FOM: Plato and XX-century platonisms

[Jeffrey Ketland]

Dear Luigi

I agree – time out. Let’s stop arguing about whether Plato was a “platonist”
, in the
modern XX-century sense. Would you agree that these are the problems?

(A) We need to *interpret* Plato’s writings, which were of course written in
an entirely
different context to that of our day. In particular, the state of
mathematical knowledge
circa 380BC in Ancient Greece was very different from that of modern
mathematical
knowledge (especially our recently-gained better understanding of the
infinite and our
better understanding of logic, truth, provability, axiomatizability,
computability, etc.).
Of course, any interpretation of Plato’s writings will inevitably be driven
by our own
inclinations, biases and so on. (But even biases can result in true
beliefs!).

(B) As you rightly say, Plato himself seemed to be deeply sceptical about
the infinite,
thinking it was contradictory. So, in that sense, I agree: talk of the
Platonic continuum
might be called blasphemy.

(C) The third problem is that modern XX-century mathematical platonism (or
mathematical realism) comes in a variety of shapes and sizes:

(i) There are versions of realism advocated by the major innovators in the
foundations
of mathematicians at the turn of the century, such as Frege, Cantor and,
later, Godel.

(ii) There are mid-XX-century versions of realism advocated by philosophers
such as
Quine and Putnam.

(ii) And there are late-XX-century versions of realism advocated by more
recent
writers including Penelope Maddy, John Burgess, Stewart Shapiro, Michael
Resnik,
Crispin Wright, JR Brown (and many others). And even these various positions
do not
form an obviously coherent whole, although they have much in common.

What Plato would have made of each of these “platonisms” is very hard to
say!
Furthermore, most of these advocates of various versions of modern
XX-century
platonism are very cautious about identifying their position with Plato
himself, and
tend to avoid the term “platonism”.

I think that part of the reason is the influence of WV Quine on the modern
philosophy
of mathematics “scene” (although there are some British philosophers of
maths, such
as Dummett and Wright, who refuse to be swayed by Quinian empiricist
trends).
Anyway, Quine’s views (as well as Frege’s and Godel’s) are usually
considered to be
worthy of major discussion. (NB: interpretation of Quine is probably as
controversial
as interpretation of Plato, Kant, Wittgenstein, etc!).

In particular, most contemporary philosophers of maths usually take
seriously Quine’s
arguments about “ontological commitments”, and also take seriously Quine’s
repeated
arguments about the indispensability of abstract (possibly even
non-constructive)
mathematics to modern empirical science. Putnam certainly did (hence the
so-called
Quine-Putnam indispensability argument) and Hartry Field certainly does
(since much
of his work is an attempt to undercut the QP argument). In any case, many,
many
contemporary philosophers of maths write from a “post-Quinian” viewpoint.

Although Quine is an ontological mathematical realist, his *epistemology*
for
mathematics is fundamentally different from Plato’s (and Godel’s). In
particular,
Quine’s epistemology for mathematics is based on a kind of complicated
bootstrap
from empirical science. In a nutshell, Quine holds that our ultimate warrant
for
believing in (abstract) mathematical objects ultimately rests on ordinary
sensory
input, rather than on some direct “grasping” or “seeing” of abstract
concepts and facts
(and this “direct seeing” is usually attributed to Plato). And some
commentators think
that Quine's epistemology for mathematics is itself a reductio ad absurdum,
since it
seems to follow that our warrant for believing 2 + 3 = 5 depends upon our
warrant
for believing theoretical physics!

Of all modern mathematical realists, it is probably true to say that Godel’s
epistemology is the closest to Plato’s own position (as it is usually
interpreted), since
Godel thought that we had a separate mental mechanism or faculty (aside from
ordinary sense perception) for perceiving abstract mathematical concepts.
However,
many modern mathematical realists (again, largely influenced by Quine’s
naturalism:
there must be natural, sensory, mechanisms for acquiring knowledge) tend to
be
sceptical of such non-physical mechanisms for perceiving mathematical
objects, facts
and concepts.

Jeff

Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at nottingham.ac.uk