FOM: Wilson's dual view of foundations

[Joe Shipman]

>>1.  To what extent is this dual view of foundations accurate and/or
relevant?<<

This way of looking at it seems both accurate and useful to me.

>>2.  Is it indeed possible that "an ontological system could be devised
that was sufficient to represent the objects and reasoning of
mathematics but wasn't simultaneously an axiomatization of some given
pre-theoretic notion(s)"?<<

Yes, but not in a way that would be actually workable and useful to
ordinary mathematicians.  The advantages of having the ontological
system axiomatize pre-theoretic notions are that it can thus be seen to
be consistent, and that it can be developed smoothly and intuitively.

>>3.  How much of the success of set theory as a f.o.m. is ontological
and how much is axiomatic?  How much is due to its integration of the
two?<<

The axiomatic side is a tremendous success story; the ontological side
has been slower to make headway, but as large cardinal axioms are shown
to be increasingly relevant to ordinary mathematical practice it will
become equally significant.

>>4.  Is the integration mentioned in question 3 largely fortuitous, or
is it (contra question 2) a fundamental aspect of all foundations?<<

It is somewhat fortuitous.  The development of first-order logic and
type theory allowed ordinary mathematics to be done without much extra
in the way of ontology; the technical advantages of ZFC which led to its
adoption as the canonical f.o.m. are partly ontological but not in a way
that has impacted mathematical practice much.  One exception: the
acceptance of the Axiom of Choice may have been facilitated by the
naturalness of the picture of the Von Neumann set-theoretical universe
it provides.

>>5.  Where does naturalness fit into the picture?  As for ontology,
there is of course an advantage to being able to represent mathematics
in a simple and direct way.  But once a representation and its
properties are established, the representation need not (and indeed
should not) trouble us further.  In other words, if forging a
representation is a one-time job, why should we discount a foundational
scheme because it's a little harder to set up the basic representations,
since one this is done, mathematics can (for the most part) proceed as
usual. <<

Excellent point; though the unnaturalness of the set-theoretical
representation is exaggerated, there are really only two places where
the standard account is less than graceful:
1) Kuratowski definition of ordered pair (a,b)={{a},{a,b}} where {x}
means {x,x}
2) Von Neumann identification of integers with finite ordinals 0={},
n+1=U({n,{n}})
These components of the standard account are there for technical
reasons, because the "pure" set theory that results is easier to work
with and prove theorems about; but taking ordered pair as a primitive
operation and the integers as Urelements would give an equivalent theory
that tracked mathematical practice quite naturally.

>>And as for axiomatization, if a foundational scheme proposes to
capture some pre-theoretic notions from a certain point of view,
shouldn't it be judged on whether or not it has captured these notions
accurately from this point of view and not on whether other notions or
points of view are expressed naturally in terms of the ones at hand?<<

It should be judged on both -- successful representation of the
pre-theoretic notions gives us assurance that the system is consistent
and allow it to be developed more smoothly and intuitively, but the
relation to other notions and points of view should at least be natural
enough to allow easy interpretability, or there will be resistance to
metamathematical application of the foundational scheme.

-- Joe Shipman