The Quasi-Empirical Epistemology of Mathematics

This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that (1) Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; (2) what can be called the quasi-empirical epistemology of mathematics is not correct; (3) analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.

The Relationship between pre-service elementary school mathematics teachers’ beliefs about epistemology of mathematics, teaching and learning, and mathematics assessment

<div><p>This study aims to examine the relationship between epistemological beliefs, teaching-learning beliefs and assessment beliefs in mathematics education. This research is a quantitative study with a correlational study. Data collection using the survey method with a cross-sectional design. The participants were 71 pre-service elementary school , mathematics teachers. The data on beliefs were collected through means of a questionnaire. The data collected from the questionnaire were then analyzed quantitatively through descriptive and inferential statistics. Descriptive statistics utilizes the mean value, maximum value, and standard deviation values. Inferential statistics use the product-moment correlation as well as path analysis. The research results show that there is a positive and significant correlation between static and dynamic beliefs on epistemology of mathematics, and the constructivist beliefs on mathematics teaching and learning, with the productive beliefs on mathematics assessment. In addition, there is seen to be a functional influence between both epistimological beliefs (both static and dynamic), as well as beliefs on teaching and learning (constructivist) and beliefs about mathematic assessment (productive). The results of this research signify the importance of considering one’s beliefs about the epistemology of mathematics and mathematics teaching and learning when constructing their beliefs regarding mathematics assessment.</p></div>

UMA PROPOSTA DE INVESTIGAÇÃO HISTÓRICO-EPISTEMOLÓGICA SOBRE SEQUÊNCIAS RECORRENTES DE 2ª ORDEM

A formação de professores no Brasil não pode prescindir de um componente histórico-matemático e evolutivo. Nesse sentido, se torna imprescindível ao professor compreender a natureza intrinseca sobre o conhecimento matemático e seus processos ou itinerários de evolução e de irrefreável generalização. Dessa forma, o presente trabalho apresenta os dados preliminares de uma investigação amaparada pelos pressupostos de uma Engenharia Didática de Formação, em desenvolvimento no Brasil. O trabalho revela a uma importante cooperação científica envolvendo pesquisadores portugueses, sobre o asssunto de sequências recorrentes de 2ª ordem e aponta a contribuição de pesquisas desenvolvidas no período de 2015 – 2020, no Programa de Pós-graduação em Ensino de Ciências e Matemátca, do Instituto Federal de Educação, Ciências e Tecnologia do Estado do Ceará – IFCE. Por fim, o trabalho apresenta algns indicadores que devem demarcar um importante cenário para a formação (inicial e continuada) de professores no Brasil.Palavras-chave: História e Epistemologia da Matemática; Sequências recorrentes; Formação de Profesores de Matemática; Engenharia Didática de Formação.A PROPOSAL FOR HISTORICAL-EPISTEMOLOGICAL RESEARCH ON THE 2ND ORDER SEQUENCESAbstract. Teacher training in Brazil cannot do without a historical-mathematical and evolutionary component. In this sense, it is essential for the teacher to understand the intrinsic nature of mathematical knowledge and its processes or itineraries of evolution and irrepressible generalization. Thus, the present work presents the preliminary data of an investigation supported by the assumptions of Didactic Engineering of Training, under development in Brazil. The work reveals an important scientific cooperation involving Portuguese researchers, on the subject of recurrent 2nd order sequences and points out the contribution of research developed in the period 2015 - 2020, in the Postgraduate Program in Science Teaching and Mathematics, of the Institute Federal Institute of Education, Science and Technology of the State of Ceará - IFCE. Finally, the work presents some indicators that should outline an important scenario for the training (initial and continuing) of teachers in Brazil.Keywords: History and Epistemology of Mathematics; Recurring strings; Mathematics Teacher Training; Didactic Engineering Training.

Truth, Paradoxes, Freedom, Applications, and Knowledge

This chapter begins by showing that with the problems of mathematical existence and determinate truth solved, a sophisticated inferentialist theory of mathematics leads to mathematical conventionalism. A philosophical worry harkening back to the Carnap/Quine debate is addressed before a number of issues in the philosophy of mathematics are given conventionalist treatments. The chapter discusses how conventionalists can handle the set-theoretic paradoxes, the freedom of mathematics, the many applications of mathematics to the physical world, and then provides a naturalistic epistemology of mathematics, even addressing the epistemology of consistency. In almost all of these cases, the discussion in the chapter shows that a conventionalist theory deals with these issues in a more satisfying way than other approaches to the philosophy of mathematics.

Philosophical Challenges for Scientism (and How to Meet Them?)

Scientism is expounded. Then its two major challenges are stated and responses to them sketched. The first challenge is to its epistemology of mathematics-how we know the necessary truths of mathematics. The second challenge is to the very coherence of its eliminativist account of cognition. The first of these problems is likely to be taken more seriously by philosophers than by other advocates of scientism. It is a problem that has absorbed philosophers since Plato and on which little progress has been made. The second is often unnoticed, even among those who endorse scientism, since they don’t recognize their own commitment to eliminativism and so do not appreciate the threat of incoherence it poses. It is important for scientism to acknowledge these challenges.

The Question of Platonism

This book defends the existence of abstract mathematical objects. Should this be regarded as a defense of Platonism? Platonism involves an analogy between mathematical and physical objects. Although mathematical objects are counterfactually independent of us, just like paradigmatic physical objects, there are other respects in which mathematical objects are strikingly different from physical objects: by giving rise to the phenomenon of indefinite extensibility and by having a shallow nature. The view here is therefore not a full-blown form of Platonism. However, the shallow nature of mathematical objects has the advantage of enabling an epistemology of mathematics where our mathematical beliefs are appropriately sensitive to the truth of these beliefs.

Dissertações e teses em História e Epistemologia da Matemática: contribuições para a abordagem da Geometria Espacial no Ensino Médio

The present article adopts as an object of study dissertations and theses in History and Epistemology of Mathematics of Brazil in the period between 1990 and 2010 of the stricto sensu postgraduate programs that present historical information on topics related to Space Geometry (High School), The objective is to investigate the didactic potential of the Research. We also suggest didactic referrals that can somehow be used in our teaching practice, provided that there is a treatment of this information before its application in the classroom so that the content can be taught in the best way possible according to students’cognitive potential and also to the reality of each school environment. In order to select the theses and dissertations, we used the three current trends in the history of mathematics, according to the cartography of the researches in this area, carried out by Mendes (2014). To identify if a study covered High School content , we used as a parameter the areas of knowledge of the National High School Examination (SEM). This article may be useful to widely disseminate these publications so as to enable the dissemination of the mathematical contents that can approach the concepts of Space Geometry in High School, for once these dissertations and theses are defended, they are forgotten in libraries.