scholarly journals Philosophy of Mathematics and Ontological Commitment

2000 ◽  
Vol 33 (1) ◽  
pp. 69-80
Author(s):  
Joseph Vidal-Rosset
Keyword(s):  
Ontological Commitment ◽  
Philosophy Of Mathematics

Quine and the Web of Belief

2009 ◽  
Author(s):  
Michael D. Resnik
Keyword(s):  
Ontological Commitment ◽  
Philosophy Of Mathematics ◽  
Scientific Theories ◽  
Indispensability Arguments ◽  
Ontological Relativity ◽  
Systematic Philosophy ◽  
Web Of Belief ◽  
The Relationship ◽  
The Web

This article focuses on Quine's positive views and their bearing on the philosophy of mathematics. It begins with his views concerning the relationship between scientific theories and experiential evidence (his holism), and relate these to his views on the evidence for the existence of objects (his criterion of ontological commitment, his naturalism, and his indispensability arguments). This sets the stage for discussing his theories concerning the genesis of our beliefs about objects (his postulationalism) and the nature of reference to objects (his ontological relativity). Quine's writings usually concerned theories and their objects generally, but they contain a powerful and systematic philosophy of mathematics, and the article aims to bring this into focus.

scholarly journals Ontological Indifference of Theories and Semantic Primacy of Sentences

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dirk Greimann
Keyword(s):  
Empirical Evidence ◽  
Scientific Theory ◽  
Ontological Commitment ◽  
Philosophy Of Mathematics ◽  
Ontological Commitments ◽  
Ontological Reduction ◽  
Structuralist View ◽  
New Interpretation

Abstract In his late philosophy, Quine generalized the structuralist view in the philosophy of mathematics that mathematical theories are indifferent to the ontology we choose for them. According to his ‘global structuralism’, the choice of objects does not matter to any scientific theory. In the literature, this doctrine is mainly understood as an epistemological thesis claiming that the empirical evidence for a theory does not depend on the choice of its objects. The present paper proposes a new interpretation suggested by Quine’s recently published Kant Lectures from 1980 according to which his global structuralism is a semantic thesis that belongs to his theory of ontological reduction. It claims that a theory can always be reformulated in such a way that its truth does not presuppose the existence of the original objects, but only of some objects that can be considered as their proxies. Quine derives this claim from the principle of the semantic primacy of sentences, which is supposed to license the ontological reductions he uses to establish his global structuralism. It is argued that these reductions do not actually work because they do not account for some hidden ontological commitments that are not detected by his criterion of ontological commitment.

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.

A Priority and Application: Philosophy of Mathematics in the Modern Period

2009 ◽  
Author(s):  
Lisa Shabel
Keyword(s):  
Early Modern ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Natural World ◽  
Pure Mathematics ◽  
Philosophical Account ◽  
Mathematical Ontology ◽  
To Come ◽  
Modern Mathematics

The state of modern mathematical practice called for a modern philosopher of mathematics to answer two interrelated questions. Given that mathematical ontology includes quantifiable empirical objects, how to explain the paradigmatic features of pure mathematical reasoning: universality, certainty, necessity. And, without giving up the special status of pure mathematical reasoning, how to explain the ability of pure mathematics to come into contact with and describe the empirically accessible natural world. The first question comes to a demand for apriority: a viable philosophical account of early modern mathematics must explain the apriority of mathematical reasoning. The second question comes to a demand for applicability: a viable philosophical account of early modern mathematics must explain the applicability of mathematical reasoning. This article begins by providing a brief account of a relevant aspect of early modern mathematical practice, in order to situate philosophers in their historical and mathematical context.

Nominalism

2009 ◽  
Author(s):  
Charles Chihara
Keyword(s):  
Philosophy Of Mathematics ◽  
Contemporary Philosophy

Undoubtedly, the most enlightening published work dedicated to giving knowledgeable readers an overview of the topic of nominalism in contemporary philosophy of mathematics is A Subject with No Object by John Burgess and Gideon Rosen. This article begins with a brief description of that work, in order to provide readers with a solidly researched account of nominalism with which the article's own account of nominalism can be usefully compared. The first part, then, briefly presents the Burgess–Rosen account. A contrasting account is given in the longer second part.

scholarly journals Modal Structuralism with Theoretical Terms

Erkenntnis ◽  
2021 ◽  
Author(s):  
Holger Andreas ◽  
Georg Schiemer
Keyword(s):  
Philosophy Of Mathematics ◽  
Formal Semantics ◽  
Peano Arithmetic ◽  
Theoretical Terms ◽  
Structural Interpretation ◽  
Oxford University ◽  
Structuralist Approach ◽  
Modal Semantics ◽  
Oxford University Press

AbstractIn this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic.

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