[FOM] Platonism and Formalism and mixtures thereof

[A.P. Hazen]

   Arnon Avron tells us that he takes a "formalist" attitude toward set
theory, but is a "platonist" when it comes to the natural numbers.  This
sort of hybrid philosophical attitude is perhaps more common than one would
guess from the traditional protrayal of Platonism and Formalism as opposing
views might suggest.  I think many philosophers (and many mathematicians
reflecting philosophically on their subject) take a more "realist" view of
the relatively small and concrete mathematical structures, and a less
"realist" view of the abstractions of the higher infinite.  Different
people, however, seem to draw the line in different places: perhaps the
real Formalist (Kanovei) could be described as drawing the same line, but
so narrowly that the "platonistic" core has shrunk to a point?
   Some people  draw the line  further out than Avron.  Poszgay, in his
contribution to the 1967 AMS symposium on Axiomatic Set Theory (published
in "Proceedings of Symposia in Pure Mathematics" volume XIII, part 1)
advocated what he called "liberal intuitionism": basically he seemed
willing to accept all of ZF, but with the proviso that only intuitionistic
logic be allowed for statements quantifying over ALL sets.
   Views more like Avron's have been advocated by Velleman ("Constructivism
liberalized," in the "Philosophical Review," vol. 102 (1993), pp. 59-84):
the law of excluded middle seems fine to him when you are talking about the
natural numbers, but non-denumerable systems oght to be taken with
intuitionistic salt.
   My first logic teacher was Frederic Fitch (a survivor of the "heroic
age" of mathematical logic, who had published in volume 3 of the JSL).
Once, at an evening seminar in a series sponsored by the philosophy
department where scholars from other disciplines talked about philosophical
aspects of their own work, Abraham Robinson described the Platonisitic and
Constructivistic attitudes, and then asked Fitch for comments.  Fitch said
that he agreed with most of what Robinson had said about the two
contrasting views, but that he, Fitch, personally thought of himself as
both a Platonist and a Constructivist.
   I ***think*** what he meant was that he was a Platonist about numbers
(or strings of symbols), but less of one about such further abstractions as
set theory.  This showed in his practice: he was happy to reason
classically (inferring, for example, "Either A or for all x Fx" from "For
all x, Fx unless A is true") in his metamathematics, but he had no
correspondingly platonistic attitude to the systems whose consistency he
proved by this reasoning: of the system of his book "Symbolic Logic" he
once said (in an aside in a lecture) hat he had not been trying to build a
constructive system, but just the strongest system he was able to prove
consistent.
    So perhaps we can liken the Formalists and Platonists to two political
parties... but there seems to be a large "swinging vote" which goes with
one party on some issues and the other on others!
----
Allen Hazen
Philosophy Department
University of Melbourne