FOM: Realism, Conceptualism, Nominalism

[Joseph Vidal-Rosset]

In order to take part in the discussion on philosophy of mathematics
(Mycielski & Kanovei), I take the freedom to post a mail on Quine's
work. What follows is an extract of a paper which can be download
from my web page:
http://www.u-bourgogne.fr/PHILO/joseph.vidal-rosset/ (to download the
paper use the link "Philosophy of Mathematics and Philosophical
Systems"(pdf).)

I give here the main point:

Philosophical debates would gain in clarity if the terminology used
by philosophers was strictly defined and shared by everybody. In one
of his most famous papers Quine has defined three ontological
positions in philosophy of mathematics: realism, conceptualism, and
nominalism. These are the three main old types of ontological
commitments which roughly speaking have their filiation in three
contemporary surveys of the philosophy of mathematics, namely:
logicism, intuitionism, and formalism:

>"Realism, as the word is used in connection with the medieval
>controversy of universals, is the Platonic doctrine that universals
>or abstract entities have being independently of the mind; the mind
>may discover them  but cannot create them. Logicism, represented by
>Frege, Russell, Whithead, Church and Carnap, condones the use of
>bound variables to refer to abstract entities known and unknown,
>specifiable, and unspecifiable, indiscriminately.
>Conceptualism holds that there are universals but they are
>mind-made. Intuitionism, espoused in modern times in one form or
>another by Poincar=E9, Brouwer, Weyl, and others, countenances the use
>of bound variables to refer abstract entities only  when those
>entities are capable of being cooked up individually from
>ingredients specified in advance. [...]
>Formalism, associated with the name of Hilbert, echoes intuitionism
>in deploring the logicist's unbridled recourse to universals. But
>formalism also finds intuitionnism unsatisfactory. The formalist
>might, like the logicist, object  to the crippling of classical
>mathematics; or he might, like the nominalists of old, objetc to
>admitting abstract entities at all, even in the restrained sense of
>mind -made entities. The upshot is the same: the formalist keeps
>classical mathematics as a play of insignificant notations." (QUINE,
>"On what there is", in From a Logical Point of View, HUP, 1953, pp.
>14-15)

(...)

We need in fact not one criterion but two in order to understand this
tripartite classification. The ontological commitment of a
mathematical theory can be decided by knowing if

(a) the theory quantifies over sets which canot be reduced to
elements (i.e. the language of the theory cannot be translated into a
first order logic like Lower Predicate Calculus;)
(b) at least one axiom of the theory is impredicative or involves a
impredicative definition.

The Zermelian set theory holds (a) and (b), then it is a Platonist
(or a realist theory). Assuming (a) but rejecting (b), Wang's set
theory is conceptualist. Chihara interprets Wang's system in a no
class theory, and he believes that such a set theory succeeds in
denying (a) and (b), showing that a mathematics theory is able to be
strong enough and nominalist as well.

(I add here that according to the Quinean classification the great
majority of Mathematicians is realist (or Platonist) without knowing
it.)