Philosophy of Mathematics

Author(s):  
Denis Bonnay

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.

Author(s):  
Janet Folina

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.


Author(s):  
José Ferreirós

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.


2018 ◽  
pp. 13-38
Author(s):  
N. Ceramella

The article considers two versions of D. H. Lawrence’s essay The Theatre: the one which appeared in the English Review in September 1913 and the other one which Lawrence published in his first travel book Twilight in Italy (1916). The latter, considerably revised and expanded, contains a number of new observations and gives a more detailed account of Lawrence’s ideas.Lawrence brings to life the atmosphere inside and outside the theatre in Gargnano, presenting vividly the social structure of this small northern Italian town. He depicts the theatre as a multi-storey stage, combining the interpretation of the plays by Shakespeare, D’Annunzio and Ibsen with psychological portraits of the actors and a presentation of the spectators and their responses to the plays as distinct social groups.Lawrence’s views on the theatre are contextualised by his insights into cinema and its growing popularity.What makes this research original is the fact that it offers a new perspective, aiming to illustrate the social situation inside and outside the theatre whichLawrenceobserved. The author uses the material that has never been published or discussed before such as the handwritten lists of box-holders in Gargnano Theatre, which was offered to Lawrence and his wife Frieda by Mr. Pietro Comboni, and the photographs of the box-panels that decorated the theatre inLawrence’s time.


2021 ◽  
Vol 21 ◽  
pp. 273-294
Author(s):  
Gabriele Baratelli ◽  

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall.


PARADIGMA ◽  
2019 ◽  
Vol 40 (2) ◽  
pp. 196-217
Author(s):  
Luis Andrés Castillo ◽  
Juan Luis Prieto G. ◽  
Ivonne C. Sánchez ◽  
Rafael Enrique Gutiérrez

En los últimos años, los profesores han tratado cada vez más de utilizar los recursos didácticos para apoyar sus clases, un hecho que se ha agregado con la llegada de las tecnologías digitales. En este contexto surgió la tendencia de que los profesores se convirtieran en creadores de este tipo de recursos, pero dejando de lado la sistematización y el deber de compartir con sus compañeros la experiencia de desarrollar este tipo de recursos. Por esta razón, el presente trabajo describe una experiencia concreta de elaborar un simulador con el software GeoGebra para estudiar una situación de tiro libre en el fútbol, utilizando nociones de movimiento parabólico. La elaboración del simulador incluye la resolución de siete "tareas de simulación", que destacan los objetos y los procesos matemáticos que justifican las técnicas de construcción utilizadas por los autores. Finalmente, se presentan algunas reflexiones derivadas de la experiencia de desarrollo del simulador, que consideran, por un lado, el conocimiento matemático movilizado en la producción del simulador y, por otro lado, las acciones y capacidades requeridas para tal elaboración.AbstractIn recent years, teachers have increasingly sought the use of didactic resources to support their classes, a fact that has been added with the arrival of digital technologies. In this context, the tendency has emerged that teachers have become creators of this type of resources, but leaving aside the systematization and the duty to share with their peers the experience of developing this type of resources. For this reason, the present work describes a concrete experience of developing a simulator with GeoGebra software for the study of a situation of free kick in football, using notions of parabolic movement. The elaboration of the simulator includes the resolution of seven "simulation tasks", which highlight the objects and mathematical processes that should justify the construction techniques used by the authors. Finally, some reflections derived from the simulator development experience are presented, which consider, on the one hand, the mathematical knowledge mobilized in the production of the simulator and, on the other hand, the actions and capabilities required for such an elaboration.ResumoNos últimos anos, professores têm procurado cada vez mais o uso de recursos didáticos para apoiar suas aulas, fato que foi acrescentado com a chegada das tecnologias digitais. Nesse contexto surgiu a tendência que os professores se tornaram criadores desse tipo de recursos, mas deixando de lado a sistematização e o dever de dividir com seus pares a experiência de desenvolver este tipo de recursos. Por este motivo, o presente trabalho descreve uma experiência concreta de elaboração de um simulador com o software GeoGebra para o estudo de uma situação do tiro livre no futebol, utilizando noções do movimento parabólico. A elaboração do simulador inclui a resolução de sete “tarefas de simulação”, que destacam os objetos e processos matemáticos que devem justificar as técnicas de construção utilizadas pelos autores. Finalmente, apresentam-se algumas reflexões derivadas da experiência de desenvolvimento do simulador, que consideram, por um lado, o conhecimento matemático mobilizado na produção do simulador e, por outro lado, as ações e capacidades necessárias para tal elaboração. 


