Making and Breaking Mathematical Sense

2017 ◽  
Author(s):  
Roi Wagner
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Social Constraints ◽  
Cognitive Studies ◽  
Human Endeavor ◽  
Institutional Contexts ◽  
Underlying Reality ◽  
Fixed Ideal ◽  
Changes Over Time ◽  
Mathematical Signs

In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do—and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications? This book combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, the book challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics' exceptional consensus is due to the subject's underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

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.

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.

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.

Mathematical Reasoning

2020 ◽  
Vol 57 (4) ◽  
pp. 74-86
Author(s):  
Vitaly V. Tselishchev ◽  
Keyword(s):  
Cognitive Abilities ◽  
Mathematical Reasoning ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal System ◽  
Logical Structure ◽  
Formal Proof ◽  
Formal Proofs ◽  
Logical Conclusion

The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments of the proponents and opponents of the Thesis, “conceptualists” and “formalists”, is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of “interesting” mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat’s Last Theorem (Y. Rav). Formalists say that a conceptual proof “points” to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an “interesting” theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of “mentalists” and “mechanists” on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself.

Book reviews

1998 ◽  
Vol 52 (2) ◽  
pp. 363-379
Author(s):  
Colin A. Russell
Keyword(s):  
Eighteenth Century ◽  
Seventeenth Century ◽  
Royal Society ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
X Rays ◽  
Robert Boyle ◽  
Von Liebig ◽  
Principles Of Geology

Eleven book reviews in the July 1998 edition of Notes and Records : Robert Boyle. A free enquiry into the vulgarly received notion of nature , Edward B. Davis and Michael Hunter (eds.). P. Mancosu, Philosophy of mathematics and mathematical practice in the seventeenth century . Patricia Fara, Sympathetic attractions: magnetic practices, beliefs, and symbolism in eighteenth–century England . Alan Cook, Edmond Halley: charting the heavens and the seas . Jan Bondeson, A cabinet of medical curiosities . J.M.H. Moll, Presidents of The Royal Society of Medicine . W.H. Brock, Justus von Liebig: the chemical gatekeeper . Charles Lyell's Principles of Geology (1830–33) , J. Secord (ed.). X–rays––the first hundred years , Alan Michette and Slawka Pfauntsch (eds.). Jack Morrell, Science at Oxford 1914–1939 . John Polkinghorne, Beyond science. The wider human context .

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