Monday, December 29, 2008

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I think we have discussed this before. The arguments are valid, but depend on assumptions that are not questioned in the arguments (e.g. the Axiom of union depends on the questionable assumption that absolutely everything is a set).

It is not a creature of darkness if its existence is required to explain basic mathematical truths. The question is whether basic mathematical truths (those requiring induction, in particular) do require it.

I once tried to put it this way: you would think that what a set is, essentially, is something to which other objects bear the "element of" relation. But the empty set is not such an object, since by definition it has no elements. And you can't say it could have elements, because if it doesn't, it couldn't.

Alex,

Nice and pithy and exactly right.

O writes, "e.g. the Axiom of union depends on the questionable assumption that absolutely everything is a set." I guess I don't see this.

But if we make the is-an-element-of relation primitive we can think of it as a table with objects labelling the rows and sets labelling the columns and with a tick in the cell at row r and column c iff object r is an element of set c. Suppose the table contains a column with no ticks---after all, why should this possibility be excluded?---then the relation is telling us that there is an empty set.

David Brightly -- hmmm...that's an interesting image. It makes me think of Cantor's diagonal argument -- but I can't quite see why it should.

Bill,

How can we dispel the darkness surrounding the empty set? The main charge against it in your post is that

"the existence of a set depends on the existence of its members. The set consisting of me and my cat cannot exist unless both man and cat exist: if either of us should cease to exist, the set would cease to exist. It exists because we exist, not vice versa. [...] In the case of the null set, however, there is nothing on which the null set can depend for its existence."
My suggestion is that it's this realist (and presentist?) view that absorbs the light. For if we take Cantor's idea outlined in the subsequent post that a set is the result of a (possible) mental act of collecting then the difficulties fade away. If I'm interested in my genealogy it makes sense to talk about the set of my ancestors. Granted, there is a presupposition here that there are such things as people and a relation of descent between them. Your point, perhaps, is that there first must be (or must have been?) things in existence before we can find collections of things. But that seems to be digging deeper for foundations than we need. For if Cantor is right then sets are a way of thinking about multiplicity when a universe of objects is already given. So I can think of my ancestors as a subset of the universe of all the people who have ever existed. Again it makes sense to partition my ancestors into two further sets, those living and those dead. Many people have living ancestors. I don't. If you deny us the empty set then we cannot talk in general about our living ancestors using the language of sets, for I am an exception, apparently having no set of living ancestors. But this, of course, is just another mathematician's motivation for admitting the empty set---we want to be able freely to take complements, and the complement of the universe is the empty set.

To summarise, we differ in that your view appears to be that sets must be in the world, external to ourselves, and somehow dependent on the objects in the world. My view is that they are mental constructs, abstracted from a world in which multiplicity is a given. I don't know how sustainable my view is---it may well collapse under further probing---but it doesn't find the empty set a creature of darkness. And it's this (and the effectiveness of the mathematical system erected upon it) that suggests to me that your realist view is mistaken conceptualisation of sets. But then I'm a mathematician, not a philosopher, and can only hope this isn't a silly answer to a philosopher's question.

Alex,
It's the mathematician's standard view of a relation.

David

On some consideration I wonder if your objection does not beg the question. For that is exactly what is at issue -- whether the table *should* contain a column with no ticks.

Alex,

Yes, I can see that I might be seen as begging the question. But the point I'm trying to make is that there is a perfectly good way of thinking about a relation that does allow for tickless columns (or rows). Consider the relation is-a-child-of. If we think of it defined between children on the left (labelling the rows) and parents on the right (labelling the columns) then there can be no tickless rows or columns. But this introduces an undesirable asymmetry. If we think of the relation as defined between people we will find tickless columns (childless people) and if we restrict to living people we will also find tickless rows (children whose parents have died). If we allow flexibility over the domain on which the relation is defined we gain substantially. For example, if we think of the child relation R as specified symmetrically between people on the left and people on the right, then we can define the product relation of R with itself, and this is exactly the is-a-grandchild-of relation between people. Further, we can calculate this relation by multiplication of boolean matrices. We can't do this if we insist on distinct domains on the left and right. The lesson I take from this is that there is no right or wrong way of thinking about relations, but there are more and less useful and effective ways. With my apologies if this stuff is old hat.

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