Mathematical Structuralism, Modal Nominalism, and the Coherence Principle

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.

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Carnap’s Structuralist Thesis

The Prehistory of Mathematical Structuralism ◽

10.1093/oso/9780190641221.003.0015 ◽

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.

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Coherence Principle

Multimedia Learning ◽

10.1017/9781316941355.009 ◽

2020 ◽

pp. 143-165

Keyword(s):

Coherence Principle

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Erich Reck* and Georg Schiemer.** The Prehistory of Mathematical Structuralism.

Philosophia Mathematica ◽

10.1093/philmat/nkaa028 ◽

2020 ◽

Vol 28 (3) ◽

pp. 416-420

Author(s):

Jean-Pierre Marquis

Keyword(s):

Mathematical Structuralism

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

Philosophia Mathematica ◽

10.1093/philmat/nkaa029 ◽

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.

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