Structure

If naturalness isn‘t natural, then an analysis of fundamentality in terms of perfect naturalness starts to look less promising. The obvious next place to search for an adequate conception of fundamentality is in Sider‘s (2011) extension of naturalness theory. Recall that, according to Sider, x is fundamental =df x is

part of the world‘s structure (where ‗structure‘ here refers to Sider‘s primitive theoretical posit).

The problem for naturalness theory that I identified above comes about primarily because naturalness theory seems to break down when it is applied beyond first order properties. This we saw to be explicitly the case in our analysis of the duplication role, but it was lurking in the background of the discussion of supervenience, and to some extent in the discussions of empiricism and laws and reference magnetism too. The problems with giving a definition of perfect naturalness can be diagnosed along the same lines.

Sider‘s (2011) proposal extends naturalness theory ‗beyond the predicate‘, freeing it from the constraints that seem to generate the problems discussed above. Recall that the primitive structural operator Sider introduces (2011: 92) applies to any arbitrary portions of the language, including to quantificational phrases and even to logical connectives. For anything we might wish to express, Sider‘s extension of naturalness theory provides a mechanism for us to consider whether or not it is part of the fundamental structure of the world.

If we ask the analogous question to the one we have been considering here of structure (is ‗structure‘ structural?) we should expect Sider to answer ‗yes‘. (The ways in which the unnaturalness of naturalness is unpalatable has a straightforward extension to structure). The question is easier to ask of Sider than it is of the friend of naturalness, because Sider‘s posit is designed to attach to any portion of the language

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(including operators like ‗structure‘). Sider indeed considers this question explicitly, and his initial response is simply ‗yes‘; structure is structural (2011: 137).

Sider‘s primary reason for embracing the structuralness of structure is the methodological one discussed in the previous section. If structure were not structural, then structure itself would fail to be explanatory. Any definition of structure in structural terms would have to be long and disjunctive, since the class of structural notions is so heterogeneous (it includes, for example, mass, spin, set membership, disjunction, and universal quantification). Nothing unifies the class of structural notions besides their all being structural (Sider, 2011: 140).

We are faced, Sider claims, with a choice: ‗either adopt extreme realism about structure—holding that structure is itself structural—or else give up altogether on explanations that invoke structure, which is tantamount to giving up on structure itself‘ (2011: 140). It is vital then for Sider‘s proposal that structure is indeed structural.

As Schaffer notes, Sider (2011: 128) is explicit about what his primitive is; ‗structure is absolute: I say ―is structural‖ rather than ―is more structural‖‘. Sider‘s structure operator attaches to single expressions rather than relating pairs of expressions, and he has good reason for taking the absolute rather than the relative notion of structure to be primitive. Sider subscribes to a principle he calls purity: ‗fundamental truths involve only fundamental notions‘ (2011: 106). As we have seen, he also takes structure to be structural. If a relative notion of structure were primitive, then there would be fundamental truths that related the fundamental to the non-fundamental. Fundamental truths would then involve non- fundamental notions, violating purity.20

Because structure is an extension of Lewisian naturalness, we should anticipate that a similar argument to the argument I make against naturalness above could be mounted against structure. A thorough investigation is likely to reveal that structure is not structural. Constraints on space prevent me from undertaking that investigation here, though I‘ll briefly mention some reasons for thinking that structure isn‘t structural. I‘ll appeal in part to an argument made by Schaffer (2014) which purports to show that

20 _{The threat here is what deRosset (2013) refers to as ‗the collapse‘ – facts that relate the fundamental to the nonfundamental}

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the role Sider carves out for the notion of structure is played by a relative notion of structure, and not the absolute notion Sider himself takes to be primitive. First, I‘ll introduce some of Sider‘s own reasoning for the conclusion that fundamental posits are to be avoided in general. If, as I will argue, Sider‘s primitive does not in fact play the role carved out for it, then Sider‘s own ideological parsimony ought to lead him to abandon the notion of structure.

As with naturalness, grounding, and other such theoretical posits, appeal to structure is largely justified by the role that structure plays in metaphysics. Sider admits (2011: 141) that his own very strong preference for simplicity militates against many primitive notions. So strong is this preference of Sider‘s that he thinks even well-established primitive concepts such as modality and causation fall by the wayside. He gives just one good reason for opposing, for example, modal primitivism: ‗ideological economy‘ (2011: 267). As Sider himself admits, when it comes to structure these scruples ‗go out the window‘ (2011: 141).

Sider responds, in a way he concedes is not entirely persuasive, with the claim that structure cannot be reduced without loss (2011: 141). Of course, defenders of primitive modality and primitive causation can say the very same thing about their own posits, but we can set aside their indignation for now. Instead, I want to argue that the notion of absolute structure does not play the role Sider carves out for his primitive. Given Sider‘s strong taste for ideological parsimony, this diminishing in the applications of Sider‘s primitive are strong grounds for questioning commitment to it.

