There is a widespread misconception amongst people, who are not particularly good at mathematics that mathematicians can do mathematics, by which I mean that a mathematicians can do the whole range of subdisciplines that are collected together under the term mathematics. Nothing could be further from the truth. It is claimed that there is a fundamental divide between mathematicians who think in diagrams, i.e., at the entry level geometry, and those who think in symbols, i.e., algebra and analysis. Not that I’m a mathematician but I certainly think in symbols, not diagrams. At a higher level, most mathematicians have a special discipline where they are at home and excel and other disciplines that that they have difficulties even comprehending. When I studied mathematics at university the introductory courses were allotted in rotation to the professors. My introductory analysis lectures were held by a really sweet guy, who was a world-renowned finite group theory specialist, where world renowned means that the fifty people in the world in his field all knew him. One day he came into the lecture hall and said, “today we should start with step functions, but I’ve never understood them, so we’ll do something else instead.”
Although I studied mathematics at university up to about BSc level, worked in a research project into mathematical logic for a number of years, and have tutored school leaving/university entrance level (A-level, Abitur, Baccalauréat etc.) mathematics for the last twenty years, I do not consider myself a mathematician. Having said that, I definitely have my strengths and weaknesses in different mathematical disciplines. I could always do basic calculus without even think about it, in fact it was my love of calculus that led me into the history of mathematics, when I discovered that Newton and Leibniz had both invented/discovered (choose your preferred term) calculus independently of each other. I later discovered that this wasn’t true but that’s another story. An entry level discipline which I could never get my head around was probability theory and statistics and I used to groan inwardly when I had to teach them to one of my private pupils.
When I studied mathematics at school before the second ice age, I did A-level maths, probability theory and statistics were not part of the curriculum. However, teaching Abitur in Germany over the last four years probability theory and statistics were very much part of the curriculum. Today, there is a widespread and very dynamic discussion in many lands about changing, extending, and improving the mathematics courses taught in schools at all levels, in order to combat a perceived mathematical illiteracy. See, for example, the Tory government’s call for maths to eighteen for school kids in the UK. One prominent argument in these discussions is for the reduction or removal of much of the tradition diet of algebra, geometry, and calculus and replacing it with a much-expanded emphasis on probability and statistics because these are the areas of mathematics that people need to understand and even use in everyday life.
It is in fact true that we stumble across probability theory and statistics on a daily basis in the media, on the news, in advertising etc. often misused and mostly misunderstood by the people reading it. Probability turns up in the weather forecast, there’s a X% chance of rain, in sport betting, which has become a vast industry, in other forms of gambling, but also in areas of science and medicine. What is the probability of blah, blah blah… Here the quoted probabilities are derived from statistical analysis. Today, it is perfectly normal for all aspects of human existence to be analysed statistically. From trivial things like what percentage of the population is left-handed, to serious topics such as what is the probability of someone developing a particular type of cancer. We have statical analyses of opinion polls, election results, school exam results and …
We are so inundated with statistical information, oft presented as a probability, that we simple accept it without really thinking about it. However, where do these two areas of mathematics come from, when were they developed and why? Although there are earlier simple examples of the calculation of probabilities, both probability theory and statistics first emerged in the Early Modern Period. Not surprisingly, probability theory first emerged in the calculation of odds in gambling. The first major work on probability theory was written by one of my favourite Renaissance polymaths, Gerolamo Cardano (1501–1576), who was a passionate, and at time professional, gambler. His book, Liber de ludo aleae (Book on Games of Chance), written in the 1560s but first published in 1663, also includes advice on how to cheat. In their correspondence in 1654, Pierre Fermat (1607–1655) and Blaise Pascal (1623–1662) discussed various aspects of probability theory after being asked how the pot should be divided in an interrupted game. Christiaan Huygens (1629–1695) came across this correspondence in Paris and wrote the most coherent and at that time, most advanced book on probability theory his De Ratiociniis in Ludo Aleae (On reasoning in games of chance). Originally written in Dutch, it was translated into Latin and published by Frans van Schooten Jr. in 1657. The mathematics of probability was firmly established in the early eighteenth century by Jacob Bernoulli (1655–1705) with his posthumously published Ars Conjectandi (The Art of Conjecturing) in 1713, which covers combinatorics and probability, and Abraham De Moivre (1667–1754) with his The doctrine of chances: or, a method for calculating the probabilities of events in play in 1718 with an expanded second edition in 1738, and a further expanded edition published posthumously in 1756.
Statistics, however, developed from the start through a desire to count people, a development that had a long and complex prehistory before it began to become formalised on a very simple level in the second half of the seventeenth century. That formalisation took place in the work of John Graunt (1620–1674), Edmond Halley (1656–1741), John Arbuthnot (1667–1735), Gregory King (1648–1712), Charles Davenant (1656–1714), and William Petty (1623–1687), who gave the counting of people the early name of Political Arithmetick.
