Geoffrey Hellman* and Stewart Shapiro.**Mathematical Structuralism. Cambridge Elements in the Philosophy of Mathematics, Penelope Rush and Stewart Shapiro, eds

2020 ◽  
Vol 28 (2) ◽  
pp. 277-281
Author(s):  
Andrea Sereni
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Structuralism

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 “If Numbers Are to Be Anything At All, They Must Be Intrinsically Something”: Bertrand Russell and Mathematical Structuralism

2020 ◽  
pp. 303-328
Author(s):  
Jeremy Heis
Keyword(s):  
Philosophy Of Mathematics ◽  
Bertrand Russell ◽  
Positive Role ◽  
Mathematical Structuralism ◽  
Philosophy Of Arithmetic

Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.

Philosophy of Mathematics

2018 ◽  
Author(s):  
Denis Bonnay
Keyword(s):  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
Detailed Account ◽  
Mathematical Structuralism ◽  
Critical Approaches ◽  
The One

Philosophy of mathematics deals both with ontological issues (what is it that mathematics studies?) and epistemological issues (how is mathematical knowledge possible?). This chapter reviews the main answers given to these two sets of issues, stressing how interrelated they are. It starts from the classical opposition between empiricist, rational, and critical approaches to set the sage and poses the question of mathematics’ relationship with experience as well as the one of the respective roles of intuition and logical principles. A detailed account of two anti-realist programs (finitism and intuitionism) is provided. Arguments in favor of realism are presented, and distinct realist views are distinguished. Having confronted the epistemological difficulties of various realist views, the last part of the chapter deals with naturalist perspectives and mathematical structuralism.

scholarly journals REALISMO E INDETERMINATEZZA NEI FONDAMENTI DELLA MATEMATICA

2019 ◽  
Author(s):  
Francesca Boccuni
Keyword(s):  
Philosophy Of Mathematics ◽  
The Past ◽  
Mathematical Structuralism ◽  
Arbitrary Reference ◽  
Semantic Indeterminacy

Two of the most influential foundational approaches in the philosophy of mathematics of the past thirty years, i.e. neologicism and ante rem mathematical structuralism, suffer from metaphysical and semantic indeterminacy. My present aim is to offer an explanation of this phenomenon and provide a solution in terms of arbitrary reference.

scholarly journals Poincaré and the Prehistory of Mathematical Structuralism

2020 ◽  
pp. 273-302
Author(s):  
Janet Folina
Keyword(s):  
Subject Matter ◽  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Strong Version ◽  
Mathematical Structuralism ◽  
Abstract Structure

The view that mathematics is about abstract structure is quite deeply rooted in mathematical practice, with further philosophical views about the nature of structures emerging more recently. This essay argues, first, that Poincaré’s views about structure are properly philosophical, since they go beyond basic claims about the general subject matter of mathematics. Second, it proposes that these further views align Poincaré with a strong version of structuralism—one typically associated with realism. This raises a question since he is a constructivist about mathematics, supporting a broadly Kantian conception of mathematical knowledge and existence. There is thus an apparent tension in Poincaré’s philosophy of mathematics, and a third goal is to resolve this tension.

scholarly journals Explication as Elimination: W. V. Quine and Mathematical Structuralism

2020 ◽  
pp. 421-442
Author(s):  
Sean Morris
Keyword(s):  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Mathematical Objects ◽  
Mathematical Structuralism ◽  
Natural Way

This chapter examines the development of and motives for Quine’s particular form of mathematical structuralism. It will argue that Quine, unlike many contemporary mathematical structuralists, does not appeal to structuralism as a way of accounting for what the numbers really are in any robust metaphysical sense. Instead, his structuralism is deeply rooted in an earlier structuralist tradition found in scientific philosophers such as Russell and Carnap, which emphasized structuralism as a critique of more metaphysical approaches to philosophy. On this view, a philosophy of mathematics answers, in a sense, only to mathematics itself. An account of mathematical objects requires only that the entities—whatever they are—serving as the mathematical objects satisfy the relevant postulates and theorems. Here we also see how Quine’s early work in the foundations of mathematics leads in a natural way to the more general naturalism of his later philosophy.

Mathematical Knowledge and the Interplay of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
Philosophy Of Mathematics ◽  
Advanced Mathematics ◽  
New Approach ◽  
Agent Based ◽  
Epistemology Of Mathematics ◽  
Actual Experience ◽  
Elementary Math ◽  
Mathematical Tradition

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

A Second Pilgrim’s Progress

2017 ◽  
Author(s):  
Mark Wilson
Keyword(s):  
Set Theory ◽  
Applied Mathematics ◽  
Philosophy Of Mathematics ◽  
Pilgrim's Progress ◽  
Open Questions

Influenced by Quine, self-styled naturalist projects within the philosophy of mathematics rest upon simplistic conceptions of linguistic reference and how the inferential tools of applied mathematics help us reach empirical conclusions. In truth, these two forms of descriptive enterprise must work together in a considerably more entangled manner than is generally presumed. In particular, the vital contributions of set theory to descriptive success within science have been poorly conceptualized. This essay explores how a less onerous “naturalism” can be conceived on this corrected basis. A useful distinction between “mathematical optimism” and “mathematical opportunism” is introduced, which draws our attention to some open questions with respect to the concrete representational capacities of applied mathematics.

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