Mathematical and Scientific Structuralism: Connections and Tensions

Elaine Landry (University of California, Davis) - I will use a structuralist account of mathematics to critically examine the claim, typically made against structural realists, that one cannot speak of structures without objects, because relata are prior to relations. I will argue that such priority claims result from a general conflation of mathematics with metaphysics and of metaphysics with mathematics. Specifically, I will show that many of these priority arguments arise from attempts to provide either a metamathematical or metaphysical framework for mathematical structuralism. If, however, one shifts to providing a linguistic framework, and views mathematical structuralism as a methodological stance, then one can focus on the fact that mathematical structuralism does indeed show us that it is conceptually possible to speak of structures without objects. But too, against the many pronouncements of ontic structural realists, one must be careful not to conflate what is conceptually possible with what is physically possible. Thus, accepting that mathematics is not metaphysics, one ought to adopt the more modest position of the methodological structural realist.

Kerry McKenzie (University of California, San Diego) - Mathematical structuralism and scientific structuralism both aspire to provide a metaphysics of objects based on the structure of theories. But separating them is the fact that only the latter must accommodate the historical fact of radical theory change. Epistemic structuralism was introduced in an attempt to secure scientific knowledge in a period when further such change is anticipated. But its metaphysical counterpart, ontic structural realism, consists of the thesis that ‘structure is ontologically fundamental,’ and thus fashions itself as though we already know about the fundamental. Since we can only make such assertions when we are confident that there will be no further significant theory change in science, it seems that structuralism as a general philosophy of science is guilty of a form of incoherence. In the face of this, I suggest that structuralist metaphysics should proceed with a different agenda: that of articulating what the value of metaphysics is while we still want for a theory of the fundamental. By analogy with science, such a project would seem to demand that we state what progress in metaphysics consists of and whether we can hope to achieve it. In this paper I will argue that there is no obvious answer to either question, and certainly no easy inference from progress in science to progress in metaphysics. For what makes it possible to say that science makes progress is the availability of a meaningful notion of approximation – something facilitated by the fine-grained nature of the mathematical language in which scientific theories are couched and the clear purposes of scientific enquiry. But I will argue both of these features seem inapplicable to metaphysics, construed as an activity aspiring to the highest generality and lacking in independent purpose. In the face of this criticism, two options are open to the contemporary structuralist who insists on engaging in metaphysics. Either they develop a notion of metaphysical progress highly distinct from that applicable in science – something that seems of questionable coherence given structuralism’s avowed naturalism. Alternatively, they confine its purpose to a project more normally viewed as belonging to the philosophy of mathematics: that of describing the conditions that make possible the applicability of mathematics, and hence that make possible the only body of theory to which progress can unambiguously be ascribed.

Otávio Bueno (University of Miami) - Structural realism, as a realist form of scientific structuralism, crucially relies on there being suitable interpretations of mathematical structures (particularly those used in successful applications in the sciences) in order to derive ontological conclusions about the structure of the world. I argue that mathematical structures are importantly underdetermined for this task: not only are the same mathematical structures compatible with radically different physical interpretations, but also significantly different mathematical structures can be used to account for the same physical phenomena. The result is that, strictly speaking, no significant ontological conclusions from mathematical structures alone can be drawn. These considerations challenge the ontological interpretation of mathematical structures that is at the heart of structural realism. I then provide an alternative form of scientific structuralism, of an empiricist rather than a realist sort, that resists any such reification of mathematical structures and thereby is able to accommodate this ubiquitous structural indeterminacy in a principled way.