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Foreword

Variation, Evolution, Heredity, and Mathematics in the 21st Century

All three non-adaptive forces of evolution—mutation, recombination, and random drift—are stochastic… and can generally only be understood in probabilistic terms. It is well-known that most biologists abhor things mathematical, but the quantitative details really do matter.

—Michael Lynch

In the twenty-first century, the concepts of variation, evolution, and heredity have influenced science, technology, and the public imagination in ways that could never have been imagined by their developers. Evolution has been embraced by the public to explain complex transformations in everything from organisms to economies, and in the process, has divided the public spheres on issues as diverse as religion, science, and public policy. Variation and heredity have similarly become part of our modern social and cultural awareness. As our capabilities to modify genes in plants and animals grow, so do the difficulties of our deliberations over whether and to what extent we should bioengineer our way to a better world.

Though we easily recognize how these ideas influence our social and cultural landscape, most of us rarely consider how they transform science. This task falls to historians, philosophers, and sociologist of science. However, even scholars in these fields have not considered all of the consequences of these notions. One of the effects that has not been explored is the impact of these ideas on the development of argument in the biological sciences. This book examines how the concepts of variation, evolution, and heredity, introduced by Charles Darwin and Gregor Mendel, transformed argument in the biological sciences by encouraging the growth of mathematical argumentation.

Mathematics and Modern Investigations of Variation, Evolution, and Heredity

Unlike scientific ideas, which regularly filter into the public’s awareness, mathematical aspects of scientific argument tend to develop quietly and anonymously. Despite their low profile, they are ubiquitous and in modern investigations of variation, evolution, and heredity. By going behind the scenes of current research in these fields, it is possible to illustrate just how important they are.

The extent to which these research fields have come to rely on mathematics is nothing short of extraordinary. At the dawn of the twentieth century, very few researchers investigating these phenomena would have been employing mathematical methods or arguments. However, in the twenty-first century, these methods pervade their work. This pervasiveness is evidenced by the spectacular growth in the last fifty years of mathematical fields of study related to these phenomena, including: population genetics, molecular genetics, biostatistics, bioinformatics, computational biology, and quantitative genetics. The ubiquity of mathematics is also evident in the range of subjects that are being examined using quantitative methods. According to Alan Templeton, a professor of population genetics at Washington University in St. Louis, mathematical models are currently being used in a variety of research areas, including “wildlife conservation projects, research assessing what it means to be human, and investigations tracking the historical development of disease.”

One publicly salient application of mathematics to the study of variation, evolution, and heredity has been the use of DNA to track the historical migrations of human populations as they spread out of Africa. This subject has been the focus of attention in a number of works in the popular media, including books such as Steven Olson’s Mapping Human History and Brian Sykes’s The Seven Daughters of Eve, websites like Wikipedia’s “Human Evolutionary Genetics,” and televised specials like PBS’s Journey of Man. In all of these media, however, the role of mathematics in the science is invisible. Closer scrutiny of these popularizations, though, offers a sense of the true extent to which mathematics contributes to the science that captures the public’s imagination.

In the television documentary Journey of Man, for example, English geneticist Dr. Spencer Wells travels the world tracing the hereditary path of our human ancestry by following the physical route by which it migrated out of Africa. As is the case for most popularizations of science, the main focus of the documentary is on the human story. Though it gets second billing, science does appear throughout the documentary. Before Wells leaves on his trip, for example, he visits his geneticist mentor, Luca Cavalli-Sforza, to talk about the foundations of research into human genetic variation. He also takes breaks in his travels to explain key scientific points, such as what scientists know about gene change over time and how it helps them establish relationships between modern humans and their progenitors.

However, the mathematical work that makes identifying these relationships possible receives only the briefest of acknowledgements. Interspersed throughout the documentary are visual images of peaked line graphs on computer screens. In addition, Wells makes brief reference to “the clear data” that has sent him to Africa in search of the Kalahari bushman whose genetic heritage represents the starting point of the human journey of geographic expansion and genetic diversification. The obliqueness and briefness with which the documentary treats the contributions of mathematical argument creates the impression that it played almost no substantive role in understanding the spread of our ancestors. In reality, however, Wells’s trip would not have had a scientifically supportable itinerary without quantitative data and mathematical methods for managing, comparing, and analyzing that data.

For example, to establish the chain of genetic ancestry from fixed mutations in the Y chromosome—which Wells uses as the scientific basis for his travels—thousands of blood samples taken in the field would first have to be processed. In the initial phases, chemical and physical procedures would be used to extract and precipitate DNA. Once the DNA had been extracted, it would be “unzipped,” bonded to other known bits of DNA, and run through a process of electrophoresis where it would be separated out and read by a laser.

