FOM: use of 'platonism' in f.o.m.

[Charles Parsons]

On the origins of the term "platonism" in f.o.m.

Its currency may derive from Bernays' paper of 1935, "Sur le platonisme
dans les mathématiques". Goedel's lecture of 1933 mentioned by Martin Davis
was delivered five months earlier than the lecture that Bernays's paper was
based on, but Goedel didn't publish and his lecture seems to have remained
little known. I had never heard of it before we got going in the vork on
volume III of his works.

Still, it's hard to believe that the term wasn't used in the literature
with something like the meaning that has become current, for some fairly
strong level of realism about mathematical objects.

Robert Black usefully calls our attention to the remarks about Ramsey in
Carnap's 1931 paper on the logicist foundations of mathematics, and to
Goedel's 1932 review of the paper. What's striking about the Carnap paper
is that the phrase Black quotes is the only reference to 'platonism' or the
'platonic'; it seems really just to be introduced for rhetorical effect and
not to correspond to what anyone would affirm under that name. Also, the
view he attributes to Ramsey is not one about classes or sets but about
properties.

I think Black misreads Goedel when he says that Goedel in the 1933 lecture
"is repeating there something from his 1932 review of Carnap." The review
limits itself to summarizing what Carnap says. Goedel writes:

Auch F. P. Ramsey habe diesen Standpunkt zu begruenden versucht, doch seien
seine Argumente wegen des ihnen zugrunde liegenden Begriffsabsolutismus
(Platonismus) unannehmbar" (Works I 244).

I quote the German because the subjunctive makes clear that this is simply
a report of what Carnap says.

In the 1933 lecture, on the other hand, Goedel is speaking in his own voice
and makes the statement that the axioms of set theory "necessarily
presuppose a kind of platonism." There is a hint of a distinction between
impredicativity about sets and impredicativity about properties, but he
still puts the discussion of impredicativity mainly in terms of properties,
to that extent following Carnap.

Carnap and Goedel introduce talk of "platonism" in the context of
discussions of impredicativity, and Bernays also makes this connection. The
suggestion that a polemical use of the term was in the air in the Vienna
Circle seems right to me, but it may have been Carnap's apparently offhand
use of it in 1931 that inaugurated its use with this more specific meaning.

Many writers include disclaimers to the effect that they are not claiming
that the view they are calling by that name is Plato's.

Charles Parsons