FOM: Mysteriousness of membership: completed
Allen Hazen
a.hazen at philosophy.unimelb.edu.au
Mon Jun 12 08:30:02 EDT 2000
(My post on this yesterday broke off in the middle. This is the complete
version; sections (1) and (2) have minor changes; (3) and (4) are now
written. Apologies for earlier post.)
I think David Lewis's "Parts of Classes" is a good, LITTLE (main text 120
pages of relatively small format), book, one which I recommend to my
students and friends... and not JUST because I helped with the Appendix!
Since it is very different from most mathematical expositions, and some its
philosophical assumptions may seem bizarre to many readers, I'd like to
give some reasons for reading it. (1) is minor, (4) relates to a 6 June
2000 post by Randall Holmes, (2) answers a question about Lewis's
philosophical orientation, and (3) is where I think he has helped clarify
thinking about set theory in a useful way.
---
(1) Pure literary pleasure. Lewis's style is clear and simple, and presents
a model of how to IN ENGLISH things that most logicians would resort to
symbols for.
---
(2) Why does Lewis find memebership mysterious? He ISN'T criticizing set
theorists: his attitude to mathematics as practiced is if anything overly
deferential, and he is NOT saying that mathematicians' understanding of
membership is insufficient for mathematical purposes. The "mysteriousness"
is a problem for the philosophical understanding of the motivation of the
axioms, and presupposes a certain philosophical perspective: aspects of
Lewis's version of "realism."
Presupposed aspect A: Lewis takes it that set theory should
be thought of as literally TRUE. He is not satisfied with a formalist or
"deductivist" ("if-then-ist": it's what he criticizes as "fictionalism")
philosophy of mathematics, largely because it seems insufficiently
respectful of the content of mathematics.
Presupposed aspect B: Quantifiers mean the same thing in
mathematical and non-mathematical contexts. Set theorists say THERE ARE
sets (with such-and-such properties) and zoologists say THERE ARE animals
(of such-and-such kinds). They are talking about utterly different kinds of
things (abstract mathematical things versus empirically discovered physical
things), but the words THERE ARE mean the same thing in the two sciences.
(This thesis is sometimes called, by lovers of philosophical jargon, the
thesis of "the univocity of being"; Lewis I think inherits it from Quine:
cf. "On What There Is" and "Two Dogmas of Empiricism" in Quine's "From a
Logical Point of View.")
Given these presuppositions, there is a mysteriousness about mathematical
objects like sets. We have a good idea what animals are like, and a good
idea how we find out about them (we see and touch them), but we don't have
such an idea of what sets are like-- all set theory tells us about them is
that they are things with members, and then details about what things bear
the membership relation to which sets-- and we don't have a good account of
how we find out about them. (Giving the latter, after all, is a perennial
TASK for the philosophy of mathematics.) Since the concept of a set and the
concept of the membership relation are two aspects of the same thing, the
mysteriousness of sets can with equal appropriateness be called a
mysteriousness of membership: to say we never SEE or TOUCH sets is pretty
much the same as saying we never OBSERVE the membership relation holding
between two objects.
It is a mysteriousness that concerns the epistemology of the subject-- the
philosophical reasons that can be given for accepting the axioms as true--
and not its mathematical development. An analogy: the dwarves in Tokien's
"The Lord of the Rings" make things out of "mithril". Do I know what
mithril is? Do I understand what Tolkien wrote about it? Well, yes in the
sense that I can follow the story: mithril is a silvery metal, enough
harder than real silver to be usable for swords. But could I identify it
metallurgically? No; it seems likely that Tolkien had no real metallic
element or alloy in mind when he wrote. It seems to be a curious fact about
mathematics (one that any good philosophical account of mathematics should
explain!) that productive work in the science does not require the expert
to have an understanding of the meanings of the basic terms that goes much
beyond my understanding of that of "mithril." (Task for philosophy of
mathematics: explain why mathematics is so different from metallurgy.)
---
(3) But suppose that fact about mathematics (which "structuralist"
philosophies of mathematics claim to be able to explain) doesn't concern
you. Is there something about the foundations of mathematics that Lewis
says that is interesting independent of that? I think there is an
interesting conceptual feature of set theory that Lewis's re-axiomatization
of ZF helps make visible. Set theory, it seems to me, does two
distinguishable things for mathematicians.
First thing ("descriptive" or "logical"): Set theory allows the
formalization of a good deal of higher-order reasoning. (A first-order
quantification is one in which the variable would be replaced, in a
specific instance, by the name of a particular object. A higher-order
quantification is one in which the variable would be replaced, in a
specific instance, by a predicate/condition/formula.) By quantifying over
SETS of objects we can express many higher-order claims about objects.
Second thing ("constructive" or "postulational"): set theory allows
us to "construct" (the metaphor is in common use, though I'm sure it gives
constructivists the shivers!) new objects. Do you want a counterexample to
some conjecture? Take a topology defined ... over the power set of the
product-algebra of .... The set theoretic existence axioms give us what we
need.
Lewis's reformulation of ZFC (actually of a stronger system:
second-order MK) is useful, I think, in that it separates the two things.
He defines a "framework" that expresses the higher-order stuff without
postulating sets. The framework consists of "mereology" (the theory of
parts and wholes, as in Leonard & Goodman, JSL 1940) and "plural
quantification" (as in Boolos, Jornal of Philosophy 1984 (=Logic, Logic, &
Logic, ch. 4)). Each by itself is formally similar to monadic second-order
logic; combined and assuming infinitely many objects they give the
framework essentially the expressive power of third-order logic. (This is
explained in the Appendix to Lewis's book; I have given more details in J.
Philosophical Logic 1997.) Lewis also claims-- I personally have doubts,
but many people find it plausible-- that this framework is,
philosophically, perspicuous and non-mystifying. By using it as his
underlying logic, he is able to give a very elegant formulation of the
existence assumptions of set theory; it seems to me that a good test for
when set theory is being used "logically" might be whether the "framework"
can do the job without the postulates.
---
(4) Holmes, in his FoM post of 7 June ("Are sets made up of their
members?") describes an interpretation of set theory in terms of
(isomorphism classes) of graphs in a background second-order logic (graphs
being represented by their edge relations). On this interpretation some of
the ZF axioms are theorems of second-order logic (I think, but haven't
checked, that these axioms are close to what Montague calls "Rank Free" set
theory in his paper in Crossley & Dummett, eds, "Formal Systems and
Recursive Functions"); the rest are consequences of natural assumptions
about the number of objects in the universe. Lewis, under the name of a
"structuralist" interpretation, discusses something very similar (with
proof sketches).
Allen Hazen
Philosophy Department
U of Melbourne
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