FOM: misunderstandings?

Cristian Cocos cristi at
Sun Jun 4 21:40:24 EDT 2000

holmes at wrote:

> Simpson:

> I certainly do think that a
> set is made up of its elements.  What is wrong with this view?  Why
> does Holmes say that it is not based on reflection?
> Holmes reply:
> It is self-evident that a set is not made up of its elements.

Not *mereologically* at least. But I guess Simpson uses the phrase 'made up of'
the way almost everybody (except perhaps Lewis and Holmes) uses it in this
context, i.e. not as 'made of parts' but as 'made of elements'. See for example
mereotopological analysis versus conceptual analysis: *different* analytical
perspectives of the *same* thing. There is nothing wrong with that. Indeed, 'made
up of' could suggest in the majority of cases a *mereological* decomposition. But
it shouldn't always... Not in this case anyway.

> Otherwise, what is the difference between x and {x}?  The relation of
> part to whole for sets is the subset relation, not the membership
> relation.  It is "almost" true that a set is "made up" of its elements
> -- it is actually "made up" of the singleton sets of its elements.  It
> remains to figure out what the singleton construction does...  (This
> argument is developed by David Lewis in his book Parts of Classes -- I
> came up with it independently while analyzing the metaphysical
> underpinnings of my favorite set theory).

Lewis doesn't like the membership relation; he considers it too 'mysterious' and
he tries to reduce it to mereological fusion. Of course he won't be able to
achieve this. Lewis' merit is that he has managed to push the frontiers of
mereology within set theory to the extreme, at the same time isolating the
ontological originality of the membership relation in its purest form and
emphasizing its irreducibility: Lewis has shown how much of set theory is
mereology and how much is not.

As for the reasons of his aversion to the membership relation, I am still not very
clear... I have never considered it problematic, nor have I come across
mathematicians that do. I find it to be one of the basic/primitive epistemic
tools: a mechanism that, ultimately, *constitutes* us, human beings, as cognitive
agents. What is it to explain besides that? What is so mysterious about that?

Cristian Cocos

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