### 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

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

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

So, how to respond. Suppose instead that we think of the notion of "represents" as being

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".]

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

a model $\mathcal{A}$Notice that this definition contains a "represents* the world if and only ifthere arerelations $R_i$ such that $\mathcal{A}$ is isomorphic to these.

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

*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 worldThis is not being defined. It is being adopted as a primitive, at least for the time being. One obtains two derived notions, byrelativeto relations $R_1, \dots$.

*ramsifying*with respect to relations. First, quantifying over all relations, one defines "r-represents",$\mathcal{A}$ r-represents the world if and only ifIt 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 athere arerelations $R_1, \dots$ such that $\mathcal{A}$ represents the worldrelativeto $R_1, \dots$.

*Lewisian notion*of representation by ramsifying with respect to all__natural__relations:$\mathcal{A}$ Lewis-represents the world if and only if there areThis doesnaturalrelations $R_1, \dots$ such that $\mathcal{A}$ represents the worldrelativeto $R_1, \dots$.

*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".]

ReplyDeletea model represents the world just if the world has large enough cardinality.It seems like another way to clarify this is to focus on the word "just". If it is read in the ordinary English sense of "only", then this is simply giving a necessary condition for representation, and a rather obvious one at that -- it the cardinality of the model is not big enough, then it's not clear at all that it can adequately represent the world.

It's only if you read "just if" as "exactly when" that the sentence starts seeming suspect.

Hi Sara,

ReplyDeleteOops - yes, it's meant to mean "if and only if". Sorry if it's a bit confusing! So, yes, it's "exactly when".

In what I called the Glub Lemma, the reasoning goes in both directions. So, it's a sufficient condition too: if the world is at least as large as |dom(A)|, then A represents* the world.

This is the central problem raised by Newman, in his criticism of Russell's theory of representation. (The quote from Newman is in the previous post.)

Cheers,

Jeff