Monday, June 16, 2008

Nominalistic Content

In "A Role for Mathematics in Phyiscal Theories" I argued that fictionalists have a problem when it comes to specifying the nominalistic content of our best scientific theories. David Liggins recently drew my attention to Gideon Rosen's explanation of nominalistic adequacy in his 2001 article "Nominalism, Naturalism, Epistemic Relativism". Does this approach block my argument?

Here is Rosen's account:
Let's say the concrete content of a world W is the largest wholly concrete part of W: the aggregate of all of the concrete objects that exist in W ... S is nominalistically adequate iff the concrete core of the actual world is an exact intrinsic duplicate of the concrete core of some world at which S is true -- that is, just in case things are in all concrete respects as if S were true (p. 75).
Here is a summary of my challenge to fictionalism: (i) the fictionalist must present something like Field's axioms if he is to explain which parts of the full content get into the nominalistic content. But (ii) giving these axioms would involve taking a stand on features of the concrete world that went beyond our evidence for the mathematical scientific theory. So, (iii) there was no epistemically responsible way for the fictionalist to specify how the nominalistic content differed from the full content.

A fictionalist might agree with the demand in (i), but think that Rosen's approach resolves the issue without appealing to Field-style axioms. I am not sure how this will work, though. If we use the real numbers to represent temperature, how does Rosen's test apply? For example, suppose we consider a law about thermal expansion. If that is part of my theory, what does it mean to say that the law is nominalistically adequate? Let's take two potential things that may or may not get in there: (a) instantiated temperatures are dense, (b) there is no lowest temperture. Both of these can be expressed in a nominalistic language provided we have Field's temperature predicates around. So, I think they are about the concrete world and should be relevant to nominalistic content.

Now I suggest that even if we accept Rosen's test, this is no help in resovling the question of the nominalistic adequacy of the law. I do not know if (a) and (b) are part of the nominalistic content of the law or how this is determined. This is the sense in which the commitments are indeterminate for the fictionalist. (I am not saying I explained this very well in the paper, but this is at least how I am thinking about it now.)

Suppose a fictionalist responded that whatever indeterminacy there is for the nominalistic content also arises for the full content. So, there is nothing here to tell against fictionalism and in favor of some kind of realism. My view is that a realist who can accept the mapping account can specify the full content with reference to these mappings. For this law, it would be something like "For any iron bar, if the temperature were to be increased by amount t, then the length of the bar would increase by alpha * t". Here the antecedent and the consequent involve mappings between objects with physical properties and mathematical objects. This clearly does not require (a) or (b). So, because we can apeal to mappings or relationships between physical properties and mathematical objects, we can resolve some apparent indeterminacies in the full content.

Why can't the fictionalist say the same thing? Maybe he can, but it seems that no fictionalists have explained how this would work beyond some toy examples involving counting. So, maybe the best way to see my discussion is as a challenge to the fictionalist to explain how she can match the realist in giving determinate contents to our scientific statements and theories. On my story, the commitment to realism comes in explaining the full content. The fictionalist either needs to bring in this explanation or else directly specify the nominalistic content by other means. I have not shown that both of these strategies are hopeless, but I think the burden is on the fictionalist to work it out.

Another reply is that the fictionalist need not satisfy my demand to explain how the full content relates to the nominalistic content or to clarify the nominalistic content directly. This is the reply I try to deal with in the article by saying that the fictionalist must explain what he is committing himself to in accepting a given statement or theory. Otherwise, he is not facing up to Quine's challenge on ontological commitment.

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