Abstract:  Consider an evolutionary process.  In most genetic inheritance and in human cultural systems each new offspring is atypically assigned to be produced by a specific pair of the previous population. This form of mathematical arrangement is called a surjection.   We have thus briefly described the mechanics of genetics – physical mechanics describes the possible forms of loci, and normal genetic statistics describe the results as viability of offspring in actual use.  But we have also described much of the mechanics of mathematical anthropology.  Understanding that what we know as cultural inheritance is also the result of finding surjections and their consequences, is useful in understanding, and predicting, biological, as well as cultural evolution. especially in cultures for the expected populations. The review below shows us that in describing evolution, two sorts of statistics are required: those that follow specific inheritance of actual surviving individuals from specific matings, and those which follow population averages on a larger classes of individuals. 

The unewest part of this discussion are the comments and citations to the work of Lila Gatlin in Parts 8 and 9 of the text, plus the observations of Petoukhov and Ballonoff in Section 10.  (This version of text submitted on July 17, 2020derlying probabilities of each in a population, given the underlying surjections, is that the mathematics of these probabilities can be understood by finite Hilbert spaces.   This does not mean “quantum mechanics” predicts both biological and cultural evolution, thought citations below shows that quantum physics is in fact responsible for the underling group theory that creates biological evolution.  We especially note the brilliant work of L. Gatlin in the 1960’s describing how biological evolution is a necessary part of physics. The parallels to cultural evolution are instead in the relation to how mathematical group theory can affect the potential structuring of subsequent generations.)