The level at which natural selection “acts” has been a much debated issue in evolutionary biology (Okasha 2006), especially in connection with the evolution of altruism. Two major kinds of explanations have been proposed: group selection theory (GST) and inclusive fitness theory (IFT). Marshall (2011) surveys the most common objections to the latter, claiming that the two approaches are in fact formally equivalent. We discuss exclusively his claim of equivalence. Rewriting of the Price equation (Price 1970) yields: 1) a version of Hamilton’s rule (a basic tool in IFT); 2) a partition of selection in between-group and within-group selection (typical of GST explanations). Thus, 1) and 2) are formally equivalent. If we assume that 1) holds, then also 2) holds, and conversely. Is this the same as saying that IFT and GST are formally equivalent? The author concedes that IFT cannot be identified with Hamilton’s rule. Thus the claim of formal equivalence must be, at least as far as this article goes, re-evaluated. On the other hand, to prove formal equivalence it requires an explicit set of axioms to be given. This request seems honestly unreasonable. Even in the case where such a formal model could be described, it remains dubious that many, or any, would agree on it. The word “theory”, as in “inclusive fitness theory” and “group selection theory”, should be better understood as a set of different but related explanations (for IFT alone, in addition to Hamilton’s rule and the Price equation, population genetics and evolutionary game theory have been useful approaches, among others). In particular, formal models are currently being used with a more modest goal: to describe more limited and well defined (agreed upon) aspects of the world, in order to produce testable predictions. On a final note, even if any two of these models will be found to be formally equivalent, we may have gained a more thorough “understanding” in the process. Mathematics also, after all, is more than the axioms that we put into it.

Bibliography

**Okasha, S**. (2006), Evolution and the Levels of Selection, Oxford: Oxford University Press.

** Marshall, J.A.R.** (2011), Group selection and kin selection: formally equivalent *approaches, Trends in Ecology and Evolution, 26 (7): 325-332.*

** Price, G.R.*** (1970), Selection and covariance, Nature, 227 (5257): 520-521.
(I thank all the participants in our last reading group for discussing and sharing ideas).*