Online research in philosophy

Forthcoming in A. Bokulich & P. Bokulich (eds.), Scientific Structuralism, Boston Studies in the Philosophy of Science, Springer. Abstract: Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any general confidence we might have in the structural claims made by our theories. Thinking about the role of mathematics in science may also complicate other versions of realism.

We present a precise form of structural realism, called group structural realism , which identifies ‘structure’ in quantum theory with symmetry groups. However, working out the details of this view actually illuminates a major problem for structural realism; namely, a structure can itself have structure. This article argues that, once a precise characterization of structure is given, the ‘metaphysical hierarchy’ on which group structural realism rests is overly extravagant and ultimately unmotivated.

Forthcoming in A. Bokulich & P. Bokulich (eds.), Scientific Structuralism, Boston Studies in the Philosophy of Science, Springer. Abstract: Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any general confidence we might have in the structural claims made by our theories. Thinking about the role of mathematics in science may also complicate other versions of realism.

According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea of ante rem structures have appeared. Some argue that it is impossible to give identity conditions for places in homogeneous ante rem structures, invoking a version of the identity of indiscernibles. Others raise issues concerning the identity and distinctness of places in different structures, such as the the natural number 2 and the real number 2. The purpose of this paper is to take the measure of these objections, and to further articulate ante rem structuralism to take them into account.

Abstract: According to Luciano Floridi (2008) , informational structural realism provides a framework to reconcile the two main versions of realism about structure: the epistemic formulation (according to which all we can know is structure) and the ontic version (according to which structure is all there is). The reconciliation is achieved by introducing suitable levels of abstraction and by articulating a conception of structural objects in information-theoretic terms. In this essay, I argue that the proposed reconciliation works at the expense of realism. I then propose an alternative framework, in terms of partial structures, that offers a way of combining information and structure in a realist setting while still preserving the distinctive features of the two formulations of structural realism. Suitably interpreted, the proposed framework also makes room for an empiricist form of informational structuralism (structural empiricism). Pluralism then emerges.

We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understanding what physics is telling us about the world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects isn’t easy. But there seems to be no further problem for understanding mathematical structure. Modern theories of physics are formulated in terms of these mathematical structures. In order to understand “structure” as used in physics, then, it seems we must simply look at the structure of the mathematics that is used to state the physics. But it is not that simple. Physics is supposed to be telling us about the nature of the world. If our physical theories are formulated in mathematical language, using mathematical objects, then this mathematics is somehow telling us about the physical make-up of the world. What is..

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.

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.

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.

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.