Today, we use the term demography derived from the Ancient Greek demos meaning people, society and graphía meaning writing, drawing, description, and meaning the statistical study of populations. The American historian Ted McCormick, who teaches at Univesité Concordia in Montreal earlier wrote William Petty and the Ambitions of Political Arithmetic (OUP, 2009), using the manuscripts of Sir William Petty (1623-1687) to show how a mixture of alchemical and natural-philosophical ideas were brought bear governing colonial populations in Ireland and the Atlantic, as well as confessional and labouring populations in Britain (taken from his university webpage), which I haven’t read but I’ve now added to the infinite reading list. He has now followed up with Human Empire: Mobility and Demographic Thought in the British Atlantic World, 1500–1800,[1] which details the gradual development of demography in Britain, Ireland and the American Colonies in the Early Modern Period.
Before I write more about McCormick’s book in any detail, a couple of important notes about it. Although this book relates the historical developments over a couple of centuries that led to the first low level uses of statistics by the scholars I named above, it is in no way whatsoever a history of mathematics text. In fact, actual numbers are strikingly absent from McCormick’s narrative. Rather it examines the social, political, environmental, cultural, philosophical, and economic circumstances that led authorities and individual to consider it necessary to enumerate elements of the population. Secondly, having said this, it should be fairly obvious from my general description that this is also in no way a popular book, but rather a deeply and intensively researched academic book.
The word demography was first coined in the nineteenth century, but societies have been indulging in demographic thought at least since the emergence of the earliest civilisations. McCormick’s book might well be regarded as an extended case study into the structure and content of such thought in Britain and its colonies over four centuries. McCormick himself illustrates the ancient origins of the discipline, in his introduction, with references to the Bible. He also touched briefly on Aristotle’s thoughts on the topic, as these were of course relevant to those engaged in debates in the Early Modern and Modern periods. He writes the following:
Aristotle thus presented population not just as a measurable number of inhabitants but also, more saliently, as the living material of the city-state. Its size would be constrained by the territory that it occupied. It should be limited, too, by the counterpoised imperatives of magnitude and order in the context of polity conceived as an organic unit – a body politic – with a constitution. More significant than absolute size was the relative proportions of the body’s parts: The balance between citizens, slaves and foreigners, [my emphasis]and between soldiers, husbandmen and artisans (page, 29).
Reading these lines and in particular the clause I have emphasised I was instantly reminded of current political and cultural debates that are currently raging in many countries about exactly that balance, (ignoring the slaves of course) or as many see the lack of what they see as the correct balance–too many foreigners, migrants, asylum seekers… take your pick. This was the first time that such parallels between the historical debates that McCormick outlines in great detail and the actual political debates of our times, but it was by no means the last time. Again and again, I found myself thinking this is all too familiar. My feelings were confirmed when in the closing pages of his book McCormick remarked:
To put it another way: what might the early modern history of demographic governance tell us about the persistence or reemergence of concerns about the mobility, mixture and mutability of populations in the nineteenth, twentieth and twenty-first centuries) Instances of such concerns are not only numerous but also fundamental to received interpretations of modern history (page 249)
The main text of the book covers four phases or periods of debate about perceived demographic topics. The first concerning the fifteenth century reminded me of my days as a field archaeologist. I spent several seasons, both Easter and summer, working on the major excavation of a deserted medieval village, or DMV, in the north of England. DMVs, shrunken medieval villages, SMVs and expanded medieval villages, otherwise known as towns, are the result of a major shift in population distribution during the High Middle Ages largely brought about by the enclosures. That is turning agricultural land, which peasants farmed to make a subsistence living, into pasture for sheep grazing, thereby forcing the ploughmen to abandon their villages and try to seek employment elsewhere.
The socio-political debate that McCormick covers, about this depopulation, concerns the reformers, who wished to restore the honest ploughman to his rightful place in society. At this time the talk, however, is not of populations, but of the much more imprecise multitudes. In the second section of the book, we still have to do with multitudes but here, in the Elizabethan era, it is not a debate about a positive part of the body politic, the humble ploughman, but concerns about negative sections of that body, vagrants and the poor. Once again, the main participants in the public debate are reformers offering and discussing potential solutions what they see as the proliferation of undesirable elements in society.
This second phase moves out of England into Ireland where the English had problems with various aspects of various parts of the island’s population.
