How did numbers become part of the furniture of everyday life? That is, how did mathematics move from being an esoteric subject of which only the learned were aware to appearing everywhere and influencing everyone: public opinion polls, pie charts of the economy, the idea of the "average man," and rates of crime, disease, and death? I. Bernard Cohen's The Triumph of Numbers relates, in 180 pages, many important episodes in this history. This little book is the product of a lifetime of reflection by one of the century's leading historians of the exact sciences. Cohen sent off the proofs of this book one week before his death in 2003. His description of these episodes is easily accessible to non-mathematical readers, but also should interest mathematical ones. I give some of his most striking examples.
Even pre-modern societies needed numbers. For instance, the ancient Egyptians, in building the pyramids, solved an immense accounting problem: How many individual stones were needed, and how long would it take to move them to the pyramid's site? How many workmen were required, how much money would it take to pay them all, and how much would they need to be fed? Again, the Hebrew Bible tells us how King David was punished by God for undertaking a census, and this Biblical episode made governments hesitate to take censuses as late as the eighteenth century.
In seventeenth-century science, not only does Cohen describe familiar examples like the scientific use of numbers by Kepler and Galileo, but we learn how William Harvey demonstrated the circulation of the blood by a quantitative argument about how much blood can pass through the one-way-valved heart in one beat, showing that in less than an hour the heart pumps more blood than the body contains — hence it must be the same blood. It was also in the seventeenth century that societies first began systematically collecting quantitative data about population, trade, agricultural output, illnesses and deaths. Here one pioneer was John Graunt, who compiled mortality tables for London and then became the first to estimate a total population based on reasoning from real numerical data. Building on Graunt's ideas, William Petty envisioned a new statecraft based on numerical data, a subject he called "political arithmetic."
In the eighteenth century, Thomas Jefferson used mathematical reasoning to find the fairest way to apportion congressional seats to the states when the census numbers didn't come out even, while Benjamin Franklin used risk-benefit analysis to argue for the value of smallpox inoculation (the pre-vaccination practice of giving someone a mild case of the real disease). Surprisingly, it is even more recently that statistical methods were first used to test medical treatments: only in the 1830s, as Cohen relates, did Dr. Pierre Louis use such methods to show that the ancient practice of bloodletting did not cure disease.
Only in the nineteenth century was statistics used in a serious way to study society. Here the key figure was the Belgian Adolphe Quetelet, who was the first to find that the frequency of many human traits, like height and chest circumference, when graphed produced the "normal curve." Also, it was Quetelet who coined the term "average man" (homme moyen). Quetelet also emphasized that crime rates and suicide rates were more or less constant every year, replacing the idea of capricious individual actions with the possibility of a quantitative social science. Cohen observes that this approach was opposed by social reformers like Charles Dickens, who believed that a science based on averages masked the suffering of the poorest members of society.
Cohen's last chapter, on Florence Nightingale, although it pays homage to her nursing skills and her making nursing into a respectable profession in Britain, more interestingly outlines her use of statistics and her ability to explain statistical findings to a wider public. Especially striking is the way she demonstrated that British soldiers during the Crimean war were much more likely to die of medical neglect and disease than from enemy action. Using similar types of data, she argued for sanitary reform both in Britain and in India. For her plan for uniform hospital statistics, Florence Nightingale was elected to the London Statistical Society; she was its first woman member. She imagined Queen Victoria as typifying the influential, but statistically untrained audience for her arguments, and for this reason used multi-colored pie charts to make her findings both clear and memorable.
Although the book has an overriding theme, it can be faulted for not being tightly organized. Also, it isn't always clear why some subjects get much space, others little, and some that seem equally relevant get none at all. Many of the examples come from the English-speaking world. There is a bit of repetition, both within individual chapters and between them. And the story doesn't get past the nineteenth century, although computers rate a brief mention. Overall, "fascinating but uneven" seems a fair description.
Nevertheless, there is a lot here to interest a "Read This!" audience. The topics covered are important. We see three things advancing together: the increasing complexity of society and of large organizations, progress in mathematics and in statistics, and the social need for an informed citizenry. There's a useful index and a 10-page bibliography to help. And when we are done reading this short but immensely learned book, we will have learned a lot of things that may be known to historians of science, but aren't — and should be — known to mathematicians and teachers of mathematics.
Judith V. Grabiner is the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College in Claremont, California. Her thesis advisors were Dirk Struik and I. Bernard Cohen.