The Statistics of Sin in Hominid Populations: Checking the Math on De Cruz’s Use of the Price Equation (Part One)

April 19th, 2018 by

The Statistics of Sin in Hominid Populations: Checking the Math on De Cruz’s Use of the Price Equation (Part One)

2018 marks the third year of the Analytic Theology project at Fuller. This year’s theme is theological anthropology. It would have been unfortunate if the year had passed without anyone addressing the question of humanity’s struggle with sin. Happily, this topic was boldly engaged by Dr. Helen De Cruz in a presentation titled, “Transmission of Original Sin: A Cultural Evolutionary Model.”

Dr. De Cruz’s presentation proposes a social transmission model of original sin to answer two questions in hamartiology. First, how is sin transmitted from generation to generation (i.e the mechanism of transmission)? Second, what should we make of the doctrine of original guilt?  How can humans be blameworthy, if at all, for the sins committed by humans before us. You can watch her presentation online here:


De Cruz’s presentation is organized into six sections. Part One introduces the topic of sin theologically by contrasting Augustinian and Eastern orthodox approaches to original sin.  Part Two presents three models of transmission: an Augustinian model, a Federalist model, and a social transmission model. De Cruz further focused on two leading examples of a social transmission models, that of Friedrich Schleiermacher and Walter Rauschenbusch. In Part Three she enumerates recent empirical studies that lend support to a social transmission model. These include research from developmental psychology and sociology on topics such as over-imitation, promiscuous normativity, and child development.  Part Four of the presentation turns to paleoanthropology to locate the earliest possible point in hominid history offering evidence of the capacity for “God consciousness” necessary for moral responsibility. In other words, if a transmission model of sin is correct, at what point might the transmitting of sin have begun? In Part Five of the presentation De Cruz uses the Price equation to mathematically model the spread of negative or positive behaviors throughout the early human population. She closes with implications of the model for ascribing guilt and blame to individuals.


Given our limited space, I want to focus on the more novel aspect of De Cruz’s talk; her use of the Price Equation to model the transmission of sin throughout early hominid culture.  In her defense, De Cruz made superb use of presentation time to cover a wide variety of issues. She states clearly that she can only briefly explain how the Price equation portrays the Social Transmission account of sin.  However, in her talk, she refers us to a 2004 article by Joseph Henrich as the place to look for an explanation of how the equation works. Having looked carefully at Henrich’s paper, two significant questions arise about how she uses the equation. This first blog (Part One) will attempt to help the reader understand enough of Henrich’s project to (a) see and (b) think through a possible faulty assumption De Cruz is making in her otherwise impressive presentation.  A second blog (Part Two) will continue to unpack Henrich’s project so as to raise a possible (and more serious) mistake De Cruz may have made in her use of the price equation.

Joseph Henrich’s 2004 article analyzes the gradual decline of tool usage by the indigenous population in Tasmania between the last ice age and the arrival of Europeans.[1] He attempts to give an account of the factors affecting this decline of more advanced cultural artifacts (e.g. the ability to make complicated hunting tools) in this population vis-a-vis other indigenous groups on the Australian. At the end of the last ice age, sea levels rose, cutting off the Tasmanians from the mainland. Gradually they lost the ability to make advanced tools while mainland populations did not.  To account for the decline of skills Henrich employs a version of the Price Equation, an equation introduced by George Price in the 1970’s as a way to account mathematically for the rise or fall of traits in evolving populations.[2] The equation has wide applicability in population stirs and various sciences.

∆z̅= Cov(f,z) + E(f∆z)

First the equation, briefly. The bottom line is the equation’s answer, represented here by ∆z̄. This is the change over time (∆; e.g. from one generation to the next) in the average ability of a population (i.e. the bar over the z̄) to use an advanced skill like making a special fishing net.[3] If ∆z̄ is above zero for multiple generations in a row, that means that from generation to generation the skill is spreading through the population. If it is below zero, multiple generations in a row, the population is forgetting how to use that skill. The equation captures this sort of change over time.

