On History of Mathematics (2000-)

"I can attest to the benefits brought by the use of history of mathematics through my personal experience. The study of history of mathematics, though it does not make me a better mathematician, does make me a happier man who is ready to appreciate the multi-dimensional splendour of the discipline and its relationship to other cultural endeavours. It does enhance the joy derived from my job as a mathematics teacher when I try to share this kind of feeling with my class." (Siu Man-Keung, "The ABCD of Using History of Mathematics in the (Undergraduate) Classroom", 2000)

"The history of algebra shows us that nothing is more unsound than the rejection of any method which naturally arises, merely because of one or more apparently valid cases in which such a method leads to erroneous results. Such cases should indeed teach caution, but not rejection. For if the latter had been preferred to the former, negative quantities, and still more, their square roots, would have been an effectual bar to the progress of algebra. And think of those immense fields over which even the rejecters of divergent series now roam without fear! Those fields would not even have been discovered, much less cultivated and settled." (Gavin Hitchcock, "A Window on the World of Mathematics, 1870 Reminiscences of Augustus De Morgan - a dramatic presentation", 2000)

"The history of mathematics contains a wealth of material that can be used to inform and instruct in today's classrooms. Among these materials are historical problems and problem solving situations. While for some teachers, historical problem solving can be a focus of a lesson, it is probably a better pedagogical practice to disperse such problems throughout the instructional process. Teachers who like to assign a "problem of the week" will find that historical problems nicely suit the task. Ample supplies of historical problems can be found in old mathematics books and in many survey books on the history of mathematics. These problems let us touch the past but they also enhance the present. Their contents reveal the mathematical traditions that we all share. Questions originating hundreds or even thousands of years ago can be understood, appreciated, and answered in today's classrooms. What a dramatic realization that is!" (Frank J Swetz, "Problem Solving from the History of Mathematics", 2000)

"There are quite divergent opinions about the role the history of mathematics could play in the presentation of mathematics itself. A very common attitude is simply to ignore it, arguing that a deductive approach is better suited for this purpose, since in this way all concepts, theorems and proofs can be introduced in a clearcut way. On the other extreme, a rather naive attitude is to follow the historical development of a mathematical discipline as closely as possible, presumably using original books, papers, and so on. It is clear that both methods have serious defects." (Constantinos Tzanakis, "Presenting the Relation between Mathematics and Physics on the Basis of their History: a Genetic Approachl", 2000)

"Using history of mathematics in the classroom does not necessarily make students obtain higher scores in the subject overnight, but it can make learning mathematics a meaningful and lively experience, so that (hopefully) learning will come easier and will go deeper. The awareness of this evolutionary aspect of mathematics can make a teacher more patient, less dogmatic, more humane, less pedantic. It will urge a teacher to become more reflective, more eager to learn and to teach with an intellectual commitment." (Siu Man-Keung, "The ABCD of Using History of Mathematics in the (Undergraduate) Classroom", 2000)

"Using the history of mathematics as an introduction to a critical and cultural study of mathematics is one of the most important challenges for mathematics teachers and for students. There are many possibilities in mathematics education for the use of history [...]" (Lucia Grugnetti, "The History of Mathematics and its Influence on Pedagogical Problems", 2000)

"The most amazing event in the history of Greek mathematics has to have been the discovery of irrational numbers. This was not merely a fact about real numbers, which didn’t exist yet. It was a blow to Pythagorean philosophy, one of the main tenets of which was that all was number and all relations were thus ratios. And it was a genuine foundational crisis: the discovery of irrational numbers invalidated mathematical proofs. More than that, it left open the question of what one even meant by proportion and similarity." (Craig Smoryński, "History of Mathematics: A Supplement", 2008)

"The history of mathematics can be studied chronologically, thematically, topically, and biographically. I have used in this course elements of each approach." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The issue of rigorous foundations for calculus began with gropings in the early seventeenth century and concluded with a 'final' resolution in the 1870s. This rather slow evolution toward a logical grounding is not atypical in the history of mathematics. Rigor, formalism, and the logical development of a concept, result, or theory usually come at the end of a process of mathematical evolution. In the case of calculus, mathematicians achieved very impressive results during the seventeenth and eighteenth centuries by intuitive, heuristic reasoning, and therefore had no compelling reasons to put their subject on firm foundations. This does not mean that there was no concern during these two centuries for the logic behind the algorithms of calculus; and there were attempts, albeit unsuccessful, to supply it." (Israel Kleiner, Excursions in the History of Mathematics", 2012)

"The history of mathematical ideas is often very difficult to untangle, and because ideas evolve gradually over a long period of time it is impossible to draw an exact boundary between a given theory and its offspring. It is consequently a painful task to credit some developments to a small number of authors." (Barnaby Sheppard, "The Logic of Infinity", 2014)