scholarly journals Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
José Antonio Pérez-Escobar ◽  
Deniz Sarikaya
Keyword(s):  
Applied Mathematics ◽  
Philosophy Of Mathematics ◽  
Family Resemblance ◽  
Theory Building ◽  
Clear Cut ◽  
Pure Mathematics ◽  
Extended Notion ◽  
Evaluative Standards

AbstractIn this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake.

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.

scholarly journals Fractional Derivatives and Integrals: What Are They Needed For?

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 164 ◽  
Author(s):  
Vasily E. Tarasov ◽  
Svetlana S. Tarasova
Keyword(s):  
Power Law ◽  
Applied Mathematics ◽  
Mathematical Problem ◽  
Frequency Dispersion ◽  
Point Of View ◽  
Fading Memory ◽  
Fractional Operator ◽  
Distributed Lag ◽  
Pure Mathematics ◽  
New Research

The question raised in the title of the article is not philosophical. We do not expect general answers of the form “to describe the reality surrounding us”. The question should actually be formulated as a mathematical problem of applied mathematics, a task for new research. This question should be answered in mathematically rigorous statements about the interrelations between the properties of the operator’s kernels and the types of phenomena. This article is devoted to a discussion of the question of what is fractional operator from the point of view of not pure mathematics, but applied mathematics. The imposed restrictions on the kernel of the fractional operator should actually be divided by types of phenomena, in addition to the principles of self-consistency of mathematical theory. In applications of fractional calculus, we have a fundamental question about conditions of kernels of fractional operator of non-integer orders that allow us to describe a particular type of phenomenon. It is necessary to obtain exact correspondences between sets of properties of kernel and type of phenomena. In this paper, we discuss the properties of kernels of fractional operators to distinguish the following types of phenomena: fading memory (forgetting) and power-law frequency dispersion, spatial non-locality and power-law spatial dispersion, distributed lag (time delay), distributed scaling (dilation), depreciation, and aging.

scholarly journals Horace Lamb and the circumstances of his appointment at Owens College

2012 ◽  
Vol 67 (2) ◽  
pp. 139-158 ◽  
Author(s):  
Brian Launder
Keyword(s):  
Applied Mathematics ◽  
South Australia ◽  
Critical Juncture ◽  
Classical Fluid ◽  
Sequence Of Events ◽  
Pure Mathematics ◽  
Research In Education

This paper examines a succession of incidents at a critical juncture in the life of Professor Horace Lamb FRS, a highly regarded classical fluid mechanicist, who, over a period of some 35 years at Manchester, made notable contributions in research, in education and in wise administration at both national and university levels. Drawing on archived documents from the universities of Manchester and Adelaide, the article presents the unusual sequence of events that led to his removing from Adelaide, South Australia, where he had served for nine years as the Elder Professor of Mathematics, to Manchester. In 1885 he was initially appointed to the vacant Chair of Pure Mathematics at Owens College and then, in 1888, as an outcome of his proposal for rearranging professorial responsibilities, to the Beyer Professorship of Pure and Applied Mathematics.

scholarly journals Theological Underpinnings of the Modern Philosophy of Mathematics.

2016 ◽  
Vol 44 (1) ◽  
pp. 147-168
Author(s):  
Vladislav Shaposhnikov
Keyword(s):  
Twentieth Century ◽  
Modern Philosophy ◽  
Philosophy Of Mathematics ◽  
Main Concern ◽  
Early Twentieth Century ◽  
Pure Mathematics ◽  
And Mathematics

Abstract The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

Berlin Roots – Zionist Incarnation: The Ethos of Pure Mathematics and the Beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem

2004 ◽  
Vol 17 (1-2) ◽  
pp. 199-234 ◽  
Author(s):  
Shaul Katz
Keyword(s):  
Middle East ◽  
Applied Mathematics ◽  
Jewish People ◽  
Controversial Issue ◽  
Academic Needs ◽  
German Mathematician ◽  
Pure Mathematics ◽  
Hebrew University

