Tomasz Bigaj On essential structures and symmetries Abstract The main goal of the paper is to revisit the concept of a symmetry for relational structures in the light of the ontological position dubbed “essentialist structuralism”. It is argued that the standard definition of the notion of a symmetry commits us to the existence of non-qualitative, haecceitistic differences between possible worlds. An alternative notion of symmetry is developed, based on the distinction between essential and contingent structures. It is claimed that this new concept is better suited for the doctrine of structuralism, and moreover it offers a new perspective on some well-known problems in the foundation of physical theories, such as the problem of permutation invariance in quantum mechanics and diffeomorphism invariance in general relativity. 1. Symmetries of relational structures In recent years the ontological doctrine of structuralism has gained widespread popularity. Structuralism, also known under the moniker Ontic Structural Realism, can be traced back to the observation that physical objects do not exist in isolation, but instead constitute greater wholes via links of mutual interconnections. These wholes are identified as relational structures in which objects participate and which are the proper subjects of scientific investigations. Structural realists insist that structures are ontologically more fundamental than their participating objects. This vague claim can receive many inequivalent interpretations, ranging from radical eliminativism (there are no objects, only structures) to the relatively mild statement that the criteria of identity and distinctness for objects should somehow involve qualitative relations they participate in. 1 In my (Bigaj 2014) I proposed yet another interpretation of structuralism which focuses on the following much-discussed problem in the metaphysics of modality: how to identify individual objects in possible scenarios (across possible worlds). I suggested there that we should combine the essentialist approach to this question with the structuralist-motivated assumption that relational structures are ontologically prior to objects participating in them. The result of this fusion is a position I call “essentialist structuralism”, which states roughly that the identification of objects in alternative possible worlds should be done with the help of selected qualitative structures dubbed “essential”. 2 In this paper I will further probe this idea by formalizing criteria of transworld identity of objects in terms of essential structures and their isomorphism- or homomorphism-based similarities. The starting point of my discussions will be the concept of a symmetry of a relational structure. I will argue that the textbook characterization 1 The literature on different variants of Ontic Structural Realism is vast and growing. Instead of giving here a semi- complete list of relevant publications, I will refer the reader to the recently updated overview (Ladyman 2014) and the works quoted therein. 2 To my knowledge, this interpretation of structural realism has never been subject to extensive studies. Notable exceptions that should be mentioned here are (Gołosz 2005) and (Glick 2015).