Group Structural Realism (Part 4)

Part 1 | Part 2 | Part 3 | Part 4
The Higher Structures Problem.

My worry about structural realism starts with the observation that a relation is a very general concept. Relations can describe not only objects, but also other relations. Consequently, it is a general fact about a structure (which is made up of relations) that it itself often admits a structure (made up of relations between the relations). The result, I argue, is this:

Structural realism is forced to either risk incoherence, or else adopt an overly extravagant and uninformative account of reality.

What does the ‘structure of a structure’ really mean? In the case of GSR, we have the following important sense in which symmetry groups are describable terms of their own symmetry group structure.

The ‘symmetry group structure’ describing a group G itself is called the automorphism group, Aut G. An automorphism of a group G is a mapping from G to itself that preserves group structure. The group Aut G is formed by collecting together the set of all such automorphisms, and taking the binary operation to be functional composition. Now, to see in what sense Aut G describes the ‘symmetries’ of G, consider the following analogy with the Wigner-approach to GSR.

As in our earlier post, take the specific example of angular momentum. On the old description, we had three tiers: an individual entity (like a Hydrogen atom), an invariant quantity j, and a group SO(3) whose action on the atom left j invariant. GSR announced that we should demote the individual entity, and promote the metaphysical status of the group. But now, consider by analogy the following three new tiers: an individual group SO(3), a representation H, and a group Aut SO(3) whose action on SO(3) leaves H invariant (up to isomorphism). Just as SO(3) described the symmetries of the Hydrogen atom, so Aut SO(3) describes the symmetries of SO(3). Shouldn’t the structural realist strategy demand that we now demote SO(3), and promote the status of Aut SO(3)?

To make this analogy more concrete, let’s think about what Aut SO(3) actually looks like. Begin by presenting SO(3) as the group of rotations of a sphere, where x, y and z are orthogonal axes of rotation. Then there is an automorphism of SO(3) formed by a smooth rotation of these axes, by mapping each rotation to another rotation about a new axis:

The class of all such automorphisms forms a subgroup of Aut SO(3), which is visibly isomorphic to SO(3) itself. The rest of the automorphisms involve an orthogonal transformation of the axes that is not accessible by a smooth rotation, and so the full automorphism group turns out to be given by the semi-direct product of SO(3) and {-1, 1}.

Moreover, all of the virtues of elevating SO(3) seem to carry over when we elevate the metaphysical status of Aut SO(3) instead. Note that in both cases, some important properties are left invariant under the action of the group (that is, both can be called ‘symmetry’ groups). In the case of the electron shell, they are the properties deriving from the total angular momentum j. In the case of SO(3) itself, they are the properties deriving from the group structure14. Note also that both can be taken as the basis for a construction in which the rest of quantum theory is recovered. The only difference is, the group Aut SO(3) is ‘one level more abstract,’ so that this construction begins by constructing an invariant group SO(3), and then proceeding as usual.

An infinite regress now threatens. In general, the group Aut G will also admit an automorphism group. This gives rise to what is known as an automorphism tower, given by

G, Aut G, Aut Aut G, ….
As long as each successive automorphism group results in a distinct new group, we can continue producing new, ‘metaphysically fundamental’ structures all the way up. Since this tower can be very high, the result is a bloated, very abstract ontology. Indeed, there are even groups for which the tower can be continued transfinitely.

The Dilemma: Horn 1. The advocate of GSR would like to place the original group G at top of the metaphysical hierarchy. But there does not seem to be a well-motivated reason to choose G over Aut G. This pushes GSR to: Horn 2. We instead promote the highest automorphism group Aut G in the tower, or the ‘whole shebang.’ This introduces a tower’s worth of ‘lower down’ groups into our ontology, and risks the possibility that a ‘highest’ automorphism group doesn’t exist. Or, if it is possible to fix a structural foundation by promoting the ‘whole shebang’ of higher group structures, then it seems that our ontology is excessively extravagant and uninformative.

In summary: structural realism is forced to either risk incoherence (Horn 1), or else adopt an overly extravagant and uninformative account of reality (Horn 2).

For a summary of my comments on GSR, you can now read the paper! (PhilSci Archive)

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4 thoughts on “Group Structural Realism (Part 4)

  1. Bryan

    Hi Jim — thank you for mentioning this paper, I hadn’t seen it yet.

    Floridi’s ‘information realism’ is an interesting idea, especially if it can be worked out in precise detail. Floridi is operating at a fairly abstract level. So his impact on the scientific realism debate seems to depend on whether or not it can be said to capture the world described by our best scientific theories.

    To do that, the view will have to make some contact with a theory — I’d like to see a specific example of ‘informational structure’ in physics, for example.

    However, notice that Floridi describes an ‘ontological hierarchy,’ very similar to the one I target in my first post. I suspect that (once the details are worked out) any such hierarchy will suffer from the worry I raise. An informational structure will itself admit a structure, and then I’m off and running.

  2. Luciano Floridi

    Thank you for your interest in informational structural realism (ISR).

    I hope to engage with the debate but, until then, perhaps I may make a small contribution to it by recommending this introductory book, which seems to go well with my ISR and add “some contact with a theory”:

    Another alternative would be

    I hope this might be of interest,

    all the best,

    Luciano Floridi

  3. Bryan

    Dr. Florini,

    Thank you for the lovely resources! I look forward to seeing how your theory makes contact with the physical theories I’ve been thinking about. In particular, I’d love to hear your thoughts on which ‘informational structure’ is relevant in Wigner’s legacy.

    The trick in this business is identifying precisely which structures one has in mind. I would be happy to discuss this with you in more detail; feel free to contact me at any time.