Online research in philosophy

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics.

• This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.

In this paper I focus on the impact on structuralism of the quantum treatment of objects in terms of symmetry groups and, in particular, on the question as to how we might eliminate, or better, reconceptualise such objects in structural terms. With regard to the former, both Cassirer and Eddington not only explicitly and famously tied their structuralism to the development of group theory but also drew on the quantum treatment in order to further their structuralist aims and here I sketch the relevant history with an eye on what lessons might be drawn. With regard to the latter, Ladyman has explicitly cited Castellani's work on the group-theoretical constitution of quantum objects and I indicate both how such an approach needs to be understood if it is to mesh with Ladyman's 'ontic' form of structural realism and how it might accommodate permutation symmetry through a consideration of Huggett's recent account.

Ontic structural realism is the view that structures are what is real in the first place in the domain of fundamental physics. The structures are usually conceived as including a primitive modality. However, it has not been spelled out as yet what exactly that modality amounts to. This paper proposes to fill this lacuna by arguing that the fundamental physical structures possess a causal essence, being powers. Applying the debate about causal vs categorical properties in analytic metaphysics to ontic structural realism, I show that the standard argument against categorical and for causal properties holds for structures as well. Structural realism, as a position in the metaphysics of science that is a form of scientific realism, is committed to causal structures. The metaphysics of causal structures is supported by physics, and it can provide for a complete and coherent view of the world that includes all domains of empirical science.

Ontic structural realists hold that structure is all there is, or at least all there is fundamentally. This thesis has proved to be puzzling: What exactly does it say about the relationship between objects and structures? In this article, I look at different ways of articulating ontic structural realism in terms of the relation between structures and objects. I show that objects cannot be reduced to structure, and argue that ontological dependence cannot be used to establish strong forms of structural realism. At the end, I show how a weaker, but controversial, form of structural realism can be articulated on the basis of ontological dependence.

Ontic Structural Realism (OSR) gives ontic priority to structures over objects. In its perhaps most extreme form (captured, admittedly, by a slogan) it states that “all that there is, is structure” (da Costa and French 2003, 189). If this is true, if there is nothing but structure(s) in the world, the very idea of contrasting structure to nonstructure loses any force it might have. Actually, if the slogan is right, the very idea of characterising what there is as structure—as opposed to anything else—becomes incoherent. Traditionally, characterising something as a structure has made full sense —and has served excellent scientific and philosophical purposes—precisely because structure was understood as an entity with slots, which could be occupied by objects and whose individuation-conditions involved objects only qua slot-fillers. If objects altogether go, whatever remains can be called ‘structure’ only if we take ‘structure’ to be a term of art. Well, Ontic Structuralists are happy to ‘mimic’ talk of non-structure, or objects in particular, but they hasten to add that this mimicking does not imply any serious metaphysical commitment to them. Here are a couple of characteristic passages.

James Ladyman has recently proposed a view according to which all that exists on the level of microphysics are structures "all the way down". By means of a comparative reading of structuralism in philosophy of mathematics as proposed by Stewart Shapiro, I shall present what I believe structures could not be. I shall argue that, if Ladyman is indeed proposing something as strong as suggested here, then he is committed to solving problems that proponents of structuralism in philosophy of mathematics such as Shapiro are trying to solve. Attempting to do so, however, brings out a tacit tension in Ladyman's position. I shall argue that the upshot of this is that the ontological import that Ladyman attributes to structures is rather epistemological import properly understood.

This paper takes issue with Ontic Structural Realism (OSR). It is structured around the three elements of the title. Section 2 highlights a substantive non-structural assumption that needs to be in place before we can talk about the structure. Then, by drawing on some relevant issues concerning mathematical structuralism, it claims that (a) structures need objects and (b) scientific structuralism should focus on in re structures. But then pure structuralism is undermined. Section 3 discusses whether the world has ‘excess structure’ over the structure of appearances. The main point is: the claim that only structure can be known is false. Finally, section 4 argues directly against ORS that it lacks the resources to accommodate causation within the structuralist slogan that “all that there is, is structure”.

This paper is structured around the three elements of the title. Section 2 claims that (a) structures need objects and (b) scientific structuralism should focus on in re structures. Therefore, pure structuralism is undermined. Section 3 discusses whether the world has `excess structure' over the structure of appearances. The main point is that the claim that only structure can be known is false. Finally, Section 4 argues directly against ontic structural realism that it lacks the resources to accommodate causation within its structuralist slogan.

According to structural realism, in mature science there is structural continuity along theoretical change. A major counterexample to this thesis is the transition from the Eightfold Way to the Standard Model in particle physics. Nevertheless, the notion of structure is significantly important in comprehending the theoretical picture of particle physics, where particles change and undergo transmutations, while the only thing which remains unchanged is the basic structure, i.e. the symmetry group which controls the transmutations. This kind of view agrees with the paradigmatic case where the structure is an internal symmetry and the instantiations are the elementary particles. The metaphysical view which reflects this situation is a version of ontic structuralism.