Weyl and Automorphism of Nature. Drunken Risibility.


In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.


Diffeomorphism Invariance: General Relativity Spacetime Points Cannot Possess Haecceity.


Eliminative or radical ontic structural realism (ROSR) offers a radical cure—appropriate given its name—to what it perceives to be the ailing of traditional, object-based realist interpretations of fundamental theories in physics: rid their ontologies entirely of objects. The world does not, according to this view, consist of fundamental objects, which may or may not be individuals with a well-defined intrinsic identity, but instead of physical structures that are purely relational in the sense of networks of ‘free-standing’ physical relations without relata.

Advocates of ROSR have taken at least three distinct issues in fundamental physics to support their case. The quantum statistical features of an ensemble of elementary quantum particles of the same kind as well as the features of entangled elementary quantum (field) systems as illustrated in the violation of Bell-type inequalities challenge the standard understanding of the identity and individuality of fundamental physical objects: considered on their own, an elementary quantum particle part of the above mentioned ensemble or an entangled elementary quantum system (that is, an elementary quantum system standing in a quantum entanglement relation) cannot be said to satisfy genuine and empirically meaningful identity conditions. Thirdly, it has been argued that one of the consequences of the diffeomorphism invariance and background independence found in general relativity (GTR) is that spacetime points should not be considered as traditional objects possessing some haecceity, i.e. some identity on their own.

The trouble with ROSR is that its main assertion appears squarely incoherent: insofar as relations can be exemplified, they can only be exemplified by some relata. Given this conceptual dependence of relations upon relata, any contention that relations can exist floating freely from some objects that stand in those relations seems incoherent. If we accept an ontological commitment e.g. to universals, we may well be able to affirm that relations exist independently of relata – as abstracta in a Platonic heaven. The trouble is that ROSR is supposed to be a form of scientific realism, and as such committed to asserting that at least certain elements of the relevant theories of fundamental physics faithfully capture elements of physical reality. Thus, a defender of ROSR must claim that, fundamentally, relations-sans-relata are exemplified in the physical world, and that contravenes both the intuitive and the usual technical conceptualization of relations.

The usual extensional understanding of n-ary relations just equates them with subsets of the n-fold Cartesian product of the set of elementary objects assumed to figure in the relevant ontology over which the relation is defined. This extensional, ultimately set-theoretic, conceptualization of relations pervades philosophy and operates in the background of fundamental physical theories as they are usually formulated, as well as their philosophical appraisal in the structuralist literature. The charge then is that the fundamental physical structures that are represented in the fundamental physical theories are just not of the ‘object-free’ type suggested by ROSR.

While ROSR should not be held to the conceptual standards dictated by the metaphysical prejudices it denies, giving up the set-theoretical framework and the ineliminable reference to objects and relata attending its characterizations of relations and structure requires an alternative conceptualization of these notions so central to the position. This alternative conceptualization remains necessary even in the light of ‘metaphysics first’ complaints, which insist that ROSR’s problem must be confronted, first and foremost, at the metaphysical level, and that the question of how to represent structure in our language and in our theories only arises in the wake of a coherent metaphysical solution. But the radical may do as much metaphysics as she likes, articulate her theory and her realist commitments she must, and in order to do that, a coherent conceptualization of what it is to have free-floating relations exemplified in the physical world is necessary.

ROSR thus confronts a dilemma: either soften to a more moderate structural realist position or else develop the requisite alternative conceptualizations of relations and of structures and apply them to fundamental physical theories. A number of structural realists have grabbed the first leg and proposed less radical and non-eliminative versions of ontic structural realism (OSR). These moderate cousins of ROSR aim to take seriously the difficulties of the traditional metaphysics of objects for understanding fundamental physics while avoiding these major objections against ROSR by keeping some thin notion of object. The picture typically offered is that of a balance between relations and their relata, coupled to an insistence that these relata do not possess their identity intrinsically, but only by virtue of occupying a relational position in a structural complex. Because it strikes this ontological balance, we term this moderate version of OSR ‘balanced ontic structural realism’ (BOSR).

But holding their ground may reward the ROSRer with certain advantages over its moderate competitors. First, were the complete elimination of relata to succeed, then structural realism would not confront any of the known headaches concerning the identity of these objects or, relatedly, the status of the Principle of the Identity of Indiscernibles. To be sure, this embarrassment can arguably be avoided by other moves; but eliminating objects altogether simply obliterates any concerns whether two objects are one and the same. Secondly, and speculatively, alternative formulations of our fundamental physical theories may shed light on a path toward a quantum theory of gravity.

For these presumed advantages to come to bear, however, the possibility of a precise formulation of the notion of ‘free-standing’ (or ‘object-free’) structure, in the sense of a network of relations without relata (without objects) must thus be achieved.  Jonathan Bain has argued that category theory provides the appropriate mathematical framework for ROSR, allowing for an ‘object-free’ notion of relation, and hence of structure. This argument can only succeed, however, if the category-theoretical formulation of (some of the) fundamental physical theories has some physical salience that the set-theoretical formulation lacks, or proves to be preferable qua formulation of a physical theory in some other way.

F. A. Muller has argued that neither set theory nor category theory provide the tools necessary to clarify the “Central Claim” of structural realism that the world, or parts of the world, have or are some structure. The main reason for this arises from the failure of reference in the contexts of both set theory and category theory, at least if some minimal realist constraints are imposed on how reference can function. Consequently, Muller argues that an appropriately realist stucturalist is better served by fixing the concept of structure by axiomatization rather than by (set-theoretical or category-theoretical) definition.