The goal of this project is to construct a theory of evolution that is based only on assumptions that we have good reason to think are actually true. I refer to these assumptions as axioms to distinguish them from the simplifying assumptions that are the starting place for most models in biology. Of course, it is possible that our axioms could turn out to be wrong. Why, then, distinguish them from simplifying assumptions? The reason to make this distinction is that using only those assumptions that we think are true has consequences both for how we build our models and how we interpret them.
All scientific theories start with premises. Sometimes these premises are things that we think are actually true, but often they involve simplifying assumptions that we know are not exactly true, but that we are willing to make in order to achieve mathematical tractability. For example, models using evolutionary game theory typically ignore mutation. This is not because game theorists don't believe in mutation - they not only know that it happens but consider it an important factor in evolution. Rather, game theorists ignore mutation because they think that it is not relevant to the specific questions that they are asking, which concern the consequences of frequency dependent selection for phenotypic evolution.
Nobody uses game theory to study the distribution of neutral alleles in a population, any more than they would use coalescent theory to study frequency dependence. In each of these approaches - as in most model building in the rest of science - we first choose a particular question to ask, then identify the processes we think are important, then make appropriate simplifying assumptions to allow us to study these processes, and finally build the model.
In building an axiomatic theory, on the other hand, we start out with what we think is actually true about the system in question, then derive from this the mathematical rules that tell us what processes are important under what circumstances. The resulting mathematical descriptions can be complicated (though they can often be written in surprisingly simple form), and contain many different terms. In traditional model building this would be a drawback - all those terms complicating the interpretation of the model. For an axiomatic theory, though, the appearance of unexpected terms leads to new discoveries. Because we started out with only assumptions that we have good reason to think are actually true, each of the resulting terms must correspond to some real biological process - including some that we might never have thought of.
One such unexpected process is the amplification of the mean selection differential in small populations (described in Rice 2008). This is visible only when we relax the assumption that population size is fixed. Though nobody thinks that real populations remain exactly the same size from generation to generation, the assumption of fixed population size is used often in evolutionary theory because it greatly simplifies the math. It is only when we relax this assumption, and allow population size to be a random variable, that we see that some surprising evolutionary processes were hidden by the assumption of fixed population size.
Axiomatic theories thus allow us to discover processes that were rendered invisible by the assumptions used to make special case models (and simulations) tractable. In many fields we have no useful axiomatic theories. When we do have them, though, they serve to clarify the fundamental relationships between different processes, and facilitate the discovery of new processes that we did not expect, but that follow necessarily from the basic facts of the system.
Finally, in our papers we often first state the general equation that follows directly from our axioms, then look at simplified systems that isolate the effects of particular terms. We are thus still using simplifying assumptions in our analysis. The difference is that, because we started out deriving an axiomatic theory, the simplifying assumptions come at the end, rather than at the beginning, of the analysis - after we have seen what the exact general rules look like. Axiomatic theories, when we can derive them, thus also serve as formulas for generating special case models.
The axioms that we start with are as follows:
1 - The things that we are studying come in populations.
2 - These things leave descendants such that we could, in principle, identify which descendants came from which ancestors. (Note: An individual can count itself at a future time as a descendant. Also, ancestors and descendants need not be the same type of thing. For example, ancestors could be mated pairs of individuals and descendants could be individual organisms.)
3 - Both ancestors and descendants have phenotypes that can be quantified using real numbers.
4 - Each individual within a population has a probability distribution of possible numbers of descendants that it could leave after a specified time interval. This "fitness distribution" is causally influenced by both the individual's phenotype and its environment. (Note: This includes both biotic and abiotic environments.)
5 - Each individual has a distribution of possible descendant phenotypes. This distribution is causally influenced by the individual's phenotype, its environment, the transmission process (genetic and cultural), and the development of the descendants themselves.
Axioms
Jul 8, 2021