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Monday, May 22, 2006

Revealing reviews

Book reviews generally provide the most accessible accounts of practitioners' views of the workings of their field. The presence and lack of certain virtues are their constant themes. As an example, consider a review by Peter Johnstone of a mathematics book - Natural dualities for the working algebraist - reported here, which reveals something of his sense of the moral and intellectual virtues required of a mathematician. Here is an extract:
One might argue that the book's shortcomings are not of major importance in relation to its declared purpose. After all, it contains a wealth of detailed information on particular techniques for establishing duality theorems, and "working algebraists" (read: working universal-algebraists) will undoubtedly find it immensely useful to have all these techniques collected together in one place. I admit the force of that argument; but I also find it seriously worrying, for it carries the implication that universal-algebraists have given up the attempt to engage in dialogue with the rest of the mathematical community. As the authors remark at the end of their Preface, there is much that remains to be done in studying and classifying concrete dualities; but this book is not likely to inspire anyone outside the closed circle of universal-algebraists to take up the task. (my emphasis)
Any member of a tradition of enquiry should freely participate in such dialogue. Having noticed the absence from the bibliography of certain key references, Johnstone says he

... began studying the Bibliography more systematically, and soon realized its salient feature: all works written by category-theorists, or making serious use of categorical ideas, are excluded from it -- with the twin exceptions of Saunders Mac Lane's classic 'Categories for the working mathematician' (which, after all, the authors could hardly have left out, given their indebtedness to his title), and of Peter Freyd's 1966 paper on algebra-valued functors.
So, now an accusation of a lack of justice.


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