‘In the history of mathematics the twentieth century will remain as the century of topology.’ (Jean Dieudonné).
That remark may be something of an exaggeration; but perhaps not by very much. And, by any reckoning, philosophers of mathematics ought to be especially interested in the development of topology over the century. It provides such a rich set of case studies in the way new mathematical concepts emerge and are developed, and in the way that new problems and new methods become adopted as canonical.
The trouble, of course, is that it isn’t easy to get a handle on the conceptual development of topology. I.M. James has edited a History of Topology, which weighs in at over a thousand pages, comprising forty essays of decidedly mixed quality and interest. Dieudonné has written A History of Algebraic and Differential Topology, 1900 – 1960, another six hundred pages or more, much of it pretty impenetrable to anyone other than a serious topologist. Then there is a three volume Handbook of the History of General Topology — another daunting twelve hundred pages, and pretty difficult to extract out any nuggets of philosophical interest.
As far as I know, however, there is nothing previously published which does the job of Topology: A Conceptual History. This book doesn’t at all pretend to be a comprehensive and fine-detailed history, recording all the false starts and mis-steps and minor alley ways; it is more in the spirit of a Lakatosian rational reconstruction, done with verve and insight. And, by contrast, this approach does vividly bring out the main contours of some key conceptual developments in a way that e.g. the lumbering Handbook essays mostly don’t. Moreover we get to see this done at a level of some mathematical detail — it’s not just arm-waving — while still remaining relatively accessible (a modest amount of undergraduate mathematics should mostly suffice as background). So philosophers of mathematics will come away, for example, with at least some feel for what homotopy and homology theories are about — and hence acquire some better understanding of the way different areas of enquiry (in this case, topology and algebra) can hang together in deep ways. We also get along the way genuinely illuminating sketches of the contributions of major figures starting from Euler, Riemann and Möbius, noting a dozen more.
An impressive achievement, then, and done within a tolerable length too! So I’m going to greatly enjoy reading this book (and also following up the two or three seemingly well-chosen suggested short readings for each chapter). I plan to blog about all this here …
… if and when someone gets round to writing this needed Topology: A Conceptual History.