In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception (...) of mathematical knowledge, in particular mathematical growth. (shrink)

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...) field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (shrink)

Although there are quite a few directions in modern philosophy of mathematics that invoke some essential role for (quasi-)empirical material, this chapter will be devoted exclusively to what may be considered the seminal tradition. This enabled me to present the subject as one coherent whole and to forestall the discussion getting scattered in a diversity of directions without doing justice to any one of them. -/- Quasi-empiricism in this tradition is the view that the logic of mathematical inquiry is based, (...) like the logic of scientific discovery, on the bottom-up retransmission of falsity (by means of counterexamples) rather than the top-down transmission of axiomatic truth. Imre Lakatos, who originally introduced the distinction between Euclidean and quasi-empirical theories, construed mathematics as a quasi-empirical science following in essence Popper’s method of conjectures and refutations. His essay on Proofs and Refutations was an attempt to apply, and thereby to test, Popper’s critical method and fallibilist theory of knowledge in a field – mathematics – to which it had not primarily been intended to apply. While Popper’s proposal stood up gloriously to this test, the new application gave rise also to new insights, in particular the construal of proofs as thought experiments with a role similar to that of testing and corroborating experiments in science. -/- After examination of a number of examples of mathematical thought-experiments from this perspective, the doctrine of the relative autonomy of objective knowledge is focussed upon, a doctrine which is part and parcel of the dialectic of proofs and refutations and hence of the quasi-empiricist programme. -/- Some remarks on the educational implications of quasi-empiricism conclude this chapter. These implications have no direct bearing on the actual practice of teaching mathematics in the classroom, but are mainly, if not exclusively, concerned with the image of mathematics that is conveyed in education. (shrink)