Our first question deals with the left term in the equation (i.e. Cov(f,z)). De Cruz mimics an assumption that Henrich makes in his paper – about  Cov (f, z). This assumption may not be warranted for De Cruz. Cov(f,z) is a covariant between z and f.[4] z is a quantitative number given to each Tasmanian to represent how good he is at performing certain skill, and f is a number representing how likely others are to mimic that person in their own attempts to learn how to perform that skill. Henrich simplifies his equation by assuming that any time z is the highest out of all the other z values for the population, f can be set to a value of one. In plain English, he assumes all the Tasmanians always copied the artisan with the best skills (i.e. the highest z value). This allows him to simplify his equation to:   ∆z̅ = zh – z̅ + Δzh .[5]

In De Cruz’s presentation, she assumes that z will represent the capacity to perform the overall moral norms of a hominid community, rather than the capacity to make some particular tool or use a particular fishing skill. The Price equation can accommodate that. However, she then assumes that she too can set the f value (she uses the letter c instead)  to to one.[6]  She states in her presentation: “Assume that every member of a hominid community strives to fulfill the normative ideals of their community (the highest possible z-value in the community), zh. So, we’re not talking about some sort of absolute moral ideal but just the sort of ideal that you have within a community.”[7] Then she proceeds to set her f value to one like Henrich and zero for any lower z value. De Cruz then goes on to simplify her equation to  ∆z̅ = zh – z̅ + Δzh   like Henrich does. Is this assumption and simplification warranted in her case?

In Henrich’s case, the zh was the person in the community with the highest skills (h subscript stands for highest) at performing some tool making craft. For De Cruz, zh would have to be the person in the community with the highest display of moral norms. Now, for our question: is it right for De Cruz to assume that all hominids are interested in copying the person with the highest moral norms? It seems plausible that most hominids in a population would want to copy from the best fishing net maker or the best crawfish diver (i.e. zh ).   Can De Cruz assume that, analogously, all the hominids in a population would try to mimic the hominid that best displayed the moral norms of the community? The disanalogy seems significant enough to raise a shadow of doubt. The motivation to learn to make the best spear versus approximating a community’s most moral member” seem too different. If this worry is right, then De Cruz is not warranted in simplifying her c ( Henrich’s f) to a value of one any time zh is plugged into Cov (f, z) and zero when any community member other than zh is plugged in. If she is not warranted, she can’t simplify her Price equation to ∆z̅ = zh – z̅ + Δzh like Henrich does. If she can’t simplify, then it doesn’t seem she can make use of his subsequent steps and arrive at the final conclusion about what the Price equation “predicts.”

We could conclude our discussion here. However, a second and more worrisome problem from De Cruz’s use of the Price equation begs to be addressed. To get to that point, we have to assume that the assumption addressed above goes through. Therefore, let’s conclude this blog, instead, by granting De Cruz’s assumption so that we can turn to a more interesting question in a subsequent blog. That question is whether the Price equation, as used by Henrich, actually predicts a more Pelagian outcome of the transmission of sin.



[1] Henrich, Joseph. “Demography and Cultural Evolution: How Adaptive Cultural Processes Can Produce Maladaptive Losses—The Tasmanian Case.” American Antiquity 69, no. 02 (April 2004): 197–214.

[2] See Gardner, Andy. “The Price Equation.” Current Biology 18, no. 5 (March 2008): R198–R202.

[3] Imagine giving each person in a population of 1000 Tasmanians a number (e.g. 1 to 10) that represents their ability to make an advanced fishing net. Average all 1000 numbers.

[4] Covariance is simply a mathematical way of showing how two sets of data vary in relationship to each other. Imagine a table with monthly ice cream sales in the first column and average temperatures in the second column. These data points are related; they co-vary together. As temperatures rise, ice cream sales rise. They co-vary positively (i.e. their covariance approaches one). If in the second column we had winter-coat sales we would expect them to co-vary negatively (i.e. the covariance would approach  negative one) . As temperature rose, coat sales would drop. The closer covariance reaches zero, the less relationship there is between the data sets (e.g. relationship between masking-tape sales and temperature should give a covariance close to zero).

[5] See Appendix A in Henrich’s article.

[6] Remember that she uses c. Again, this is a quantitative number that indicates how likely someone is to mimic zh – the person in the community with the highest z value.

[7] Quote taken from De Cruz’s presentation slides.

Jesse Gentile is a new PhD student at Fuller seminary studying systematic theology. He has interests in theological anthropology, epistemology, ethics, technology and pretty much everything else. Jesse is the father of two awesome elementary school kids and is the husband of Ella (who works as a wills and trust attorney). He regularly does itinerant preaching among plymouth brethren assemblies throughout the U.S. Jesse hold’s degrees in Biblical Studies, Philosophy, and Instructional Design.

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