Online Papers - Brian Weatherson

David Lewis 83
The big payoff of the Main Thesis is that it reduces the mysteries of set theory
to a single mystery. Any class is a fusion of singletons, i.e., sets with one member.
If we understand what a singleton is, and we understand what fusions are, then we
understand all there is to know about classes, and about sets. That’s because any set
is just the fusion of the singletons of its members.
But singletons are deeply mysterious. The usual metaphors that are used to introduce
sets, metaphors about combining or collecting or gathering multiple things into
one are less than useless when it comes to understanding the relationship between a
singleton and its member. In (1993c), Lewis settles for a structuralist understanding
of singletons. He also says that he “argued (somewhat reluctantly) for a structuralist’
approach to the theory of singleton functions” in (1991), though on page 54 of (1991)
he appears to offer qualified resistance to structuralism.
One of the technical advances of (1991) and (1993c) was that they showed how a
structuralist account of set theory was even possible. This part of the work was coauthored
with John P. Burgess and A. P. Hazen. Given a large enough universe (i.e.,
that the cardinality of the mereological atoms is an inaccessible cardinal), and given
plural quantification, we can say exactly what constraints a function must satisfy for
it to do the work we want the singleton function to do. (By ‘the singleton function’
I mean the function that maps anything that has a singleton onto its singleton. Since
proper classes don’t have singletons, and nor do fusions of sets and objects, this will
be a partial function.) Given that, we can understand mathematical claims made
in terms of sets/classes as quantifications over singleton functions. That is, we can
understand any claim that would previous have used ‘the’ singleton function as a
claim of the form for all s: ...s...s..., where the terms s go where we would previously
have referred to ‘the’ singleton function. It is provable that this translation won’t
introduce any inconsistency into mathematics (since there are values for s), or any
indeterminacy (since the embedded sentence ...s...s... has the same truth value for any
eligible value for s).
Should we then adopt this structuralist account, and say that we have removed the
mysteries of mathematics? As noted above, Lewis is uncharacteristically equivocal on
this point, and seemed to change his mind about whether structuralism was, all things
considered, a good or a bad deal. His equivocation comes from two sources. One
worry is that when we work through the details, some of the mysteries of set theory
seem to have been relocated rather than solved. For instance, if we antecedently
understood the singleton function, we might have thought it could be used to explain
why the set theoretic universe is so large. Now we have to simply posit a very large
universe. Another is that the proposal is in some way revisionary, since it takes
ordinary mathematical talk to be surreptitiously quantificational. Parts of Classes
contains some famous invective directed against philosophers who seek to overturn
established science on philosophical grounds.
I’m moved to laughter at the thought of how presumptous it would be to
reject mathematics for philosophical reasons. How would you like the
job of telling the mathematicians that they must change their ways, and
abjure countless errors, now that philosophy has discovered that there are