FOM: Plato and XX-century platonisms
ketland at ketland.fsnet.co.uk
Thu Feb 17 11:52:26 EST 2000
I agree time out. Lets 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* Platos writings, which were of course written in
different context to that of our day. In particular, the state of
circa 380BC in Ancient Greece was very different from that of modern
knowledge (especially our recently-gained better understanding of the
infinite and our
better understanding of logic, truth, provability, axiomatizability,
Of course, any interpretation of Platos writings will inevitably be driven
by our own
inclinations, biases and so on. (But even biases can result in true
(B) As you rightly say, Plato himself seemed to be deeply sceptical about
thinking it was contradictory. So, in that sense, I agree: talk of the
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
of mathematicians at the turn of the century, such as Frege, Cantor and,
(ii) There are mid-XX-century versions of realism advocated by philosophers
Quine and Putnam.
(ii) And there are late-XX-century versions of realism advocated by more
writers including Penelope Maddy, John Burgess, Stewart Shapiro, Michael
Crispin Wright, JR Brown (and many others). And even these various positions
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
Furthermore, most of these advocates of various versions of modern
platonism are very cautious about identifying their position with Plato
tend to avoid the term platonism.
I think that part of the reason is the influence of WV Quine on the modern
of mathematics scene (although there are some British philosophers of
as Dummett and Wright, who refuse to be swayed by Quinian empiricist
Anyway, Quines views (as well as Freges and Godels) are usually
considered to be
worthy of major discussion. (NB: interpretation of Quine is probably as
as interpretation of Plato, Kant, Wittgenstein, etc!).
In particular, most contemporary philosophers of maths usually take
arguments about ontological commitments, and also take seriously Quines
arguments about the indispensability of abstract (possibly even
mathematics to modern empirical science. Putnam certainly did (hence the
Quine-Putnam indispensability argument) and Hartry Field certainly does
of his work is an attempt to undercut the QP argument). In any case, many,
contemporary philosophers of maths write from a post-Quinian viewpoint.
Although Quine is an ontological mathematical realist, his *epistemology*
mathematics is fundamentally different from Platos (and Godels). In
Quines epistemology for mathematics is based on a kind of complicated
from empirical science. In a nutshell, Quine holds that our ultimate warrant
believing in (abstract) mathematical objects ultimately rests on ordinary
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
that Quine's epistemology for mathematics is itself a reductio ad absurdum,
seems to follow that our warrant for believing 2 + 3 = 5 depends upon our
for believing theoretical physics!
Of all modern mathematical realists, it is probably true to say that Godels
epistemology is the closest to Platos own position (as it is usually
Godel thought that we had a separate mental mechanism or faculty (aside from
ordinary sense perception) for perceiving abstract mathematical concepts.
many modern mathematical realists (again, largely influenced by Quines
there must be natural, sensory, mechanisms for acquiring knowledge) tend to
sceptical of such non-physical mechanisms for perceiving mathematical
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
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