Wednesday, 20 March 2013

Structural Representation Claims, Newman's Objection and the Glub Lemma

Consider structural representation claims, which often take the form:
(R) The model $\mathcal{A}$ represents the world.
For example, one might see something like,
a. Assuming special relativity, a model of the form $(\mathbb{R}^4, \eta_{ab}, F_{ab})$ represents the world.
b. Spacetime models $(M, g)$ and $(M^{\prime}, g^{\prime})$ represent different worlds.
A Newman-style objection arises in connection with such claims. Recall that in a 1928 Mind review of Russell's Analysis of Matter (1927), M.H.A. Newman made the following point in criticism of Russell's structuralist view of representation:
[A]ll we can say is, "There is a relation $R$ such that the structure of the external world with reference to $R$ is $\mathcal{A}$'". Now I have already pointed out that such a statement expresses only a trivial property of the world. Any collection of things can be organised so as to have the structure $\mathcal{A}$, provided there are the right number of them. Hence the doctrine that only structure is known involves the doctrine that nothing can be known that is not logically deducible from the mere fact of existence, except ("theoretically") the number of constituting objects. (Newman 1928, p. 144; slight change in notation.)
The problem with (R) is that the notion of "structural representation'" is not adequately defined. Suppose that we define a (wrong!) notion "represents*" as follows:
A model $\mathcal{A}$ represents* the world just if there are relations $R_i$ on (possibly a subset of) the set $W$ of things in the world such that $\mathcal{A}$ is isomorphic to these. 
There is a certain kind of conceptual nominalism or anti-essentialism, which conceives of the world as structureless---as a kind of "amorphous glub", and which aims to explain the appearance of mind-independent structure as a projection of some feature, such as the "form", of inner mental representations.

[So, for example, as I understand it, in Kant, our "external intuitions" carry a certain "form", which is actually---according to Kant---what space is. Thus, space is the form of our external intuitions; and, consequently, since intuitions cannot exist independently of thought, it follows that space does not exist independently of thought. Kant's error seems to be his assumption that space is the form of these intuitions, whereas the correct view is presumably that the form of these intuitions is representational form, associated with the inner mechanisms of perception; there seems little reason to suppose that what space itself is like be the same as what its cognitive representation, particularly in perception, might be like. For our representations can be mistaken. Space might, for all we know, be 10-diemnsional; or discrete in some way; etc. Thus, Kant seems to be conflating the perceptual representation of space with space.]

Now, if a view this is right, then there are no "external" constraints for the representation to be wrong about. More specifically, here is what I call:
The Projection Lemma
Let $B$ be a set, $\mathcal{A} = (A, R_1, \dots)$ a model and $e: A \to B$ an injection. Let $\mathcal{B} = (B, e[R_1], \dots)$. Then
\begin{equation}e: \mathcal{A} \rightarrow \mathcal{B}\end{equation} is an elementary embedding.
Consequently, given a model $\mathcal{A}$ and a sufficiently large set $B$, one can "project" the "structure'' of this model $\mathcal{A}$ onto the set $B$. This implies that one can place any mathematical structure on the world one likes so long as the world has sufficiently large cardinality:
The Glub Lemma:
Let $\mathcal{A}$ be a model, and let $\kappa = |dom(\mathcal{A})|$. Then:
$\mathcal{A}$ represents* the world iff the cardinality of the world is at least $\kappa$. 
The Glub Lemma may be regarded as a dramatic trivialization result, implying that certain conception of structural representation reduces to near triviality: i.e., the correctness of a structural representation is determined solely by cardinality.

(As Roy Cook nicely put it in conversation, physics has been reduced to "counting"!)


  1. Panu Raatikainen20 March 2013 at 06:46

    If we assume that the world is structureless, "an amorphous glub", doesn't it also follow that we can cut the cake anyway we like, and in particular, always cut it in at least κ pieces?

  2. Panu,

    The cake-cutting analogy is a bit like a (monadic) partition; but Newman's point is very general - we can arrange the worldly things into any structure we like (cardinality-permitting): a Boolean algebra, a linear order, a group, etc.



  3. What about going fundamentalists?
    Suppose we align with people like Sider, who claim there is a priviledged structure to the world. (Strechting the view a bit, we may want to postulate a Lewisian hierarchy of "naturalness", but now not of properties (as Lewis originally did), but of world-structures.)
    We would then be able to soothen the effect of the Glub Lemma: representation* is not enough, we need something more like:

    A model M represents** the world just if there are relations Ri on (possibly a subset of) the set W of things in the world such that A is isomorphic to these, and such relations Ri are described by a fundamental language in the sense of Sider.

    Of course, the question would remain how are we supposed to get to know what language(s) are fundamental, and why care about them. But Sider has a book on that, and as of today I have no new ideas on that matter to share here :)

  4. Thanks Jeff. This is well known, but it is strictly a consequence of thinking about representation under the constraints of structural realism. If instead you think of representation as what we do when we model particular phenomena or systems of interest, under some description, then you have both a limited domain A and a set of restricted relations Ri that we are interested in capturing through the model B. (The relations Ri are normally part of a description which is often conventional, if not outright a fiction.) Newman's objection disappear, although there's no longer any realism to support (we've given up on realism before we start).

  5. There you have them, in Carlos Romero's reply and mine: the two standard ways to respond to Newman's objection! So quick! Cheers, M.

  6. Hi both,

    Yes, I agree with both answers. The point is not to existentially quantify the relation variables. That is, the basic notion is

    "The model A represents the world relative to R1, ..., Rn"

    This means that the underlying notion is *relative* to some sequence of relations.

    One can then ramsify with "natural relations" (as Carlos says) to define "The model represents the world", or one can suppose that these R_i are fixed by an independent convention or description, as Mauricio says.