First, we should be aware of a complication of Sider‘s view. Lewisian naturalness is a second order property of properties that play a certain, quite well defined theoretical role. This makes it reasonably straightforward to assess the extent to which the property being a perfectly natural property itself plays that role. Many of the roles that Sider carves out for structure are not roles for particular structural notions (in accordance with which we could assess the extent to which structure is itself a structural notion) rather they are roles for Sider‘s primitive itself. In these cases, we can draw on Jonathan Schaffer‘s argument in his (2014) review of Sider‘s book, which purports to show that the roles Sider identifies for his primitive are not in fact played by the absolute notion of structure.

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Just as Lewis thinks that it is the perfectly natural properties that figure in laws, Sider thinks that the best theories are cast in perfectly fundamental (i.e. absolutely structural) terms (2011: 23). And just as there is no mention of perfect naturalness by physicists, there is no mention of structure, either. Sider will argue that structure figures in the best total theory, but we have no independent reason to think so, and so are justified in claiming that absolute structure falls short of this first element of the structure role. Analogous to the role that supervenience plays in Lewis‘ account of perfect naturalness is Sider‘s characterisation of joint carving. Sider himself stresses that ‗a vivid test for whether a given expression,

E, carves at the joints is this: did God need to think in E terms when creating the world‘ (2011: 138).

Sider goes on to admit that God clearly must think in terms of perfectly structural notions like quantification, mass, and distance, but that having done so God need not also consider whether such notions are structural. Again, this suggests that contrary to what is required of it, absolute structure is not structural.

Schafer‘s (2014) criticism of Sider is encapsulated by what Schaffer calls mismatch: ‗The roles for structure are ―structural enough‖ and ―more structural than‖, but Sider‘s primitive is ―perfectly structural‖‘ (Schaffer, 2014: 126). As with Lewisian naturalness, many of the roles that structure plays do indeed seem to require a comparative notion of structure. Sider himself acknowledges this (2011: 129), but to emphasise the point I‘ll give a few examples.

One of Sider‘s primary roles for structure (and a condition that accounts for how structure is able to play many of the other roles carved out for it) is that theories couched in structural terms make for better explanations. But most of the notions involved in explanations are not absolutely structural. Sider concedes this, claiming that genuineness of explanation requires only somewhat structural notions (2011: 141). Not only does this exhibit mismatch (genuine explanations must be given in terms that are more structural than some other terms, not in absolutely structural terms) but it also demonstrates another sense in which absolute structure is not absolutely structural. The notion of absolute structure does not itself give a characterisation of genuine explanation, and so absolute structure does not play the role carved out for Sider‘s primitive. Instead, it is a relative notion of structure that is involved in our theorising about explanations; genuine explanations are couched in terms of somewhat structural notions.

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At both the first order level (which terms are involved in genuine explanations) and the second order level (what theoretical posit is involved in characterising genuine explanation) it is the comparative notion of structure that does the work.

A comparative notion of structure is also required for Sider‘s characterisation of substantivity, where a questions are nonsubstantive just in case ‗their answers turn on which of a range of equally good meanings we choose for words in those questions‘ (Sider, 2011: 46). When no candidate interpretation of the words in the question carves closer to the joints (i.e. is more structural than) any other candidate in the vicinity, our question is nonsubstantive. Sider‘s positive characterisation of substantivity derives from this; a question is substantive when it is not nonsubstantive. The notion of substantivity does a lot of work in Sider‘s system (working out whether or not a given metaphysical dispute is substantive is crucial to Sider‘s overall project) but the notion it depends on is ultimately a comparative one.

A final example comes from the very beginning of the book (2011: 1-2). Sider motivates objectivity about structure with an example involving a universe entirely full of fluid, divided by a central plane. Fluid in one half of the universe is uniformly red, and in the other it is uniformly blue. He asks us to consider a group of people who think of the universe as divided by a different, diagonal plane, and thus corresponding to their predicates, ‗rue‘ and ‗bred‘, which cut across our predicates ‗red‘ and ‗blue‘. Sider‘s claim is that whilst this imagined community make no mistake in applying their own concepts, they‘ve got

the wrong concepts. But (at least according to Sider) the book of the world isn‘t written with predicates like

‗red‘ and ‗blue‘, and so if Sider is to use his example to pump intuitions for the objectivity of structure, then he must concede that it is the idea that ‗blue‘ and ‗red‘ are more structural than ‗rue‘ and ‗bred‘ which is doing the work.

To stave off the challenge presented above, Sider owes an account of how to define a notion of comparative structure out of that of absolute structure, and so to explain how absolute structure is fundamental even though it is a comparative notion of structure that does much of the interesting work. But he admits (2011: 129) to having no such account. The best he can do is to suggest that the solution will be something akin to the Lewisian definitional chain method for making comparative judgements of naturalness (though he finds a number of faults with Lewis‘ own suggestion) and to gesture at four

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elements that will figure in a definition of comparative structure.21 _{Even if Sider were able to solve this}

problem, there is no guarantee that he would thereby be able to show that absolute structure is absolutely structural, only that it is compatible with absolute structure being primitive that a relative notion of structure might play many of the relevant roles. An application of Sider‘s own preference for a parsimonious ideology renders acceptance of structure impermissible.