Lautman and the Reality of Mathematics

Working in he 1930s, Albert Lautman described with extraordinary clarity the new understanding of mathematics of that time. He delighted in the multiple manifestations of a common idea in different mathematical fields. However, he took the common idea to belong not to mathematics itself, but to an 'ideal reality' sitting above mathematics. I argue in this paper that now that we have a mathematical language which can characterize these common ideas, we need not follow Lautman to adopt his form of Platonism. On the other hand, Lautman should be much better known than he is for pointing philosophy towards this most important feature of mathematics

Introduction. The School: Its Genesis, Development and Significance

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School – a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland

Exploring mathematical values through mathematics teachers’ beliefs and instructional practices

Mathematical values are deep affective qualities which education aims to foster through mathematics subjects in schools and are crucial components of the classroom affective environment. Mathematical values comprise teachers’ beliefs, attitudes, and their instructional practices in mathematics classrooms. This study investigated Malaysian mathematics secondary school teachers’ beliefs and their instructional practices based on four schools of philosophy of mathematics which are logicism, formalism, intuitionism and kuhnism. A quantitative research method with a survey design was used for assessment during this study. An instrument to measure the two constructs was developed based on the four mathematical philosophies mentioned earlier. The findings indicated that majority of mathematics teachers’ beliefs were inclined towards kuhnism whilst their instructional practices were inclined towards formalism. These findings imply that in practice majority of mathematics teachers in secondary school emphasized on symbols and formulas in their teaching. However, it can also be concluded that the mathematics teachers’ instructional beliefs were closely affiliated with their social norms and culture. Thus, these findings suggested that mathematics teachers’ beliefs were not congruent with their practices. Although the teachers’ beliefs were towards kuhnism, they did not portray these in their teachings, which seemed to emphasize on formalism

A suggested programme for developing 4th year primary pupils’ performance in Mathematical word problems in Kuwait

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The main objective of this study was to investigate the effect of using a suggested mathematical word-problem training programme on Primary 4 pupils' performance in mathematical word-problems. The study had a pre-post control group design. A treatment and a no-treatment group were exposed to pre-post methods of gathering data (a mathematical word-problem achievement test and a mathematical word-problem attitude scale). The treatment group was given direct and explicit training on how to solve mathematical word-problems, while the pupils of the no-treatment group received no such training; they were taught the same material they study at school. A "t" test was used to compare the means of scores of the control group pupils and those of the experimental group in the pre-post measurements. Results of the study revealed a significant improvement in the experimental group pupils’ performance in mathematical word-problems because they had attended the suggested programme. Results also revealed that experimental group subjects' attitudes towards mathematical word-problems underwent an exceptional change because they had attended the suggested programme. Their attitudes towards mathematical word-problems became more positive than before. In the light of the results of the study, some recommendations were made for improving mathematics teacher training programmes, for mathematics teaching, and for further research

Mathematical Concepts and Proofs from Nicole Oresme: Using the History of Calculus to Teach Mathematics

"Paper presented at The Seventh International History, Philosophy and Science Teaching Conference, Winnipeg, MB, Canada August 1, 2003."This paper examines the mathematical work of the French bishop, Nicole Oresme (c. 1323–1382), and his contributions towards the development of the concept of graphing functions and approaches to investigating infinite series. The historical importance and pedagogical value of his work will be considered in the context of an undergraduate course on the history of calculus.https://link.springer.com/article/10.1007/s11191-004-7937-

Oxford mathematics at a low ebb? an 1855 dispute over examination results

Between December 1855 and March 1856, a public dispute raged, in British national newspapers and locally published pamphlets, between two teachers at the University of Oxford: the mathematical lecturer Francis Ashpitel and Bartholomew Price, the professor of natural philosophy. The starting point for these exchanges was the particularly poor results that had come out of the final mathematics examinations in Oxford that December. Ashpitel, as one of the examiners, stood accused of setting questions that were too difficult for the ordinary student, with the consequence that, in Price’s view, further mathematical study in Oxford – never as robust as in Cambridge – would be discouraged. We examine this short-lived affair, and use it not only to gain insight into the status of mathematical study in Oxford in the mid-nineteenth century, but also to point towards the increasing importance of competitive examinations in British public life at that time

Introduction

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book