2020 ◽  
Vol 10 (2) ◽  
pp. 105-114
Author(s):  
Sakon Tangkawsakul ◽  
Nuttapat Mookda ◽  
Weerawat Thaikam

In this study, we adapted the school sports day to provide opportunities to relate real-life situations with mathematical knowledge and skills. The purpose of this study was to describe the way that the teachers interact with their students and the students’ responses during mathematical modelling processes. The designing of the modelling task was inspired by the Realistic Fermi Problems about the bleacher in the school sports day. The modelling task was designed by a collaboration of mathematics teachers and educators and experimented with 10th-grade students. Each experiment lasted for 45 minutes and was conducted in the one-day camp with 45 students. The results showed that the students who had no previous experience of mathematical modelling engaged in mathematical modelling processes with their friends under the guidance and supporting of the teacher. Most of them were able to think, make assumptions, collect data, observe, make conjectures and create mathematical models to understand and solve the modelling task.   


2021 ◽  
Vol 43 (2) ◽  
pp. 235-257
Author(s):  
Daniel Edler Duarte ◽  
Marcelo M. Valença

Abstract The COVID-19 pandemic has sparked controversies over health security strategies adopted in different countries. The urge to curb the spread of the virus has supported policies to restrict mobility and to build up state surveillance, which might induce authoritarian forms of government. In this context, the Copenhagen School has offered an analytical repertoire that informs many analyses in the fields of critical security studies and global health. Accordingly, the securitisation of COVID-19 might be necessary to deal with the crisis, but it risks unfolding discriminatory practices and undemocratic regimes, with potentially enduring effects. In this article, we look into controversies over pandemic-control strategies to discuss the political and analytical limitations of securitisation theory. On the one hand, we demonstrate that the focus on moments of rupture and exception conceals security practices that unfold in ongoing institutional disputes and over the construction of legitimate knowledge about public health. On the other hand, we point out that securitisation theory hinders a genealogy of modern apparatuses of control and neglects violent forms of government which are manifested not in major disruptive acts, but in the everyday dynamics of unequal societies. We conclude by suggesting that an analysis of the bureaucratic disputes and scientific controversies that constitute health security knowledges and practices enables critical approaches to engage with the multiple – and, at times, mundane – processes in which (in)security is produced, circulated, and contested.


Author(s):  
Georg Schiemer

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.


Development ◽  
1956 ◽  
Vol 4 (1) ◽  
pp. 1-33
Author(s):  
S. Gopalakrishnan Nair

Although Rathke (1834), Dohrn (1867), van Beneden (1869), and Roule (1889, 1890, 1891, 1894, 1896) have studied the embryology of Isopoda, the first detailed account and the one that is ordinarily quoted in text-books is that of Bobretzsky (1874) on Oniscus murarius. This work is informative in a general way, though the details of segmentation and germ layer formation are not accurate. Bullar's (1878) work on the parasitic isopods was largely influenced by the generalizations of Bobretzsky. Nusbaum (1891a, 1898) and McMurrich (1892, 1895) have contributed considerably to our knowledge of segmentation and post-mandibular growth in isopods but their accounts of the different fates of the germ layers left several problems of embryology unsolved. Goodrich's (1939) studies on Porcellio and Armadillidium were confined mainly to the origin and fate of the endoderm elements. Manton's (1928) paper on the development of Hemimysis serves as a landmark in the history of Crustacean embryology.


1869 ◽  
Vol 17 ◽  
pp. 308-309

At present we are acquainted with two amylic alcohols formed by fermentation. They were discovered by Pasteur, who observed that different specimens of amylic alcohol caused a ray of polarized light to rotate to dif­ferent degrees. He succeeded in devising a separation of these alcohols, which consisted in converting them into sulphamylates of barium and re­crystallizing these salts. The one alcohol is without action on polarized light, and the other rotates it. This method of separation is beset with great practical difficulties, and has, we believe, only once been repeated, viz. by Mr. Pedler. He gives no detailed account of the separation, but gives some of the leading properties of the alcohols. He found that the rotating alcohol caused a ray of polarized light to rotate 17° with a column of 500 millims. of liquid. The following are some examples of the rotations effected by eleven different samples of amylic alcohol in a column of 385 millims. For compa­rison with Pedler’s number, the observed numbers have been reduced in the second column to observations on 500 millims.


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