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.

MATHEMATICAL RIGOR AND PROOF

2019 ◽  
pp. 1-41 ◽  
Author(s):  
YACIN HAMAMI
Keyword(s):  
Mathematical Knowledge ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal Proof ◽  
Major Obstacle ◽  
Standard View ◽  
Precise Formulation ◽  
Mathematical Proofs ◽  
Primary Form

Abstract Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

Linearity and Reflexivity in the Growth of Mathematical Knowledge

1989 ◽  
Vol 3 (2) ◽  
pp. 409-440 ◽  
Author(s):  
Leo Corry
Keyword(s):  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Hard Core ◽  
Past Century ◽  
Exact Science ◽  
The Past ◽  
The Social ◽  
Scientific Fields ◽  
Great Portion

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of the body and images of mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those of any other exact science. In no other exact science can the standard methodological framework used within the discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics.

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.

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 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.

scholarly journals Showing Mathematical Flies the Way Out of Foundational Bottles: The Later Wittgenstein as a Forerunner of Lakatos and the Philosophy of Mathematical Practice

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
José Antonio Pérez-Escobar
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
History Of ◽  
Philosophy Of Mathematical Practice ◽  
The Way

Abstract This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy.

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.

The New Entities of Abbacus and Renaissance Algebra

2017 ◽  
Author(s):  
Roi Wagner
Keyword(s):  
Fifteenth Century ◽  
Middle Ages ◽  
Natural Science ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Northern Italy ◽  
Historical Narrative ◽  
Negative Numbers ◽  
Late Middle Ages ◽  
Mathematical Signs

This chapter offers a historical narrative of some elements of the new algebra that was developed in the fourteenth to sixteenth centuries in northern Italy in order to show how competing philosophical approaches find an intertwining expression in mathematical practice. It examines some of the important mathematical developments of the period in terms of a “Yes, please!” philosophy of mathematics. It describes economical-mathematical practice with algebraic signs and subtracted numbers in the abbaco tradition of the Italian late Middle Ages and Renaissance. The chapter first considers where the practice of using letters and ligatures to represent unknown quantities come from by analyzing Benedetto's fifteenth-century manuscript before discussing mathematics as abstraction from natural science observations that emerges from the realm of economy. It also explores the arithmetic of debited values, the formation of negative numbers, and the principle of fluidity of mathematical signs.

Type-theoretical checking and philosophy of Mathematics

1998 ◽  
Author(s):  
Nicolaas Govert de Bruijn
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Mutual Influence ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
General Information ◽  
Point Of View ◽  
De Bruijn ◽  
Level Of Understanding ◽  
The Way

After millennia of mathematics we have reached a level of understanding that can be represented physically. Humankind has managed to disentangle the intricate mixture of language, metalanguage and interpretation, isolating a body of formal, abstract mathematics that can be completely verified by machines. Systems for computer-aided verification have philosophical aspects. The design and usage of such systems are influenced by the way we think about mathematics, but it also works the other way. A number of aspects of this mutual influence will be discussed in this paper. In particular, attention will be given to philosophical aspects of type-theoretical systems. These definitely call for new attitudes: throughout the twentieth century most mathematicians had been trained to think in terms of untyped sets. The word “philosophy” will be used lightheartedly. It does not refer to serious professional philosophy, but just to meditation about the way one does one’s job. What used to be called philosophy of mathematics in the past was for a large part subject oriented. Most people characterized mathematics by its subject matter, classifying it as the science of space and number. From the verification system’s point of view, however, subject matter is irrelevant. Verification is involved with the rules of mathematical reasoning, not with the subject. The picture may be a bit confused, however, by the fact that so many people consider set theory, in particular untyped set theory, as part of the language and foundation of mathematics, rather than as a particular subject treated by mathematics. The views expressed in this paper are quite personal, and can mainly be carried back to the author’s design of the Automath system in the late 1960s, where the way to look upon the meaning (philosophy) of mathematics is inspired by the usage of the unification system and vice versa. See de Bruijn 1994b for various philosophical items concerning Automath, and Nederpelt et al. 1994, de Bruin 1980, de Bruijn 1991a for general information about the Automath project. Some of the points of view given in this paper are matters of taste, but most of them were imposed by the task of letting a machine follow what we say, a machine without any knowledge of our mathematical culture and without any knowledge of physical laws.

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.

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