Chris (P and M) (01)
CP> As I think you must know there is a Fregean concept of number
> (where 2 is the set of all sets with two members)... (02)
Frege's conception is one among a large number of possible ways
of thinking about number. Another conception, which is simpler
both mathematically and psychologically, is to relate numbers
to the act of counting. In any case, it is a very big stretch to
go from Frege's definition to the idea that numbers are physical. (03)
CP> There is debate about whether sets are abstract - David Lewis
> asks why we cannot say the set of cars in the car park is located
> in the car park - and if it is located, it cannot be abstract.
> Hence, there is an account of numbers that says they are not abstract. (04)
CM> Are you sure Lewis asked that? (05)
CP> The point is made in the Plurality of Worlds. He was quite
> clear that he found arguments claiming that sets were NOT concrete
> did not make sense to him. (06)
That is why many people prefer mereology as a foundation for ontology
because the partOf operator does not create a new kind of entity.
The mereological collection of cars is just as physical as any
car in the collection. (07)
But in set theory, John is distinct from {John}, and the empty
set {} is distinct from {{}} and from {{},{{}}}. The collection
of all cars is just as physical as the cars, but the set of all
cars is not the same as the mereological collection of cars. (08)
As for David Lewis, he can say anything he likes (or could when he
was alive). But if he is using ordinary English, the word 'set'
does not obey the axioms of Cantor's set theory or its successors.
Instead the word 'set' is used more like the word 'collection'
in mereological discussions. (09)
In short, I would say that Lewis was confusing the issue. He was
trying to take evidence from informal usage in English to draw
a conclusion about set theory. (010)
But Lewis is also responsible for creating an enormous amount of
confusion about possible worlds and whether they are "really real."
His writings are often stimulating, but I would never recommend
them as a basis for clarifying anything. (011)
John (012)
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