Rules, Algorithms, and Models: An Intellectual History

Rules: A Short History of What We Live By. By Lorraine Daston. Princeton University Press, Princeton, NJ, July 2022. 384 pages, $29.95.

Our lives are permeated with rules, both large and small. They range from substantial precepts like “Thou shalt not kill” and “Congress shall make no law...abridging freedom of speech” to lesser guidelines like “Don’t wear white to a wedding” and “In simplifying a mathematical equation, be careful not to divide both sides by zero.” In Rules: A Short History of What We Live By, author Lorraine Daston surveys the vast landscape of societal rules in terms of their appearance; their evolution; their creation, perception, and enforcement; and the ways in which people obey, internalize, or flout them.

Daston is an eminent historian of early modern science and philosophy whose previous works include Classical Probability in the Enlightenment and Wonders and the Order of Nature, 1150-1750 (coauthored with Katharine Park). Her new book, Rules—which arose from her 2014 Lawrence Stone Lectures at Princeton University—combines conceptual and historical analysis with a wealth of remarkable historical examples. The text focuses primarily—but by no means exclusively—on Western Europe from roughly 1650 to the present.

The book taxonomizes the space of rules across a number of dimensions. Rules can take the form of laws (with regulations as a subcategory), algorithms, or models. They can be flexible or rigid, general or specific, and thick or thin. A thick rule is accompanied by a rich body of justifications, specific cases, caveats, and exceptions; application of a thick rule to a particular case thus requires knowledge, judgment, and discretion. In contrast, a thin rule is a bare, unelaborated statement of a principle or procedure.

Daston is particularly interested in the idea of a model as a kind of rule. She argues that this view was common in the early modern period¹ but fell into disfavor around 1800, after which philosophers came to view rules and models in opposition to one another. She prefers the earlier viewpoint, writing that “In the end, the one ancient meaning of rules that seemed to go extinct around 1800 may prove to be the most enduring. Rules-as-models are the most supple, nimble rules of all, as supple and nimble as human learning.”

A multitude of historical examples illustrate Daston’s theoretical analysis and leaven her writing, including the rules of the Benedictine monastic order, sumptuary laws and dress codes, 18th-century cooking recipes, Parisian police traffic ordinances, and proposals for spelling regularization in English, French, and German. The detailed description of human computers’ construction of large astronomical and actuarial tables between roughly 1790 and 1950—as well as the influence of the introduction of mechanical calculators around 1870—will likely be of particular interest to SIAM News readers.

The book is quite wonderful in many respects. Daston is an exceptionally fine writer, and her theoretical analysis is generally outstandingly clear, insightful, and deep. Her historical vignettes are fresh, remarkable, and entertaining; full of fascinating details; and told with an appealing balance of sympathy and ironic distance. I repeatedly encountered passages that I will long remember and ponder, such as the following:

All of these contrasts boil down to one big contrast: a world of high variability, instability, and unpredictability versus one in which the future can be reliably extrapolated from the past, standardization insures uniformity, and averages can be trusted. Although the episodes recounted in this book trace a rough historical arc from the former world to the latter, there is no inexorable dynamic of modernity at work here. An island of stability and predictability in a tumultuous world, no matter what the epoch or locale, is the arduous and always fragile achievement of political will, technological infrastructure, and internalized norms. At any moment it can suddenly be overwhelmed by war, pandemic, natural disaster, or revolution. In such emergencies, thin rules suddenly thicken, rigid rules become rubbery, [and] general rules wax specific.

However, Rules does contain a few significant flaws and gaps. No doubt these issues are partially the result of the text’s origin as a series of lectures and the author’s effort to squeeze an enormous topic into a comparatively short book. Nevertheless, I found them troubling.

The analysis and historical episodes are all excellent on their own, but I am not sure that they mesh particularly well — i.e., that the episodes actually shed significant light on the analysis. For instance, Daston’s discussion of human computers is a splendid piece of historical narrative, but does it actually tell the reader anything about rules in general? Someone who needs to compute large mathematical tables via human labor will probably have to employ a rigid and systematic organization, and the labor will be both staggeringly tedious and require close mental attention. However, it is unclear whether this reality either reflects or affects the role or form of any other kind of rule.

The opposition of rules versus models has been the focus of a substantial amount of research in cognitive psychology that examines the extent and circumstances in which human cognition uses one or the other mode of reasoning. Daston might have reasonably thought that this aspect of the question was too large and peripheral for inclusion at any length, but it is at least worth mentioning.

The book has one outright error that I thought was significant. In chapter four, Daston writes “Nowadays, virtuoso mental calculators like the great eighteenth-century mathematician Leonhard Euler (1707-1783) would be more likely to be labelled idiot savants than brilliant mathematicians, the latter often making a point of emphasizing how hopeless they are at arithmetic.” She later repeats this idea in a softened but more general form:

The history of eighteenth- and early nineteenth-century mathematics boasts several calculating prodigies who later became celebrated mathematicians, including Leonhard Euler, Carl Friedrich Gauss (1777-1855), and André-Marie Ampère (1775-1836). Anecdotes circulated about their precocious feats of mental arithmetic as early signs of mathematical genius. But by the late nineteenth and early twentieth centuries, psychologists and mathematicians had come to believe that such cases were anomalous. Great mathematicians were rarely calculating virtuosi, and calculating virtuosi were even more rarely great mathematicians.

Of course, the statement about Euler—who is indisputably the most creative and inspired mathematician between Isaac Newton and Gauss—is just wrong; it is akin to claiming that present-day society would now dismiss Mozart as merely a virtuoso pianist. More importantly, the second claim about mathematicians versus calculating prodigies is hardly truer than the hypothetical corresponding claim about musical composers versus virtuoso instrumentalists. In the 18th century (as in the 20th), the categories of creative mathematician and virtuoso calculator were different and understood to be so. And in the 20th century (as in the 18th), many important mathematicians—including Srinivasa Ramanujan, Norbert Wiener, John von Neumann, Paul Erdős, and Terence Tao—were child prodigies with extraordinary capabilities for mental mathematical problem-solving (it is said that when von Neumann was six years old, he could divide two eight-digit numbers in his head). There has probably been some historical shift in the direction of Daston’s claim—a figure like Nathaniel Bowditch might have been less successful in the mathematical world of today, for instance—but it is not nearly as large as she suggests.

Yet despite these minor flaws, Rules is ultimately one of the best written, most profound, and most far-reaching works of intellectual history that I have ever read.

¹ At the time, the word “rule” was often used to mean “model.” For example, the Oxford English Dictionary quotes “He was looked on as a living rule of religious perfection” from 1756. The idea of a model as a rule is central in Platonism and Christian thought.