Structural Representation Again

The previous post discussed the Newman-style objection to a certain way of thinking about "structural representation claims". Beginning with a certain understanding of what "represents" means and a certain "amorphous glub" conception of how the world is, one gets the conclusion that

a model represents the world if and only if the world has large enough cardinality. 

I believe that only way to begin resolving this is to focus again on "represents", and try to clarify its meaning. The previous definition, of "represents*", was that,

a model $\mathcal{A}$ represents* the world if and only if there are relations $R_i$ such that $\mathcal{A}$ is isomorphic to these. 

Notice that this definition contains a "ramsification" in the definiens, "there are relations ...".
If we adopt an extensional view of relations and suppose that there is some determinacy in domain of the things that are values of first-order variables, and permit the quantifiers to range  over all relations whatsoever on this supposed domain of things, it is then not hard to show, using a Newman-style argument, the Glub Lemma:

$\mathcal{A}$ represents* the world if and only if the cardinality of the world is sufficiently large. 

This implies that the only thing we can be either right or wrong about is the cardinality of the world. I take this to tell us that we have simply explained "represents" incorrectly. Of course, one may, or may not, accept this Newman-style conclusion. Like Ted Sider, I take the conclusion to be incredible. On this doctrine, for example, physics is nothing more than counting. For all there is to the world, all there is to the target domain, representation-independently, is its cardinality. If we think of the elements of the domain of a model $\mathcal{A}$ as representing mind-independent worldly "noumena", then the conclusion is that only the number of "noumena" count.

It is not absolutely clear to me that Kant's theory of representation faces this Newman-style objection, but I suspect that it does. For I think that Kant's theory of representation, with its "external intuitions" and "a priori categories" and so on, is more-or-less the same theory as Russell's theory of representation, set out in his Analysis of Matter (1927), and was precisely the theory that Newman objected to in his 1928 Mind review.

So, how to respond. Suppose instead that we think of the notion of "represents" as being relative to some sequence of relations (matching the signature of the model $\mathcal{A}$). The notion being used is then

$\mathcal{A}$ represents the world relative to relations $R_1, \dots$.

This is not being defined. It is being adopted as a primitive, at least for the time being. One obtains two derived notions, by ramsifying with respect to relations. First, quantifying over all relations, one defines "r-represents",

$\mathcal{A}$ r-represents the world if and only if there are relations $R_1, \dots$ such that $\mathcal{A}$ represents the world relative to $R_1, \dots$.

It is this notion that is likely to lead to trivialization (because one is likely to define relative representation in terms of there being an isomorphism from $\mathcal{A}$ to the $R_i$). And one obtains a Lewisian notion of representation by ramsifying with respect to all natural relations:

$\mathcal{A}$ Lewis-represents the world if and only if there are natural relations $R_1, \dots$ such that $\mathcal{A}$ represents the world relative to $R_1, \dots$.

This does not face a Newman-objection, at least not prima facie, for the class of natural relations is going to be much more sparse than the class of all relations whatsoever (which, e.g., include all the scientifically unnatural, gerrymandered, disjunctive ones too: such as the the property of being an electron or a cheese sandwich).

This is the fairly standardly-held resolution (it is sketched by Carlos A. Romero C in a comment to the previous post). There's a lot more to say, of course, but I leave that to some other time!

[UPDATE: 22 March. I'm grateful to a comment from Sara Uckelman, who pointed out the phrasing "just if" is ambiguous in the indented statements. So I have updated the phrasing, replacing "just if" by "if and only if".]