Once the DNA was scanned and identified, mathematics would take on its essential role in the science. The information, read by laser from the DNA, would be stored in a database whose architecture would not be possible without the use of complex mathematical algorithms. Then this information would be compared to other samples in large databases, again using sophisticated algorithms. To determine the general relatedness of the sample of DNA to a population group, researchers would apply formulae to calculate the probability of the DNA’s belonging to a particular group based on the absence or presence of certain genetic markers. Finally, scientists would establish the place, say for a Kalahari bushman’s Y chromosome, in the larger sequence of genetic diversification amongst human population with another set of formulae. These formulae would be used to detect the absence or presence of key mutations in the bushman genome and to compare them to the mutations present or absent in other human populations.

Because of long-term efforts to gather genetic data and improve the speed of its analysis, scientists now have more information relevant to investigating variation, evolution, and heredity than ever before. The extent of this data and the type of inquiries it supports means that research such as the kind popularized by Wells cannot be conducted without mathematics. Its necessity is evidenced by the emergence and coalescence of a number of mathematical subfields in modern biology—such as bioinformatics, molecular genetics, population genetics, biostatistics, and statistical genetics—dedicated to meeting the needs of a quantitative science (Templeton). Researchers in bioinformatics, for example, devote their efforts to developing databases, algorithms, and statistical and computational techniques for analyzing and managing massive data sets. With the complete sequencing of the human genome and other important organisms, the amount of genetic data that needs to be organized and synthesized has grown. The human genome, for example, has between twenty and twenty-five thousand genes and other functional elements, with an estimated three billion base pairs. To collect this data set, the institutions working on the Human Genome Project collaboratively sequenced genes for fifteen years. Computer scientists in bioinformatics employ their mathematical skills to develop more powerful algorithms for ensuring that data on this scale can be properly stored and retrieved for scientific research.

Whereas some biomathematical researchers devote their talents to managing data, others use their mathematical skills to develop formulae to pose and solve important questions about variation, evolution, and heredity, such as how closely species are related and how diseases have emerged and developed over time. Molecular geneticists, for example, might test hypotheses about the degree of relatedness between organisms by developing genetic taxonomies or gene trees. These trees require special algorithms designed to calculate the proximity of organisms to one another based on their genetic divergence in some physical trait. For example, a molecular geneticist might compare the order of amino acids in the red blood cells of humans, pigs, mice, and chickens. Using one or more of a handful of standard mathematical methods for calculating relatedness between organisms, he/she would conclude that, evolutionarily speaking, humans are closer to pigs than chickens (Hartl and Jones 612–13).With these methods, molecular geneticists are beginning to provide better insight into relations of descent between organisms, including ones that would have likely eluded qualitative taxonomists, such as the water lily’s (Nuphar polysepalum) position as the genetic progenitor of the oak tree and all other seed-bearing plants (National Science Foundation).

Along with molecular biologists, population geneticists also use established mathematical algorithms to describe changes in organic populations. They rely, for example, on the algebraic Hardy-Weinberg principle as a model for the distribution of genes in a population under random mating conditions. In this endeavor, mathematics plays a central role because it is used to define a hypothetical baseline for change in the rate of alleles (the different possible gene types at a specific location on a chromosome) in a population against which the effects of natural selection, population size, mutation, migration, and random drift can be assessed. Calculations like these are essential to a number of modern applications, such as scientific breeding programs, assessments of the efficacy of screening for genetic disease factors, and the estimation of biodiversity.

Finally, a discussion of the importance of mathematics to modern investigations of variation, evolution, and heredity would be incomplete without mentioning the general value of statistics and probability in the day-to-day pursuit of scientific research. In modern biological research, investigations regularly begin and end with statisticians or biostatisticians carefully assessing the methods and results of experiments. Trained in statistics and probability, these members of a research team provide guidance to laboratory scientists on how to structure their experiments so that they limit the influence of factors which might bias their outcomes. For example, a lab’s biostatistician might advise geneticists working in disease research on techniques for random sampling to ensure that they have a data set from the general population for a genetic trait that might be used comparatively to identify genetic disease markers in a population of interest. After the tests are run and the data are collected, statisticians and biometricians are also tasked with calculating the reliability of the results and assessing the data to determine whether, if any, significant patterns emerge. These duties are so important to modern genetic research that Eleanor Feingold, a quantitative geneticist at the University of Pittsburgh, explained, “a lab of any reasonable size would have a biostatistician, a quantitative geneticist, or a statistician attached to it.”

A behind-the-scenes investigation of operations of modern research into variation, evolution, and heredity reveals: (1) that these phenomena cannot reasonably be investigated without mathematics and (2) that because of the increasing size and availability of data on these phenomena, the importance of mathematics will continue to grow. For these reasons, understanding both the role of mathematical argument in science and how that role came to be established, which are the subjects of this book, should be considered important topics of exploration.

