FOM: misunderstandings?
holmes@catseye.idbsu.edu
holmes at catseye.idbsu.edu
Fri Jun 2 16:35:04 EDT 2000
Dear All,
This is to clear up what seem to be misunderstandings of my position
on the part of Steve Simpson.
RE "benefit to the standard of living"
If Steve reads what I said carefully, he will find that I didn't actually
say that I thought that his question
Is this area of research likely to lead to new technology which can
be expected to participate in improving the human standard of
living?
was off-topic. I said that it was just as off-topic (or just as on-topic)
as the earlier question
Is the funding of scientific research a proper function of
government?
Neither question is a question about the foundations of mathematics. Both
are questions in ethics or politics, and both are relevant to the practice
of foundations of mathematics. There is no criticism of the second
question as off-topic which does not also apply to the first question --
they are questions of the same kind.
If it is legitimate to consider in this forum whether our research is
beneficial to mankind (or how to practice it so that it will become
so), it is also legitimate to consider whether our research should be
funded by the State (or how to practice it so that it will become
deserving of or more likely to receive such funding).
I didn't express agreement with any view on these questions (even the
one that I ascribed to the founders of the field).
Simpson said:
Frank, Sazonov, Holmes and Hazen all praise AFA (the anti-foundation
axiom) in destructive terms. According to them, AFA is valuable
because it ``knocks the iterative conception of set off its
metaphysical pedestal''.
I reply:
Maybe I'm being too subtle. I expressed _disagreement_ with the
points made by the others, because I don't agree that the view of sets
in ZFC-Foundation+AFA is philosophically very different from that in
ZFC. ZFC-Foundation+AFA does nothing to knock the iterative
conception of set off its metaphysical pedestal; it is (as I take
pains to point out) simply a slightly different implementation of the
same conception.
Further, I think that iterative conception of set can be given a perfectly
adequate metaphysical grounding, and is a perfectly adequate foundation for
mathematics. I don't think it needs to be knocked off a metaphysical
pedestal -- but I think that its metaphysics could profitably be put on
a different footing (see below).
I looked back at my remarks and I find it impossible to see how they
fall under Simpson's description. Mostly, I point out that
ZFC-Foundation+AFA is essentially similar to ZFC, and that ZFC
can also be presented in the way that ZFC-Foundation+AFA is presented.
Then I criticize the idea that a set is made up of its elements, without
making any criticism of ZFC foundations there, either.
Simpson said (in reply to text of mine also included):
> 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 [...]
I guess I am one of those users, because I certainly do think that a
set is made up of its elements. What is wrong with this view? Why
does Holmes say that it is not based on reflection?
I reply:
It is self-evident that a set is not made up of its elements.
Otherwise, what is the difference between x and {x}? The relation of
part to whole for sets is the subset relation, not the membership
relation. It is "almost" true that a set is "made up" of its elements
-- it is actually "made up" of the singleton sets of its elements. It
remains to figure out what the singleton construction does... (This
argument is developed by David Lewis in his book Parts of Classes -- I
came up with it independently while analyzing the metaphysical
underpinnings of my favorite set theory).
The idea that a set is made up of its elements works fine when one is
doing something like studying point sets. The set consisting of a
single point is often confused (harmlessly in this context) with that
single point. But as soon as one has to consider sets of sets, the
idea that a set is made up of its elements becomes untenable. Explain
the differences between {1,2}, {{1},{2}}, and {{1,2}} in terms of
how sets are "made up" of their elements.
The reasons why the view is not challenged (and so is unreflectively
held by many people) have to do with the way set theory is taught, I
would guess. It is also easy to hold because (as noted above) it is
"almost" correct! But the point being missed is important.
Simpson says:
I think that, in order to understand Holmes' remarks, we have to
understand where Holmes is coming from. Holmes is an advocate of an
alternative set theory known as NFU. And NF/NFU is an attempt to
formalize a concept of set which many people regard as unclear, or at
least not nearly so coherent and compelling as the standard iterative
concept of set, on which ZFC is based. Therefore, it is
understandable why Holmes might want to denigrate the standard
iterative concept of set and ``knock it off its pedestal'', by denying
that sets are made of elements.
I reply:
This is not an accurate statement of my views. I study the set theory
NFU (and related theories), and I maintain (contrary to views
expressed by Simpson and others) that NFU (with proper attention to
_its_ metaphysical foundations, which have been more neglected than
those of ZFC, as Simpson correctly points out) is capable of serving
as a foundation for mathematics in the same sense in which ZFC serves
de facto as a foundation for mathematics. In saying this, I am not
advocating any change in mathematical practice. ZFC is also an
adequate (a very good) foundation for mathematics; but it is not THE
somehow canonical foundation for mathematics (that is the only point I
am concerned to make). I am not an advocate of NFU in the sense that
I want any extension of NFU to replace ZFC in practice.
I think that ZFC is better understood in a different way than it is
currently understood (in fact, essentially in the way that the system
with AFA is best understood). I think that ZFC is best presented as
the theory of isomorphism types of well-founded extensional relations
with a "top" element, plus assumptions about how many objects there
are. This is a question of philosophical analysis of what ZFC is
about -- it has no effect on how the theory is to be used in
developing mathematics.
By the way, Lewis's or my critique of the notion of sets as being made
up of their elements applies to any set theory at all, not just to
ZFC. It doesn't constitute an attack on ZFC; Lewis takes pains to
give an account of ZFC in terms of this conception in his book.
Practitioners of Quine-style set theory are no less likely to think of
sets as being made up of their members, I would suspect.
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 math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes
More information about the FOM
mailing list