Possible uses of history in mathematical education.
This book recounts the development of the differential and integral calculus from the early 17th to the late 18th centuries, and their subsumption under the broader subject of mathematical analysis in the 19th century. It describes the progress up to the early 20th century, and also records the introduction and progress of set theory and mathematical logic during the forty years beginning around 1870. Special attention is paid to the close relationships of these topics, and their unfolding one into the other.
All these topics have been treated by historians of mathematics before, but the kind of treatment given here is unusual; for while my co-authors and I have discussed the mathematics involved in some detail, we also aimed at introducing the reader to its historical development. We hope that the book will be useful in mathematical education, and in ways that I shall now discuss.
During their undergraduate courses students learn in great detail substantial quantities of mathematics; but usually they are given little historical information on its genesis, or on the motivations which led to its creation. This tradition of teaching has its own history. As with many sciences, mathematics began to enter its modern professionalised state about two hundred years ago, and mathematical education became correspondingly institutionalised a little later. The establishment of the École Polytechnique and the reforming of the École Normale in Paris in 1795 were particularly important 'pace-setters' in educational practice for science throughout Europe, and the writing of textbooks based on courses at such institutions became a standard procedure. [Note: I wonder how the modern student would react to the teaching programmes meted out by these institutions. For example, at the Ecole Polytechnique around 1818 students were up for prayers at 5.30 a.m. and studied from 6.00 to 8.00, then after breakfast from 8.30 to 2.30 p.m., and after lunch and recreations from 4.30 (6.30 on Tuesdays) to 8.30. Bedtime was at 10.00. Sunday was the only day off this routine, though there were still studies of various kinds.] Indeed some branches of mathematics were stimulated in their development by educational needs; as we shall see, mathematical analysis was one of them.
The professionalisation of mathematics led to a vast increase in the number of research mathematicians and therefore in the amount of publishes work. In order to present to students the basis components of this expanding world in an intelligible form, teachers and textbook writers, who were not always prominent in research work, tried to present as well as they could the essentials of the particular branch of mathematics in question in an economical and rigorous form. This style of presentation had the advantages of cramming the maximum amount of reliable mathematics into the given space, and of preventing already large books from becoming unreadably long. However, it meant that the mathematics seemed to fall complete and 'perfect' onto the printed page; while perhaps clear in form, it was largely unmotivated and therefore difficult for the student to appreciate from that point of view. The same approach to education became standard in lecture courses, with similar strengths and weaknesses in the consequences for the student.
Put another way, in this tradition of mathematical education emphasis is usually laid on the accumulation of mathematical knowledge, on the amassing of the 'facts' which comprise a mathematical theory; it does not much consider the growth of mathematical understanding, the appreciation of why a mathematical theory developed and took its form, and not merely that it does have its content. Now the history of mathematics may be of benefit here; for after all, any mathematics the result of human endeavour in the past, and strands at least of its original motivations may well still be relevant or have educational value. How much use can be made of historical aspects, and the manner in which they can most profitably be employed, are difficult questions which will not be pursued here. But it should be stressed that, while students should obtain a better understanding of mathematics (in the sense of 'understanding' mentioned above), they cannot expect the mathematics to be 'easier' to do. For example, if this book is read carefully, then it will be quite taxing in places, and pencil and paper will be needed. The rewards of understanding will, it is hoped, form sufficient compensation.
The book and its readers.
It is hoped that undergraduates can use the book, especially in the later years of their course; but this is not the place from which the mathematics can be learnt. Such training must be obtained from books and courses on the calculus, mathematical analysis, set theory or mathematical logic, to which this book would serve as collateral reading. Naturally students may not be taking courses in ll these subjects; they will then pick out the chapters of closest relevance to their studies, and possibly look at the others out of general interest. The book should also prove useful to post-graduates, teachers and research mathematicians in these branches of mathematics or in other related subjects, and to all who are acquainted with these fields and wish to obtain some impression of their historical development.
I shall now point out some limitations of the book. Firstly, specialists in the history of mathematics will find some omissions of detail and over-simplifications in historical interpretation, which they must accept as consequences of the policy followed in designing the chapters. In particular, the discussions usually concentrate on the principal mathematicians and their main (usually published) writings, consigning to passing mention or even silence their relevant manuscripts and the work of lesser figures whose work was apparently, if undeservedly, neglected in its time. However, minor work is sometimes discussed when it provides interesting historical insights.
There is also a problem about languages. We have translated all quotations into English where necessary, but unfortunately no translations (or only unreliable ones) exist for many of the original sources. Further, we have cited worthwhile historical literature as locations of further information on a particular topic irrespective of the language in which it happens to be written. In fact, for the areas of mathematics with which this volume deals, much of the original material and the best historical literature was and is written in Latin, Russian, French and German.