FOM: Steel realism
Detlefsen.1 at nd.edu
Sat Jan 17 08:03:43 EST 1998
Herewith a few comments and questions on John Steel's recent posting on realism.
He writes (though not completely in the order listed):
(a) To my mind, Realism in set theory is simply the doctrine that there are
sets, and that these sets do not depend causally on us (or anything else,
for that matter).
(b) As a philosophical framework, Realism is right but not all that interesting.
(c) Ironically, it is Realism which makes the important use of Occam's
razor, by cutting away the conceptual clutter that goes with the assertion
that sets in some way depend on us (or idealized intelligent beings).
(d) (i). One ambition in foundations is to construct a universal framework
theory in which all mathematical theories can be naturally interpreted.
We want to have ONE picture, so that, as Harvey put it, "people can
work together on a common ground". If truly different "pictures" arise,
the problem of putting them together will become of immediate
importance. In fact, as far as I know this just hasn't happened. One
can get different natural interpretations of the language of set theory
by "restricting" the notion of set, but this is not a case of
incompatible "pictures" emerging.
(d) (ii) To advocate phi for inclusion in such a framework theory committs one
to the view that phi is true.
(a) is something like a definition or characterization of what JS means by
'realism'. I take it that he would extend the characterization in the
'natural' way to areas other than set theory. So, for example, realism in
number theory would be the doctrine that there are numbers (of whatever
kind is under consideration), realism in geometry would be that there are
geometrical objects, etc.. (Note: This does not deny that numbers and
geometrical objects are sets. Neither does it affirm it. The question of
whether all mathematical objects are sets is a separate question from the
question of realism.)
Given this way of characterizing realism, I see little reason to assert
(b). More precisely, I see little reason to assert the first half of (b)
... the statement that realism is right. Indeed, this is that feature of
Hilbert's viewpoint that I find most attractive. Hilbert distinguishes
between real and ideal elements (statements, proofs, etc.). He doesn't
think that realism is a plausible attitude to adopt towards ideal elements.
(He is not clear on whether it is a plausible attitude to adopt towards
real elements.) I agree. I don't see much plausibility in lumping points
and lines at infinity, to take one well-known case, together with the more
ordinary elements of geometry. One adopts the former because they promote
the dualities, and one promotes the dualities because of their
simplificatory virtues. There is no "objective pull" (to use the phrase
from Quine recently employed by rbrt tragesser) to points and lines at
infinity, however. That is what their inventors (Desargues and Monge) and
their chief developer and justifier (Poncelet) intended to mark by calling
them 'imaginary' or 'improper' elements. Poncelet took such "beings of
thought" (etres de raison, ens rationes) to typify the algebraic approach
to things and he saw the justification of introducing imaginary objects in
geometry as the same as the use of imaginary objects (e.g. complex numbers)
in algebra. They allow one to simplify the system of formal operations by
which one reasons about the real objects of the system.
I therefore don't see that realism is 'right' ... not about geometry as
extended by (e.g.) points and lines at infinity, and not about the myriad
other areas of mathematics, several of which were mentioned by Hilbert in
his foundational essays, especially those of 1925 and 1927, in which the
introduction or ideal or imaginary plays the same type of role. I think the
most important part of Hilbert's philosophy of mathematics was his
distinction between the real and the ideal in mathematics and his Kantian
conception (i.e. the conception of Kant's general critical epistemology) of
how to justify this. By the way, it is because I take this to be the "core"
of Hilbert's Program that I also believe that program not to have been
refuted by Godel's theorems ... but that is another story.
Given what I have just said, my attitude d(i) and d(ii) is probably clear.
Like Hilbert, I regard some kind of real/ideal distinction as necessary for
any plausible philosophical foundation for mathematics. That doesn't mean
that there can't or shouldn't (as per d(i)) be but ONE "picture" (since one
way of conceiving of a "picture" is as Poncelet and Hilbert saw it). It
does, however, imply that including phi in that "picture" does not
necessarily commit one to saying that phi is true. As a defense of
mathematical grammar, then, realism would seem to be a pretty blind and
undiscerning defense. It supposes, implausibly, that all uses of
mathematical language are intended for the same end, namely, something like
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