Scott Page’s Diversity and Complexity is an ambitious book. In it, he draws upon interdisciplinary research — from mathematics, biology, economics, politics, sociology and other disciplines — to provide insight for a wide range of readers into how diversity affects complex adaptive systems.
Page begins by defining a common vocabulary. Diversity within a population comes in three types: “variation in some attribute, such as variation in the length of finches’ beaks, (…) diversity of type, such as different types of stores at a mall, (…) or differences in configuration, such as different connections between atoms in a molecule.” Complexity, on the other hand, consists of “diverse rule-following entities whose behaviors are interdependent,” such as the complexity that is produced from a bazaar in which people bring goods to trade. Establishing that diversity contributes to complexity is at the heart of Chapter 1. After all, as Page notes, “if the (bazaar) had no diversity, not much would happen.”
The first visible mathematics in the book arrives in Chapter 2. Here, Page provides a discussion of diversity measures so that researchers from nonmathematical disciplines may move from qualitative to quantitative comparisons. Chapter 3 identifies and explains the factors that produce and maintain diversity. For example, the anticoagulant dicoumarol — the main ingredient in the drug currently sold under the trade name Coumadin — was originally identified by an agricultural researcher as an effective rat poison. Transfer — a mechanism for producing and maintaining diversity — occurred when a medical researcher identified that dicoumarol could also be used to avert strokes in humans. In addition to transfer, Page identifies three other mechanisms: mutation, inversion and recombination. While he acknowledges that there is no “one-size-fits-all” model for considering the factors that produce and maintain diversity, he makes a solid argument that there are commonalities across disciplines.
“Why,” Page provocatively asks at the beginning of Chapter 4, are there “thousands of species of trees but only two species of elephants?” With this question, Page undertakes a discussion of constraints on diversity. The reader learns that relative demand, interactions, variation of selection, functionality and rate of adaptation all influence the constraints on diversity in a given complex system. Moreover, the interplay of these constraints on diversity must be considered. Diversity of species on an island, for example, is affected not only by the size of the island — “bigger islands can sustain more species” — but also by the distance of the island from the shore — “islands further from the shore will have fewer species.” The concrete examples go some distance toward grounding the technical discussion. However, the examples occasionally have the effect of leaving other questions unanswered. That is, in the discussion of species on an island, where do the small and remote but notoriously species-rich Galapagos Islands fit?
Chapter 5 focuses on the role that variation plays in enhancing the robustness — a notion that includes resilience — of complex adaptive systems. Chapters 6 and 7 focus on two benefits of diversity: averaging and diminishing returns. Here, the reader learns that diversity — even without synergistic interactions — is valuable on its own. For example, many believe that symbiosis — as when e coli simultaneously feed off us and help us digest our food — is always a byproduct of diversity. Page, however, cites examples in which such interactions are not necessary in order for diversity to produce a benefit. Of course, interactions may impact diversity and Page addresses this fact in Chapter 8. He ultimately concludes his discussion in Chapter 9 with a summary and questions for further research.
In writing Complexity and Diversity, Page had multiple audiences in mind, including individuals with nontechnical backgrounds who are interested in diversity and academics from a variety of disciplines in search of a common language. While the book succeeds in providing a “big” picture for generalists, Page does not shy away from providing technical details; the Central Limit Theorem, the Law of Requisite Variety, Fisher’s Theorem and the Price Equation, among other theorems, all have cameos. Not strictly a mathematics book, Diversity and Complexity will interest mathematicians whose work already intersects with other disciplines or who are interested in understanding how their work may inform other disciplines.
Overall, Page’s enviable supply of engaging examples balance the technical details well. At the same time, he is the first to acknowledge the inherent limitations in considering diversity and complexity from varying viewpoints. “An economy is not an ecosystem,” he writes in the preface, “and the human brain is not the Internet. The Internet doesn’t have a frontal lobe and the human brain doesn’t have email (at least not yet).” Nevertheless, Page offers a diverse and complex—if I may be permitted the redundancy — account of diversity and complexity that should inform a wide range of researchers and some interested laymen besides.
Susan D’Agostino is an Assistant Professor of Mathematics at Southern New Hampshire University. She has written articles for The Chronicle of Higher Education, MAA Focus and Math Horizons. She is currently writing a math book for a general audience.