A Sketch of Armstrong's Naturalism

And some reasons to question it. 

Top of the (Sub)stack.

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Expositing Armstrong, I wrote

 

The exclusion of so-called abstract entities or abstract objects such as mathematical sets, unexemplified universals, and numbers from the roster of the real is because of their lack  of causal power.  What causal role could they play? 

 

And then I quoted Armstrong:  "And if they play no causal role it is hard to see how we can have good reasons for thinking that they exist." (2)

Woland's Cat objects:

This reasoning is missing a step, I think. Abstract entities do exist when they are contemplated by a mind: assuming minds are 'real' (i.e. part of organisms, which are part of the space-time continuum of reality), then mathematical sets etc. become real when represented in the mind. 

How would Armstrong reply?  As follows. To exist is to exist extra-mentally. That is the only way anything can exist. If so, there cannot be two or more ways or modes of existing.   He here follows, as other Australian philosophers do, his and their teacher John Anderson. Hence there is no such way of existing as existing intra-mentally, in the mind. Whatever I do when I think about something, I do not, in thinking about it, or contemplating it, confer upon it existence-in-the mind.  

The following are candidate abstract entities: the number 7, the set {7}, the proposition expressed by '7 is prime,' the property of being prime.  To say that they are abstract is to say that they are not in space or in time, and that they are 'causally inert,' which is to say that they do not enter into causal relations with anything: they neither cause nor are caused.  Armstrong rejects the whole lot of them.  Their existence is ruled out by his metaphysical naturalism according to which reality is exhausted by the space-time system and its contents.  They don't exist outside the mind and, since that is the only way anything can exist, they don't exist inside the mind either.

So what am I thinking about then I think of {Max the cat, Manny the cat}?  Sets or "classes supervene on their members -- that is to say, once you are given the members, their class adds nothing ontologically, is no addition of being." (Sketch, 8)  But then what am I thinking about when I think about the intersection of two disjoint sets? A set theorist will say: the null set, { }!  You will also recall that in set theory, the null set is a subset of every set, and a member of every power set.   Don't confuse subset and member as Armstrong does on p. 8, n. 1.

This presents a bit of a problem for Armstrong. He cannot say that the null set  supervenes on its members since it doesn't have any.  So of course he bites the bullet: he rejects the existence of the null set. "It would be a strange addition to space-time!" (p. 8., n. 1)  The more I think about this, the more problematic it seems. If there is no null set, then there are no power sets.  And if there is no null set, why should we think that there are unit sets or singletons such as {Quine} or {Max}? What is the difference between Max and the set whose sole member is him?  If Max's singleton supervenes on him, then there is no singleton!  If there are no singletons, then there is no intersection of {Max, Manny} and {Max, Maya}!

What would Woland's Cat say about that?

Memo to self: Re-read  the section "Mysterious Singletons" in David Lewis, Parts of Classes. And blog it! You are not spreading yourself thin enough!