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Inheritance is a Surjection: Description and Consequence
Paul Ballonoff
1  Ballonoff Consulting


Consider an evolutionary process. In genetic inheritance and in human cultural systems each new offspring is 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 inheritance is the result of finding surjections and their consequences, is useful in understanding, and perhaps predicting, biological, as well as human, evolution.

Keywords: surjection; biological evolution; mathematical anthropology; predicting biological evolution; predicting cultural evolution; mathematical groups
Comments on this paper
William Bruce Sherwin
Balonoff Talk
That inheritance can be thought of as a surjection seems reasonable. What is less clear is whether there are cases where treatment as a surjection gives rise to better fit between theory and data, better forecasts, etc? Are that any such examples?
Paul Ballonoff
Thanks for the interesting comment.

In part, there are some areas in human kinship for which the use of Stirling Number of the Second Kind (SNSK) allows some predictions on average family size, not otherwise available. In the text to my article, see the references to my previous articles with citations references [84. 85, 87, 88] which give some examples.

But also I suspect your questions also more more explicitly related to genetic inheritance. For those relations, the inheritance is what actual survivors of specific parents shows, not an inference from the SNSK relations. But on this, I have also now seem your own articles in Entropy in 2018 and other recent journals, and also, the useful invited paper of Pritam Chanda in this conference. Both are dealing with only genetic inheritance. Chanda cites the work of L. L. Gatlin book of 1972, which implies that use of entropy measures means evolution takes place by information, but also similar to thermodynamic results. Your article in Entropy in 2018 says entropy and information imply that one science can cover both genetic evolution and also evolution of ecology. Some of the articles cited in your papers imply that entropy measures alone can predict genetic evolution.

This implies that we might consider when both the SNSK measures on average family sizes (which also relate to communications) can interact with purely genetic measures, and cause genetic inheritance from somewhat "'self sacrificing" individuals. In your text for example is a reference to Joan Silk, an anthropologists but citing many diverse animal examples, and in which many of those examples also discuss animals that in some way assist survival through increasing the population of animals of somewhat similar genetics by association, not direct in gene transfer from the "self-sacrificing" person. In the SNSK example, we also find that we can get assistance by non-genetic transfer, such as by assisting "adopted" offspring for example. Thus in the human kinship example, and in many of Silk examples, we can get inheritance of things that make the species more "communal" in some form. I think that Silk examples imply this but don't in that text the details to prove from the details cited, that the effects of SNSK cause it; but they apparently imply it.

As I understand the process of ECEA, there may be a publication later of papers presented. My intent is to follow the line just discussed, and analyse in an addition to the paper so far presented, to submit in that later version, if the combination of the two approaches, makes both your 2018 article correct, and also show that Gatlin's 1972 book, predict the same result, and that the notion of surjection, from both the genetic inheritance but also the SNSK results (which incorporate communications) help cause us to see they are essentially the same result.

Anyway, that is my immediate thought on your question.