## Topology: A Conceptual History

‘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 earlier 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 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). So philosophers of mathematics will come away, for example, with at least some feel for. . .

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