How to Read Historical Mathematics

By Benjamin Wardhaugh

Benjamin Wardhaug's book is entitled How to read historical mathematics and this is precisely what it is about. It is an introduction to the marvellous world of the history of mathematics, aimed at the general public. You will not learn a lot either about mathematics or its history, but you will be much better prepared for reading old mathematical masterpieces after this book. Do not expect this book to give you a lot of answers – it won't. But it will teach you to ask the right questions.

It would be fitting for the title of this book to end in a question mark. Like most good science books, you have more questions when you finish it than when you started. In fact, this is the central theme of the book: what are the right questions to have in mind when reading a piece of historical mathematics? The book is structured into five chapters.

Chapter 1 is entitled What does it say? It teaches the reader to make sense out of the notation (or, frequently, lack thereof) used by early mathematicians. How does one go about translating old writing into modern mathematical terms? A few examples show convincingly that this is not a trivial task.

Chapter 2 deals with the questions that are most delectable for any historian, not only a mathematical one. How should we read an old mathematical text to find out about its author? Who wrote it, in what circumstances and why? Too often historical texts obscure their authors, and it is one of the tasks and pleasures of the historian to reveal the real people that stand behind the theorems.

Chapter 3 discusses old mathematical writings as real objects: books, manuscripts and random notes scribbled on random pieces of paper. "Disembodied" text (edited and published in a modern form) is not everything, since there are so many things that it does not reflect: the author could arrange the writing in a unique way, make corrections, drop a thought in the middle of a phrase and go back to a previous one. Besides, the history of books as material objects is an interesting pursuit in its own right.

In Chapter 4, we are reminded of the importance of the reader. It matters who read the book. What audience, if any, did the author have in mind composing his text? Was it a student textbook, a scholarly article prepared for publication, a private letter? Naturally, the proper understanding of the text will depend on this.

Finally, Chapter 5 is devoted to the most tortuous question of all: what pieces of historical mathematics should one read? In the vast multitude of texts, how can we determine what deserves attention? There are many factors to consider: the originality of the content, its influence on the course of mathematics, etc. Moreover, it is very rare indeed that when a mathematical result appeared for the first time, it was complete. The original formulation of many fundamental ideas lacked precision. Many theorems did not have proofs, at least not in the modern sense. Many of them are just not what they seem to be. It may take a very educated mathematician simply to recognise that a statement in the historical document is equivalent to a familiar theorem.

Written in an informal language, this book nevertheless feels like a textbook for history students. It pauses to let the reader think (look out for the incandescent light bulbs scattered over the text), and you get a nice homework assignment in the end of each chapter. The main conclusions, summarising the chapters, are put in boxes. You will not be lost in this book; it is easy to navigate.

This is a short work – only slightly over a hundred pages. Already because of its size, it cannot be more than an introduction. But it tickles the reader's curiosity and it opens up the field of historical mathematics, of which many people – even expert mathematicians – have only a rudimentary idea. Far from being a definitive text on the subject, this book provides an excellent starting point. It is well-suited for students and anyone who is not an expert but is interested in the history of mathematics.

Book details:

How to read historical mathematics

Benjamin Wardhaugh

hardback — 130 pages (2010)

Princeton University Press

ISBN: 978-0691140148

About the author

Ilia Rushkin is a Postdoctoral Research Fellow at the University of Nottingham. His research is in theoretical physics.