Officially inaugurated in 1925, the Hebrew University of Jerusalem was designed to serve the academic needs of the Jewish people and the Zionist enterprise in British Mandatory Palestine, as well as to help fulfill the economic and social requirements of the Middle East. It is intriguing that a university with such practical goals should have as one of its central pillars an institute for pure mathematics that purposely dismissed any of the varied fields of applied mathematics. This paper tells of the preparations for the inauguration of the Hebrew University during the years 1920–1925 and analyzes the founding phase of the Einstein Institute of Mathematics that was established there during the years 1924–1928. Special emphasis is given to the first terms in which this Institute operated, starting from the winter of 1927 with the activities of the director and one of the founders, the German mathematician Edmund Landau, and onward from 1928 when his successors, particularly Adolf Abraham Halevi Fraenkel and Mihály-Michael Fekete, continued Landau's heritage of pure mathematics. The paper shows why and how the Institute succeeded in rejecting applied mathematics from its court and also explores the controversial issue of center and periphery in the development of science, a topic that is briefly analyzed in the concluding section.

scholarly journals Theological Underpinnings of the Modern Philosophy of Mathematics.

2016 ◽  
Vol 44 (1) ◽  
pp. 31-54
Author(s):  
Vladislav Shaposhnikov
Keyword(s):  
Nineteenth Century ◽  
Modern Philosophy ◽  
Philosophy Of Mathematics ◽  
Main Concern ◽  
Cultural Expectation ◽  
Divine Knowledge ◽  
Pure Mathematics ◽  
And Mathematics ◽  
Modern Mathematics

Abstract The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern is nineteenth-century mathematics. Theology was present in modern mathematics not through its objects or methods, but mainly through popular philosophy, which absolutized mathematics. Moreover, modern pure mathematics was treated as a sort of quasi-theology; a long-standing alliance between theology and mathematics made it habitual to view mathematics as a divine knowledge, so when theology was discarded, mathematics naturally took its place at the top of the system of knowledge. It was that cultural expectation aimed at mathematics that was substantially responsible for a great resonance made by set-theoretic paradoxes, and, finally, the whole picture of modern mathematics.

scholarly journals Grassmann’s Concept Structuralism

2020 ◽  
pp. 21-58
Author(s):  
Paola Cantù
Keyword(s):  
Number Theory ◽  
Linear Algebra ◽  
Applied Mathematics ◽  
Philosophy Of Mathematics ◽  
Abstract Algebra ◽  
General Properties

After recalling some mathematical contributions that are relevant for the structuralist transformation of mathematics, such as abstract algebra, linear algebra, and number theory, this chapter reconstructs Grassmann’s philosophy of mathematics. It is claimed that he contributed to the development of methodological structuralism inasmuch as he clearly separated the study of the most general properties of connections from pure and applied mathematics, basing them on an understanding of generality as conceptual underdetermination, and on the preeminence of the notion of series over that of function. A brief comparison with contemporary philosophical structuralism will clarify Grassmann’s tendency toward a “concept structuralism” rather than an “object structuralism.”

Aerial Navigation as Applied Mathematics

1945 ◽  
Vol 38 (7) ◽  
pp. 314-316
Author(s):  
Laura Blank
Keyword(s):  
High School ◽  
Secondary Schools ◽  
Federal Government ◽  
Young Women ◽  
Applied Mathematics ◽  
Young Men ◽  
South Pacific ◽  
Pure Mathematics ◽  
Aerial Navigation ◽  
The Moment

In the fall of 1942 at the urgent request of the federal government, as an incentive to interest in piloting and navigating airplanes, many of the secondary schools of our country set into operation classes in aerial navigation, aerodynamics and meteorology. The navigation courses were in the main, the responsibility of the teachers of mathematics. They have been preparing youth including a few young women now for three years. The motive of these young folk in selecting the course is either the wish to pilot soon or that of understanding a timely subject. Many of the young men of the earlier classes are flying missions now “down under” in the South Pacific.* Some are flying their own planes from England for furloughs and then new assignments. Now the Army Air Corps is closed to admissions, and moreover, many men classified in that branch of the service for months, “on the line,” awaiting anxiously their transfer to preflight have been notified officially that they will not be needed as pilots or navigators or even bombardiers. One wonders what the effect will be on elections to a high school course in navigation. Will it develop that aerial navigation is an emergency subject, incident to the war, in secondary schools, to vanish from the curriculum in a few years, parts of it to be taken over into the courses in so-called pure mathematics? Or will navigation continue as a course optional in high school? At the moment options are holding up well.

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