This long awaited book is important and will probably be widely read. I t summarizes the recent biomathematical work of the Fields medal-winning topologist who wrote it. It applies recent work on differentiabte (C ®) mappings to a wide variety of questions of general biology, among them embryology from which the name in part derives. The book's own embryogenesis is out of the Serbelloni conferences organized by C. H. Waddington, who contributes a preface.
The work may, for the purpose of review, be divided into three sections: (a) the methodological apologia of the first two chapters; (b) the mathematical developments of Chapters 3-5 and the summary; (c) the applications (Chapters 6-13) that comprise the remaining two-thirds of the book.
For Thorn, the great success stories of quantitative science, celestial mechanics and electromagnetic field theory, are quite atypical. Most real systems include so many parameters and are so little known or even knowable in their details that one cannot hope to discuss these in similar terms. I n particular, this applies to all biological systems. The older approach to this dilemma is the well-known methodology of model construction. Thorn's approach, not quite antithetical to this, nevertheless plays down the role of detailed models. For Thorn, a better hope of a general (and mathematical) biology lies in the application of robust qualitative theorems. These he sees as essentially topological in character.
The late Nicolas Rashevsky sometimes spoke of biomathematics as still being in its pre-Newtonian era. By this he meant that radical insights and changes of essential viewpoint such as characterized the Newtonian revolution in physics have not as yet appeared in biological thinking. Thorn's argument takes the same course, but in a paradoxical way. He sees progress in mathematical biology as occurring only if we return to an earlier (his: torically pre-Newtonian) outlook. "Descartes, with his vortices, his hooked atoms, etc., explained everything and calculated nothing; Newton, with the inverse square law of gravitation, calculated everything and explained nothing."
Thorn adopts the Cartesian view. tie rejects the hypothesis, which he stresses remains a hypothesis, of physico-chemical reductionism that has come to characterize the mechanist world view, without, however, reverting to the vitalist concept. He is at pains to stress that other possibilities are available here. I-Iere the thought is not new. The direct indebtedness is to Waddington, but similar views have been expressed by Rashevsky and, notably, by A. N. Whitehead.
The mathematical sections of the book do noi make easy reading. They summarize a body of theory known under various names, but perhaps most widely under the slightly misleading term "catastrophe theory". (The French term "catastrophe" is used in its sense of ddnouement rather than its alternative meaning, "disaster". Some authors coin the English word "catastrophy" to meet this point.) This relatively recent theory is not anywhere the subject of a really elementary exposition, although there is available a recent article by Callahan in