FOM: philosophical viewpoints holmes at
Tue May 30 15:05:25 EDT 2000

Dear FOMers,

This will be a grab bag of comments on the recent fascinating conversation
on various mathematical viewpoints:

Thus Simpson:

But this immediately raises a question of political philosophy: ``Is
the funding of scientific research a proper function of government?''
And political philosophy is clearly off-topic for the FOM list.  In
order to avoid the political issue, why not put the question this way:

   Is this area of research likely to lead to new technology which can
   be expected to participate in improving the human standard of

Thus Holmes:

This is just as off-topic, for exactly the same reason.  Is the
purpose of mathematics (or of scientific research in general) the
improvement of human standards of living?  The founders of mathematics
would certainly _not_ have said so.  Of course "just as off-topic"
does not mean that I think it is off-topic :-)

Thus Simpson:

If subjectivism is not the essence of the constructivist philosophy,
what is?

Thus Holmes (me):

I don't believe that computer scientists interested in constructive
mathematics are interested in it from a "subjectivist" standpoint; if
anything, their reasons for adopting a constructivist view are quite
the opposite.  Some would claim that statements formalized in intuitionistic
logic have objective meaning in a sense in which arbitrary statements in
classical logic do not.  I don't hold this view, but I understand it.

Because _some_ constructivists historically argued in subjectivist terms
does not mean that constructive mathematics is necessarily subjectivist
whenever it is practised.

Thus Sazonov:

Anti-Foundation Axiom (AFA) gives a new view on the concept of
sets and extends it in an interesting way. (Hyper)sets are not a
result of collecting something, but rather the result of abstraction
from directed graphs.

Thus Hazen (same topic):

It seems to me that the real significance of the AFA is that
it helps knock the "Iterative Conception of Set" off its metaphysical
pedestal.  Between about 1970 and 1990, the Iterative Conception was, at
least among philosophers, often taken in a very metaphysical way: sets
>>presuppose<<, or are in some quasi-causal way >>generated by<<, their
members.  (A particularly clear statement of this metaphysical
interpretation of the iterative conception is in the text-- the
"structuralist" tendencies of the appendix are another matter-- of David
Lewis's little book "Parts of Classes.")  The graph-theoretic inspiration
of AFA has, it seems to me, helped make alternative metaphysical views
about set theory seem more plausible.

Thus Holmes (me):

I spent a summer once reading through Aczel's book with a group of
mathematicians who were quite excited about it.  I started out
interested but already demystified, and I think I convinced at least
some of my fellow participants.

The system ZFC - Foundation + AFA does rest on an iterative conception
of foundations (or at least it can be presented in that way).  Each
successor stage consists of the sets of which we can construct direct
graph "pictures" (possibly non-well-founded) using the objects at the
previous stage.  There are anti-foundation axioms (I'm thinking of one
due to Boffa with less strong extensionality built in) which may be
less vulnerable to this objection (which is an objection to their
fundamental novelty, not to their practical mathematical interest).

The system of Aczel isn't necessarily different from ZFC in being
founded on abstraction from directed graphs; ZFC itself (or more
generally, set theory in the style of Zermelo) can be understood as
the theory of isomorphism types of well-founded extensional relations
with a top element (i.e., it can be obtained by abstraction from
well-founded graphs).  (This is the natural way to interpret
Zermelo-style set theory in NFU, for example).

It's very useful to bring out (and dispute) the view that the elements
of a set somehow automatically generate the set.  I think it's worse
than that; I think that many users of set theory (I hope not any set
theorists!) actually think on some unreflective level that the set is
somehow "made up" of its elements, though they learn to avoid in
practice the technical mistakes which such a view would imply.  (As
Lewis points out, the relation on sets which resembles the relation of
part to whole is inclusion, not membership).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at
not glimpse the wonders therein. |

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