FOM: The "Mysteriousness" of Membership

[Allen Hazen]

    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) is just a footnote 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.
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   (1) Pure literary pleasure.  Lewis's style is clear and simple, and
presents a model of saying IN ENGLISH things that most logicians would
resort to symbols for.
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   (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 even the expert to have an understanding of the meanings of the
basic terms that goes much beyond my understanding of that of "mithril."
(Though perhaps only a philosopher of Lewis's realist leanings would find
it a disturbing fact!)
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   (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?