In these sections and also in the ones to follow, McCormick outlines the perceived problems with selected parts of the population, the multitudes, and then presents the solutions proposed by the various reformers. He lets the participants in the various debates present their polemic themselves, in lengthy direct quotes all delivered up in the original English of the period, with its own vocabulary, orthography, and grammar. I must admit that I found it difficult reading some of these passages that appeared almost to be in a foreign language and not the English with which I grew up. However, it pays to persevere because it gives a much clearer picture of what the participants were aiming for than a simple modern English paraphrase.
In the third section McCormick brings the political philosophers into play and the discussions on over population. Here we see the emergence of colonialism as a potential solution, as to what to do with surplus multitudes. Already practiced with little success in Ireland, we see the beginnings of the establishment of colonies in North America. The seventeenth century sees the move in the discussion from the multitudes of the earlier reformers to the perception of population and the slow move towards encapsulation through mathematics in the form of statistics driven by the writings of such as Francis Bacon and the Hartlib Circle, a precursor to the Royal Society. The latter offering up various projects for empire, colonialism, and population.
Enter William Petty. Petty, an associate of the Hartlib Circle and a founding member of the Royal Society, no longer simply delivered polemics on population and population reform but in his survey of Ireland, on behalf of Cromwell, to organise the distribution of land to Cromwell’s soldiers, as payment for their services and also to dilute the troublesome Irish population, Petty was instrumental in putting a plan into action. McCormick covers Petty’s survey and its background in great detail. Out of his work in Ireland he developed his economic theories, marking him as a pioneer in economic science, and also his Political Arithmetick. It’s worth quoting McCormick here on Petty’s innovation:
“Political arithmetic” was coined sometime around 1670. It has appeared ever since as the invention of a new, scientific and, above all, quantitative age. Its inventor, Petty, has often seemed precocious in his focus on “number, weight and measure,” his imaginative exploitation of demographic and economic figures running well ahead of the empirical data at his disposal over a century before the census. While John Graunt’s “shop arithmetique” revealed a world of relationships hidden in the rows and columns of London’s weekly bills of mortality and in the patchy parish registers of baptisms, marriages and burials, Petty promised nothing less than a new “Instrument of Government” for the Stuart kingdoms and the growing colonial empire, predicted on the collection and analysis of vast amounts of information. Most of this was numerical.
This introduction is followed by an in-depth analysis of what exactly Petty’s innovative creation was, what it meant and how it influenced future developments.
The final section of McCormick’s main narrative follows how the rhetoric about population and demography evolved throughout the eighteenth century both in Britain and in its colonies in America. In Britain the reformist mode of thought is still dominant. In America the problem of increasing the colonial population to take over the land is predominant with Benjamin Franklin taking a lead in the debate. On the one side his acknowledgement that the land was occupied by Indians when the settlers first arrived but, on the other, that it would only be settled when the colonial population had grown enough shows a level of casual racism that I found deeply disturbing. There is also an extreme and dismissive level of racism shown by Franklin and others towards the negro slave population.
McCormick closes his stimulating and fascinating narrative in the conclusion to his book with a discussion of Thomas Malthus’ 1798 Essay on the Principle of Population, a work that famous influenced by Charles Darwin and Alfred Russel Wallace in the formulation of their theories of evolution by natural selection. Darwin and Wallace do not put in an appearance here as McCormick is concerned with Malthus’ influence on the development of demography, which he presents as revolutionary:
From the perspective of the Henrician humanists, Elizabethan pamphleteers, Jacobian colonial promotors, Interregnum projectors, Restoration political arithmeticians and Enlightenment-era physicians, philanthropists and philosophers who have populated this book, T. R. Malthus’s 1798 Essay on the Principle of Populationappears scarcely less than the French Revolution, to mark the end of an age.
Carrying on, McCormick runs revue over the polemics, reforms, schemes. and projects that he has described in the previous chapters and concludes:
Into this baroque edifice slammed the wrecking ball of Malthusian principle.
He then follows up with an analysis of that Malthusian wrecking ball. McCormick’s book closes with some very open and honest general thoughts on the limitations of his own research.
Fans of footnotes will love McCormick, the book has literally tons of them listing vast amounts of sources, that are included in a twenty-two-page bibliography, the whole closing out with an excellent index. This book does not have illustrations.
This is not an easy read. It is a book packed with intensive historical information and evidence with an equally intensive analysis. However, if you have any interest in the topics covered this is a must read.
Having started out with my personal relationship to probability theory and a very brief sketch of its early history, when the philosopher of science Ian Hacking died whilst I was writing this, I immediately got his excellent The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical inference (CUP, 1975), a title that can be found in McCormick’s extensive bibliography, out of the university library. A book that I first consulted more than thirty years hence and is now my current bed time reading.
[1] Ted McCormick, Human Empire: Mobility and Demographic Thought in the British Atlantic World, 1500–1800, CUP, Cambridge, 2022