mind Brings
Mathematics into Being.New York, NY: Basic Books, 2000.

Narens, L. "A Meaningful Justification for the Representational Theory of Measurement." Journal of
Mathematical Psychology46 (2002): 746-68.

Null, G.T., and R.A. Simons. "Aron Gurwitsch's Ordinal Foundation of Mathematics and the Problem
of Formalizing Ideational Abstraction." Journal of the British Society for Phenomenology12, no. 2
(1981): 164-74.

Steven Ravett Brown

you might want to look at:

Stewart Shapiro, "Thinking about Mathematics."

Paul Benacerraf and Hilary Putnam (eds) "Readings in the Philosophy of Mathematics."

Gideon Rosen and John Burgess "A Subject with no Object."

Crispin Wright "Frege's Conception of Number as Object."

Crispin Wright and Bob Hale "Reason's Proper Study."

Hartry Field "Science Without Numbers"

Please note that only the first two listed assumes no familarity with the field and the literature, so if
you are not au fait with recent developments and theories in phil maths then you should have a look
at the Shapiro first, which should give you a good idea of the landscape before diving into the others.
The Benacerraf-Putnam collection brings together key papers in phil maths up to about 1975 so might
be worth a look too.

Rich Woodward

This is too large a field to debate in the context of a question/ answer format. For whatever
satisfaction you may get from it, I recommend Reuben Hersh's What is Mathematics, Really?, where
the author defends the viewpoint that it is a social institution. But he goes through every
mathematician of note, ancient and modern, to justify his point of view, and he uses historical
vignettes, fictional dialogues and practical (real life) working situations in illustration. Since Hersh is a
mathematician, he ought to know. A point of view radically different, though in the last resort cognate,
was put forth by G.H. Hardy in his book, A Mathematician's Apology.Hardy thought of his work as an
art form and demanded that it should have no practical application whatever, nothing other than the
production of a beautiful artefact. He was not alone in this. Morris Kline, too, has written on the social
and artistic aspects of mathematics; in fact, there is a surprisingly large literature on the metier as art.

And then there is that eloquent and magnificent plea for mathematics as the "saviour" of mankind
from the computer by Roger Penrose, called Shadows of the Mind.Problems that are "not
computable" are a cinch for us humans, Penrose claims; and this is his warrant for asserting that "in
principle" a robot cannot ever be endowed with anything like consciousness. Now here are three
books I would consider indispensable. There are many more, of course, but you have to start
somewhere; and these titles will at least introduce you to three of the major enquiries on which the
subject of "what is mathematics" thrives.

Jürgen Lawrenz

Sydney