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REVIEWS Contents Evolutionary Dynamics REVIEW
Evolutionary Dynamics (Exploring the Equations of Life) by Martin A Nowak, Harvard University, 2006, 384 pp, £22.95, ISBN 0-674-02338-2. Its pleasant to be asked to review a book one would probably have purchased anyway and this is a well-presented, accessible text with attractive typesetting, clear illustrations, and a direct, narrative tone. The author attempts ambitious compression, covering a wide variety of topics in fourteen swift, and semi-independent chapters averaging only twenty-two pages each. At the end of this book, you are left surprised by how much is achieved, and the notable omissions – such as, a proper look at genetics or at sexual selection are at least acknowledged honestly. The book closes with a well-chosen list of references: most being easily accessible from university libraries. I found the detailed chapter commentaries, relating what was said to what was referenced, more than made up for any carping I might have made at what was left out. The author's aim is to 'present those mathematical principles according to which life has evolved and continues to evolve'. It commences with Darwin's attractive quotation: 'I have deeply regretted that I did not proceed far enough to at least understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense.' Coincidentally, I had recently come across exactly the same quotation in Nahin's book ‘Dr Euler’s fabulous formula', and while it might make mathematicians smile shyly, you can’t help wondering about Darwin's conception of these great leading principles and why it was, if they were so great, that he took no further steps in the direction they were leading. The Darwin quote that I have been chuckling over for years is that 'a mathematician is a blind man in a dark room looking for a black cat which isn't there', and while this is almost certainly apocryphal, its interesting to speculate about the form 'the evolution of species' might have taken had Darwin been more temperamentally mathematical, or even whether it could have been written at all. A novel feature of this book is the importance it gives to language. This was the area that I am ashamed to say was most unknown to me, and which left me still puzzling over what I had read. The notion that Gold's learning-theory work could in some sense support Chomsky's early views about Universal grammars has sent me scurrying to Gold's original papers (readily available from the web). I’m not convinced but I am intrigued and I cannot see why one could not employ much the same argument in relation to the recurrent feeling physicists have that they are close to a theory of everything, or, more grandly, to support a realist view of mathematical structures. Other areas that are well treated include evolutionary game theory and birth and death processes in finite populations. This book covers a wide range of topics in a clear, introductory way. It would be supplanted by detailed study of any of them, but it provides an attractive and coherent view of them all. It would be a good book to give a potential post-graduate student who wanted to have some understanding of the ideas that he or she would be studying. It would also make an attractive addition to a university library or the bookshelf of anyone interested in a modern treatment of the promised lands where mathematics and biology meet and negotiate. Bob Lockhart
Gödel's Theorem. An incomplete guide to its use and abuse by Torkel Franzén, Wellesley, Mass.: A.K. Peters, 2005. ix +172 pp, £15, ISBN 1-56881-2338-8. Kurt Gödel (1906-1978) published his two theorems on the incompletability
of axiomatised first-order arithmetic (FOA) in a paper in 1931. Specialists
in mathematical logic and the foundations of mathematics absorbed them
fairly quickly, but the mathematical community in general did not pay
significant attention until the late 1950's The first theorem states that there exists at least one proposition P that can be stated in FOA such that neither P nor not-P is provable in it. The theorem (in a slightly strengthened form, due to J.B. Rosser in 1936) assumes that FOA is consistent; the second theorem (also sometimes called the 'corollary') then states that this consistency cannot be proved within FOA itself (p.34). Of course, if FOA were inconsistent then its consistency could be proved within it, since every sentence of FOA could then be proved. Part of the point of the second theorem is that the metatheory of FOA has to be richer than that of FOA, at least for the needs of this proof. The web of self-reference involved here is very characteristic of both theorems, and a major source of the misunderstandings; the author might have emphasised the distinction between theory and metatheory a little more, since recognition of its central importance is itself a notable aspect of Gödel's paper. Among the sources of misunderstanding treated by the author, the following are brought out well. (1) The theorem is concerned with (part of) arithmetic, not with axiomatised mathematical or logical theories in general, some of which are indeed complete. (2) The sense of incompleteness in the first theorem, as stated above, is called 'negation incompleteness' by logicians, and should not be confused with other senses. (3) The first theorem says nothing about whether P or not-P is provable using means outside of FOA, or whether either proposition is known in some other sense. (4) Nor does it directly address the scope, and especially the limitations, of consciousness, or of other sciences such as physics, since forms of (in)completeness may obtain there that are not based upon arithmetic. The author reviews these and several other pitfalls, and provides a useful bibliography (from which, however, Gödel's paper is missing!). My only reservation about his treatment concerns axiomatic set theories, which are important examples of incompletable mathematical theories. He says of one version (ZFC) that its axioms are 'utterly compelling' (p.105); but for example, the axiom of choice is on the list, so that this claim could be disputed. Indeed, on p.151 he rehearses some of the doubts about forms of the axiom of infinity there. While the text assumes some familiarity with the general context and techniques of foundational studies, non-specialist readers should gain a fine impression of the issues and confusions that arise when a technical theory becomes too well known! It is much to be regretted that the author died soon after his book appeared.
I. Grattan-Guinness
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They were then also diffusing
among some philosophers, as well as in the rapidly developing profession
of computer scientists; and especially from the late 1970s they were
becoming widely enough known to serve to all and sundry as an icon for
negative results, impossibilities, and so on. Familiarity bred the usual
inability accurately to understand their content and consequences. The
book under review is a welcome tourist's guide not only to the correct
but also to many incorrect interpretations of the theorems, both in
their immediate contexts and in wider circumstances.