ST JOHN’S COLLEGE, UNIVERSITY OF OXFORD
The Deep Mathematical Theory of Selfish Genes
The College intends to appoint two two-year Fixed-term Research Associates to work in its Research Centre. The college will host a further two years of the project and provide matching funding, if external funding can be obtained. The successful applicants will work with Professors Alan Grafen and Charles Batty on an abstract mathematical project that underpins and develops modern Darwinian theory.
About the project
Alan Grafen has been publishing papers on ‘formal Darwinism’ since 1999. The quickest way to appreciate the project to date is to consult
1) a simple verbal account of the work to date (A. Grafen 2007. The formal Darwinism project: a mid-term report. Journal of evolutionary Biology 20, 1243-1254.
doi:10.1111/j.1420-9101.2007.01321.x), available at
2) an elementary mathematical introduction (A. Grafen 2008. The simplest formal argument for fitness optimisation. Journal of Genetics, 87, 421-433. doi:
10.1007/s12041-008-0064-9 ) available at
3) a reasonably complete form of the argument but using finite mathematics for ‘inclusive fitness’ (A Grafen 2006 Optimization of inclusive fitness. Journal of theoretical Biology 238, 541-563), available at
4) a paper using the heavier mathematical tools likely to be required for future work, applied to the problem of allowing the existence of different classes of individual (A. Grafen 2006. A theory of Fisher’s reproductive value. Journal of mathematical
Biology 534, 15-60.), available at
The work now requires collaboration with mathematicians, in order to tackle more abstract problems, to solve them more efficiently, and to publish the results in a mathematically acceptable way. This has brought about the involvement of co-applicant Charles Batty, a pure mathematician, and this advertisement for two research associates.
The postholders will be involved in linking the mathematics of motion with the mathematics of optimisation at an abstract level. The tools will include measure theory and Markov processes over compact Hausdorff spaces. The goals of the project are to unify, formalise and extend existing results into a rigorous and general statement about the connections between natural selection, on the one hand, and optimality in the design
of organisms, including a precise definition of ‘optimal’, on the other.
The rest of this section gives some background as a shorter alternative to reading the four papers given above. The concept of fitness optimization is routinely used by field biologists, and first-year biology undergraduates are frequently taught that natural selection leads to organisms that maximize their fitness. Dawkins’ The Selfish Gene (1976) promoted a conceptual integration of modern evolutionary theory in which genes are viewed as optimising agents, which is extremely influential and widespread today and encompasses inclusive fitness theory and evolutionarily stable strategies as well as general optimality ideas. However, mathematical population geneticists mainly deny that natural selection leads to optimization of any useful kind. This fifty-year old schism is intellectually damaging in itself, and has prevented improvements in our concept of what fitness is. One underlying
cause is that the link between natural selection and fitness optimization is much more sophisticated than the usual optimization principles associated with dynamical systems, namely Lyapunov functions and gradient functions.
The aim is to formalize relevant links between the mathematics of motion (representing the known process of gene frequency change as the dominant mechanism of evolution) and the mathematics of optimization, in a rigorous way. Generality is important, as a major aim is to find mathematical arguments that match Darwin’s verbal arguments in the Origin of Species, as well as Dawkins’s verbal arguments in the Selfish Gene and later works. This requires, for example, the use of measure theory to represent populations and to represent uncertainty.
The aim is to model fully the core of the underlying verbal arguments, so that we can be confident there is ‘nothing left in them’.
The construction of such a framework will have major implications for the conceptual basis of biology. For example, the understanding of organisms as agents is given a rigorous meaning, which is of interest to philosophers. The definition of fitness, i.e. the quantity that organisms are assumed to maximize, is often controversial in biology, and this project will settle many, at least, of those questions. To take one simple controversial example, when there is uncertainty in fitness, will natural selection lead to organisms that maximize the geometric, or the arithmetic, mean over that uncertainty? Thus this highly abstract mathematical project will have significant implications at many different levels in biology. It will also be of interest to historians of science, as it will claim to show the underlying logic of Darwin’s great insight and of Dawkins’ conceptual unification.
For those seeking more information on the work to be done, excerpts from the grant application are provided at the end of this document.