Rhetoric, Mathematics, and Science

Although modern research in variation, evolution, and heredity would be impossible without mathematics, there was a time when these phenomena were explored largely without it. The focus of this book is the one hundred-year period between the publication of The Origin of Species and the emergence of modern programs of population and quantitative genetics in the nineteen fifties and sixties. During this critical period of development, mathematics and its capacity to generate reliable knowledge about organic populations was disputed. The goal of this text is to explore some of the reasons why mathematical argument was resisted in these early periods, and how it was advocated for either successfully or unsuccessfully by natural researchers who wanted to advance its credibility and explore the possibilities for its use.

To examine the use of and debates about mathematics in this formative period, this investigation turns to the methods and tools of rhetoric, a field of research and analysis devoted to the study of human communication, argument, and persuasion. With the aid of concepts and methods from this field, the book examines choices in language, organization, and argument in discourse located within specific social, epistemological, and cultural/historical contexts. Examining these dimensions of discourse in context permits characterizations of the goals and beliefs of arguers, the perceptions they have of their audiences, and the suitability of their choices in argument and communication. By investigating these facets of argument and persuasion, this book aims to better understand mathematical argument in a scientific context as well as explore what this relationship reveals about the practical value of rhetorical tools and concepts in understanding it.

Although the text is written primarily with philosophers, historians, sociologists, and rhetoricians of science in mind, every effort has been made to accommodate a broader educated audience of readers. Non-specialist readers who follow the subjects of mathematics, genetics, and evolution will likely find their interests reflected in the choice of topics and figures being investigated in this book. Well-known researchers such as Darwin and Mendel will be discussed, and fresh perspectives on their work as mathematical argument will be examined. Chapter 3, for example, explores in detail not only Mendel’s mathematical arguments in his famous paper, “Experiments in Plant Hybridization,” but also the historical context in which he makes these arguments. Assessing these dimensions of Mendel’s work reveals his reliance on the mathematics of probability as a source of invention for his pea experiments as well as his overconfidence that by using mathematical arguments he could persuade his audience to accept the general validity of his hereditary law.

Chapter 2 looks at Darwin’s work from a seldom–examined, mathematical perspective and reveals the extent to which the self-proclaimed mathematical bumbler relied on quantitative evidence and arithmetically informed arguments to invent and support some of his central conclusions in The Origin of Species. An examination of Darwin’s letters, diaries, his “big species” book (the original manuscript from which The Origin of Species was abstracted), and his arguments in The Origin of Species reveal that Darwin hoped to place biology on par with the physical sciences by giving it a solid, mathematical foundation. Other chapters in the book are devoted to less-well-known—but no less important or interesting—figures such as Francis Galton, Karl Pearson, and R.A. Fisher, all of whom play important roles in the development of a mathematical science of variation, evolution, and heredity.

In addition to persons of interest, this book also explores topics in science and mathematics with broad appeal. For example, it engages with the perennial issue of how scientific knowledge is validated. By examining the successes and failures of the scientists featured in the text, it suggests that when a scientific paradigm cannot be relied on to establish the appropriateness of mathematical arguments, arguers can turn to beliefs and values outside of science. Further, it explores the question. “How reliable and successful are mathematics in describing real phenomena?” This query is central to our current public and scientific concerns as we turn to polls, statistics, and probabilistic assessments with increasing frequency to make decisions about products, cures, risks, and candidates. What this investigation reveals is that nineteenth and early twentieth century scientists approached new mathematical methods and their conclusions with a healthy dose of skepticism. They offered legitimate resistance to mathematically informed theories of variation, evolution, and heredity that had insufficient evidence or blatantly disregarded important aspects of biological phenomena. In a few instances, however, their rejections were premature and prejudiced by either their ignorance of mathematics or their personal commitments to other methods of analysis.

Whether you read this book for the characters or the concepts, the goal is the same: to scrutinize, using the tools of rhetoric, the texts, arguments, and contexts involved in the development of the relationship between biological investigations of variation, evolution, and heredity and mathematics from the middle of the nineteenth to the beginning of the twentieth century. In investigating these phenomena, it endeavors to show how Darwin’s and Mendel’s ideas about them influenced the transformation of biology from a predominantly qualitative science to one with a vital, mathematical component. It also reveals how difficult this transformation—which we now take for granted as the very essence of our modern sciences of genetics and evolution—really was. By approaching science from a rhetorical perspective as a process of argument and deliberation rather than a product (as it is so often presented in the popular media), we can develop a greater appreciation both for the value of mathematics as a source for knowledge about nature and for the difficult and sometimes circuitous path by which that confidence is earned.