Related papers: Varieties of mathematical understanding
In the article the problems of the systems of computer mathematics use as a tool for the students learning and research activities support are investigated. The promising ways of providing access to the mathematical software in the…
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of…
The topic of diversity is an interesting subject, both as a purely mathematical concept and also for its applications to important real-life situations. Unfortunately, although the meaning of diversity seems intuitively clear, no precise…
We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…
The emergence and continued reliance on the Internet and related technologies has resulted in the generation of large amounts of data that can be made available for analyses. However, humans do not possess the cognitive capabilities to…
I analyse differences in style between traditional prose mathematics writing and computer-formalised mathematics writing, presenting five case studies. I note two aspects where good style seems to differ between the two: in their…
We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed…
The purpose of this note is to raise two different questions, which are rarely if ever considered, and to which, it seems, we lack convincing, systematic answers. These questions can be posed as: - Why do we compute? - What do we compute?…
Artificial computing machinery transforms representations through an objective process, to be interpreted subjectively by humans, so the machine and the interpreter are different entities, but in the putative natural computing both…
Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathematics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral…
I am most honoured to have the privilege to present the Foreword to this fascinating and wonderfully varied collection of contributions, concerning the nature of computation and of its deep connection with the operation of those basic laws,…
This article aims to demonstrate how the approach to computing is being disrupted by deep learning (artificial neural networks), not only in terms of techniques but also in our interactions with machines. It also addresses the philosophical…
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.
Doing mathematics implies three levels of manipulation: manipulating the abstract, manipulating symbols and manipulating logic. Teaching mathematics therefore involves the teacher proposing situations in which pupils can explore a small…
Recent developments in computer programming and in mathematics suggest that there is a strong case for a new way of introducing programming to enhance the learning of school mathematics. The article describes a collaboration of mathematics…
This book provides a comprehensive and accessible introduction to the emerging field of AI for mathematics. It covers the core principles and diverse applications of using artificial intelligence to advance mathematical research. Through…
We commonly think of mathematics as bringing precision to application domains, but its relationship with computer science is more complex. This experience report on the use of Racket and Haskell to teach a required first university CS…
We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience…
The recent development of quantum computing, which uses entanglement, superposition, and other quantum fundamental concepts, can provide substantial processing advantages over traditional computing. These quantum features help solve many…
The present paper presents some reflections of the author on divergent series and their role and place in mathematics over the centuries. The point of view presented here is limited to differential equations and dynamical systems.