Quine’s Non-ontological Structuralism (and Mine)

In ‘Structure and Nature’ Quine advocates a ‘global ontological structuralism’, one that sees all objects, even nonmathematical ones, as positions or nodes in a structure. The term ‘ontological’ here is misleading and would be better deleted. For Quine is no ontological structuralist; in the same paper he makes it clear that he is not proposing a structuralist ontology or even an ontological doctrine in which structures are eliminated in favour of systems instantiating them. Resnick’s own mathematical structuralism evolved from a sui generis ontological form to a Quinean non-ontological form. In this chapter Resnick discusses the evolution of Quine’s structuralism and how it shaped his own.

Structuralism and Mathematical Practice in Felix Klein’s Work on Non-Euclidean Geometry†

It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.

Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.

Carnap’s Structuralist Thesis

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.