Erich H. Reck and Georg Schiemer, editors, The Prehistory of Mathematical Structuralism

2021 ◽  
Author(s):  
Andrea Sereni
Keyword(s):  
Mathematical Structuralism

Indistinguishable elements and mathematical structuralism

Analysis ◽  
2007 ◽  
Vol 67 (294) ◽  
pp. 112-116 ◽  
Author(s):  
José Luis Bermúdez
Keyword(s):  
Mathematical Structuralism

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

2020 ◽  
pp. 352-382
Author(s):  
Wilfried Sieg
Keyword(s):  
Proof Theory ◽  
Incompleteness Theorem ◽  
Mathematical Structures ◽  
General Characterization ◽  
Mathematical Structuralism ◽  
Mathematical Experience

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.

scholarly journals Carnap’s Structuralist Thesis

2020 ◽  
pp. 383-420
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Structural Properties ◽  
Philosophy Of Mathematics ◽  
Mathematical Structure ◽  
Mathematical Structuralism ◽  
Definition Of ◽  
Philosophical Debates ◽  
Implicit Definitions

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.

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

2020 ◽  
Vol 28 (3) ◽  
pp. 360-384
Author(s):  
Francesca Biagioli
Keyword(s):  
Euclidean Geometry ◽  
Mathematical Practice ◽  
Transformation Groups ◽  
Mathematical Research ◽  
Mathematical Structures ◽  
Research And Practice ◽  
Felix Klein ◽  
Non Euclidean Geometry ◽  
Mathematical Structuralism

Abstract 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.

The Tools of Metaphysics and the Metaphysics of Science

2020 ◽  
Author(s):  
Theodore Sider
Keyword(s):  
General Position ◽  
Structural Realism ◽  
Mathematical Objects ◽  
Metaphysics Of Science ◽  
Mathematical Structuralism

Metaphysics is sensitive to the conceptual tools we choose to articulate metaphysical problems. Those tools are a lens through which we view metaphysical problems; the same problems look different when we change the lens. There has recently been a shift to "postmodal" conceptual tools: concepts of ground, essence, and fundamentality. This shift transforms the debate over structuralism, in many ways. For instance: structuralist theses say that "patterns" are prior to the "nodes" in the patterns. In modal terms it is clear what this means: the nodes cannot vary independently of the pattern. But it's far less clear what its postmodal meaning is. One expects it to mean that the pattern is fundamental, the entities in the pattern, derivative. But what would a fundamental account of reality that speaks only of patterns and not objects in the patterns look like? I examine three structuralist positions through a postmodal lens. First, nomic essentialism, which says that scientific properties are secondary and lawlike relationships among them are primary. Second, structuralism about individuals, a general position of which mathematical structuralism and structural realism are instances, which says that scientific and mathematical objects are secondary and the pattern of relations among them is primary. Third, comparativism about quantities, which says that particular values of scientific quantities, such as having exactly 1000g mass, are secondary, and quantitative relations, such as being-twice-as-massive-as, are primary. Finally, I take a step back and examine the meta-question of when theories are equivalent, and how that impacts the debate over structuralism.

How Can Abstract Objects of Mathematics Be Known?†

2019 ◽  
Vol 27 (3) ◽  
pp. 316-334
Author(s):  
Ladislav Kvasz
Keyword(s):  
Human Cognition ◽  
Abstract Objects ◽  
Mathematical Objects ◽  
Epistemic Access ◽  
Mathematical Structuralism ◽  
Instrumental Practice

Abstract The aim of the paper is to answer some arguments raised against mathematical structuralism developed by Michael Resnik. These arguments stress the abstractness of mathematical objects, especially their causal inertness, and conclude that mathematical objects, the structures posited by Resnik included, are inaccessible to human cognition. In the paper I introduce a distinction between abstract and ideal objects and argue that mathematical objects are primarily ideal. I reconstruct some aspects of the instrumental practice of mathematics, such as symbolic manipulations or ruler-and-compass constructions, and argue that instrumental practice can secure epistemic access to ideal objects of mathematics.

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