Computational reverse mathematics and foundational analysis

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only $\omega$-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a $\Pi^1_1$ complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than $\Pi^1_1\text{-}\mathsf{CA}_0$.Comment: Submitted. 41 page

Using Graphing Calculators to Integrate Mathematics and Science

The computational, graphing, statistical and programming capabilities of today’s graphing calculators make it possible for teachers and students to explore aspects of functions and investigate real-world situations in ways that were previously inaccessible because of computational constraints. Many of the features of graphing calculators can be used to integrate topics from mathematics and science. Here we provide a few illustrations of activities that use the graphing, parametric graphing, regression, and recursion features of graphing calculators to study mathematics in science contexts

Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 235(2011), doi: 10.1016/j.cam.2011.04.041This article discusses the oscillatory behaviour of the differential equation of mixed type

Improving The Quality Of The Mathematics Education: The Malaysian Experience

Improving the quality of teaching and learning of mathematics has always been a major concern of mathematics educators. The four recurring and inter-related issues often raised in the development of a mathematics curriculum are: “What type of mathematics ought to be taught?”, “Why do we need to teach mathematics?”, “How should mathematics curriculum be planned and arranged?” and “ How can teacher ensure that what is transmitted to the pupils is as planned in the curriculum?”.The relatively brief history of mathematics education in Malaysia can be said to have developed in three distinct phases. In the first phase, the traditional approach, which emphasized mainly on basic skills (predominantly computational), was the focus of the national syllabus. In the late 70’s, in consonance with the world-wide educational reform, the modern mathematics program (MMP) was introduced in schools. Understanding of basic concepts rather than attaining computational efficiency was the underlying theme of the syllabus. Finally, in the late 80’s the mathematics curriculum was further revised. It is part of the national educational reform that saw the introduction of the national integrated curriculum (KBSM) both at the primary and secondary levels. This mathematics curriculum, which has undergone several minor changes periodically, is presently implemented in schools. The curriculum also emphasizes on the importance of context in problem solving. These three syllabi, as in any other curricular development, can be seen to have evolved from changing perspectives on the content, psychological and pedagogical considerations in teaching and learning of mathematics. In this paper, I will trace the development of the Malaysian mathematics curriculum from the psychological, content and pedagogical perspectives in relation to the recurring issues. I will argue that the development has in many ways attempted to make mathematics more meaningful and thus friendlier for students both at the primary and secondary levels. There has been also a marked improvement on the quality of mathematics education in Malaysi

Analytical and numerical investigation of mixed-type functional differential equations

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234 (2010), doi: 10.1016/j.cam.2010.01.028This journal article is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments

The numerical solution of forward–backward differential equations: Decomposition and related issues

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234,(2010), doi: 10.1016/j.cam.2010.01.039This journal article discusses the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions