[FOM] Corfield's book

Alasdair Urquhart urquhart at cs.toronto.edu
Mon Oct 13 10:26:22 EDT 2003

I was surprised by the fact
that a few people engaged in heated
discussions on FOM of David Corfield's recent
book "Towards a Philosophy of Real Mathematics"
without having read it.  

Anyway, I have now read the book cover to cover,
and I found it interesting and stimulating.  
I recommend it to FOM subscribers as an
enjoyable and provocative book that presents
a somewhat heterodox viewpoint on the philosophy
of mathematics.  I don't always agree with it,
but I think I may have more points of agreement
than disagreement with the author.

Corfield's book has two main aspects.  It
is a polemical argument for a re-orientation in
the philosophy of mathematics.  This is largely in
the first, introductory chapter.  The remaining
chapters are a series of rather loosely connected
studies intended partly as examples of the kind of
philosophy of mathematics that Corfield favours.
There are chapters on computer-assisted proofs,
Bayesianism in mathematics, Lakatos's philosophy
of mathematics, and a final chapter on higher-dimensional

Corfield's big heroes are Lakatos and Polya.  Since
I share his strong admiration for Polya (I am not such 
a big fan of Lakatos), I am already prejudiced in his
favour.  His main complaint about contemporary philosophy
of mathematics is its narrowness of focus and lack of
engagement with current mathematics*.  He is a sworn enemy
of "neo-logicism" and the "foundationalist filter."
By the latter, he means a lack of interest in anything
beyond the ideas of the period from 1880 to 1930,
as the following quote illustrates:

	By far the larger part of activity in what
	goes by the name "philosophy of mathematics"
	is dead to what mathematicians think and
	have thought, aside from an unbalanced
	interest in the 'foundational' ideas of
	the 1880-1930 period, yielding too often
	a distorted picture of that time (p. 5).
Corfield advocates a "practice-oriented" philosophy of
mathematics, rather in the style of recent philosophy of

I had occasion myself a few years ago to complain in a review
of how little attention philosophers of mathematics pay
to what mathematicians actually do, so I think Corfield
is definitely on to something here.  

There are quite a lot of passages in the book that could
be read as showing hostility to logic and formal research
in the foundations of mathematics, but I believe that
they are rather to be interpreted as directed against 
what Corfield takes to be a narrowness in the current philosophy
of mathematics community.  This passage (contrasting
philosophy of physics with philosophy of mathematics) 
illustrates this very clearly:

	The prospective philosopher of mathematics
	quickly gathers that some arithmetic, logic
	and a smattering of set theory is enough to
	allow her to ply her trade, and will take 
	some convincing that investing the time in
	non-commutative geometry or higher-dimensional
	algebra is worthwhile.  One of the main purposes
	of the book has been to argue against this (p. 235).

Anyway, I'd like to repeat that the book is interesting and
stimulating.  I'd like to hear from other FOM subscribers
who have read it.

* He makes an exception of writers like Penelope Maddy,
for example.

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