David Brightly in a comment far below writes:
Bill says, at 03:21 PM.
...a family cannot be reduced to a number of persons; it is not a mere manifold, or a mere many...
The same is true for a pair (of shoes) and other collective terms. They all imply relations of some sort between their members. But this is not so for a mathematical set. No such relations are implied. A set is a pure manifold. Bill also says, at 03:42 PM,
...it also makes no sense to say that an abstract object -- which is what math sets are -- has Gomer surrounded.
I dispute this: mathematical abstract objects have relations between their parts and sets qua sets do not, so sets cannot be abstract objects. These are the chief reasons for thinking that sets are pure plural referring terms. Artificial, structured, linguistic extensions engineered by us for clarity of mathematical thought and expression.
Respondeo
I am afraid I find most of the above unacceptable. First of all a pair of shoes is not a term, collective or singular. Terms are linguistic items; shoes are not. It is true, though, that mathematical sets abstract from any relation their members bear to one another. But this is not something I denied.
It is also not the case that "a set is a pure manifold." That cannot be right because a set can have sets as elements (members). The Power Set axiom of axiomatic set theory states that for any set S, there exists a set P such that X is an element of P if and only if X is a subset of S. Thus the power set P of a set S is the set of all of S's subsets. So the power set of {1,2} = {{1}, {2}, {1,2}, { }}.
The subsets of S are elements of the power set which fact shows that sets are distinct from their members and are therefore not pure manifolds. Sets are distinct from their members in that they count as objects in their own right. So if there is a set of Ed's dancing shoes, then this set is distinct from the left shoe, the right shoe, and the two taken collectively. This is one of the ways a set differs from a mereological sum/fusion. The sum of Ed's shoes is just those shoes, not an object distinct from them.
Mereology is "ontologically innocent" as David Lewis puts it, whereas set theory is not:
Mereology is innocent . . . we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many. Set theory is not innocent. . . . when we have one thing, then somehow we have another wholly distinct thing, the singleton. And another, and another . . . ad infinitum. (Parts of Classes, Basil Blackwell 1991, p. 87)
Brightly argues: "mathematical abstract objects have relations between their parts and sets qua sets do not, so sets cannot be abstract objects." This makes no sense to me since the mereological notion of parthood does not belong in standard axiomatic set theory. It is at best analogous to the elementhood and subset relations which are essential to set theory. The sum/fusion of Max and Manny has them and their cat-parts as parts. The set {Max, Manny} does not have as elements their whiskers, claws, tails, etc.
If there are sets, they are abstract objects, which is to say that they do not exist in space or time. I can trip over my cat, but I can't trip over my cat's singleton. Where in space is the null set? When did it come to be? How long will it last? Is it earlier or later than the null set's singleton? Is the nondenumerably infinite set of real numbers somewhere in spacetime? These questions involve category mistakes.
I said that if the Hatfields have Gomer surrounded, that cannot be taken to mean that the set having all and only the members of the Hatfield clan as its members has poor Gomer surrounded. What I said is obviously true.
So I conjecture that David is using 'abstract' in some idiosyncratic way. I await his clarification.
>First of all a pair of shoes is not a term, collective or singular. <
Unreasonably picky. Brightly clearly meant "a pair of shoes".
Posted by: oz | Thursday, May 04, 2023 at 05:59 AM
He probably did. But in a discussion which turns on subtleties, one must strive for precision of expression. The use-mention distinction must be observed.
If I am being unreasonably picky -- and perhaps I am -- you are unreasonably ignoring the substantive points I made.
Posted by: BV | Thursday, May 04, 2023 at 11:42 AM
Thanks, Bill, for the extended response. I am trying to nudge people away from thinking of sets as objects towards thinking of the notion of 'set' as an extension of the referential capacity of natural language. On my view '{1, 2, 3}' is a referring term that plurally refers to 1, 2, and 3. If I then say, 'Let S = {1,2,3}', I am introducing a new referring term, 'S', that also refers plurally to 1, 2, and 3. We would normally express this by saying 'S is the set containing exactly 1, 2, and 3', reinforcing the 'sets as objects' mode of thought.
When I say 'mathematical abstract objects have relations between their parts'I don't intend 'part' in the mereological sense. Take the requirements for a (number) field. Among other requirements, there has to be an addition operation and a multiplication operation and multiplication must distribute through addition. I think of these operations as 'parts' or 'aspects' of the field. There are no such 'parts' or 'aspects' required for a set.
One difficulty that you raise for this view is that math sets do seem to have a bit of structure. A set can have a set as an element in addition to ur-elements. But if 'the Hatfields and the McCoys' makes sense as a referring term in natural language, it seems we can make sense of '{{1, 2, 3}, {4, 5, 6}}' as a referring term in an artificially extended language. Set notation avoids the ambiguities of natural language plural referring terms that I drew attention to here.
Posted by: David Brightly | Thursday, May 04, 2023 at 03:47 PM