The printed number and the rise of the statistical program

This is an essay I wrote for the module History of Science in January 2021.

**Introduction **

Engaging philosophically is something I consider a social endeavour though it doesn’t exclude periods of individual solitude and a fair amount of despair. In my case, students and teachers from another country lived something that triggered me. Teachers were deprived of one of their traditional responsibilities, that of assessing students, and students, who once had a person to address themselves regarding their grades, now had a black box, which, at best, would “reply” is just executing a set of rules.

It’s May 2020, and due to COVID-19 pandemic, it was announced students in England wouldn’t sit for GCSEs, AS and A levels exams (see note 1). As the exams were cancelled, the question of how to evaluate and grade students popped up. The Office of Qualifications and Examinations Regulation (Ofqual), whose mission is to “maintain standards and confidence in qualifications in England”, proposed to address the situation through the use of an algorithm.

Seeing pictures of students protesting and holding cards “Teachers know my potential, algorithm does not” or “my grades, not my postcode” in BBC (A-levels and GCSEs, 2020) made me wonder why choosing an algorithm to decide student’s grades was better suited than teachers’ judgement. And what was this statistical model made of anyway? According to Ofqual, teachers were asked to provide data for each student and for every subject . It included an estimated grade, taking into consideration previous data, such as mock-tests and class assignments, and a ranking comparing every student at the school within the same estimated grade. One of the assumptions was grades attributed by the algorithm would be consistent with schools and students’ previous performance, even without exams taking place. The organisation affirmed the statistical model aimed at maintaining the standards both between centres and over time in order to prevent grade inflation, but alerted to the fact there could be issues for students whose grades fall outside the pattern of results in that centre in recent years. Another organisation, the Royal Statistical Society (RSS), whose motto is Data | Evidence | Decisions, alerted “Ofqual to the statistical issues relating to exam grading and assessment in 2020” (Witherspoon, email to Ofqual, 2020, April 9) and proposed to help by bringing in an advisory expert group to the discussion. The grade results provided by the algorithm provoked protests all over the country and escalated up to the point of calling the Prime Minister to intervene (Adams, Stewart, 2020, August 16).

This situation illustrates well our condition of statistical subjects, where many parts of our lives are under the influence of a probabilistic and statistical umbrella, even if these inferences and judgements may be wrong (Wissenschaftskolleg zu Berlin, 2011, January 6). This essay seemed an adequate opportunity to expand reflection on the subject, therefore I will be covering what were some of the contingencies in the origin and development of probabilities and how they became part of the way we think, hence of who we are. Untangling this narrative is like analysing a plot: what were the conditions in place that created a nutritious ground to these ideas? Who were the characters who fed these ideas and how were they related? Which moments, either of eruption or apparent stagnation, were important to it becoming a thing? It’s not my aspiration to expose in a small essay what others have done so brilliantly in a journey of a lifetime but I will adventure on referring to some of the most relevant moments which allow us to understand the students’ probabilistic and statistical situation.

This essay has three parts. First, I will start by explaining the origins of classical probability in a deterministic conception and its rise as a new mathematical program and new rationality. Second, I will move towards the enumeration of people and their habits and how it changed the idea of determinism, eroding it. Here I will focus on the law of large numbers, normal distribution and the birth of the average man. Finally I will conclude by summarising the main points and suggesting further research to address our probabilistic and statistical thinking.

**The Mechanics of the heavens and order on Earth **

In the first half of the seventeenth century, natural philosophers such as Galileo thought nature had causes which were determined by God and that they could be expressed through the language of mathematics. Then, in order to be able to describe nature mathematically, it was first necessary to understand its causes ( Gigerenzer, Swijtink, Porter, Daston, Beaty, & Kruger, 1989, p. 11). It’s in this rational context that “Philosophiæ Naturalis Principia Mathematica” by Sir Isaac Newton thrives as the first unified program of modern physics, bringing together the disciplines of astronomy, physics and mathematics. Newton helped to create the idea of an universe where there was a single force, universal gravitation, determining the motion of objects: from planets to falling objects everything was predetermined. Universal gravitation enabled retrodictions and predictions of the positions of the planets, considering its present configuration.

The devising of Newton’s theory had an intimate connection on how probabilities started to be understood. Traditionally seen as a qualitative measure of an opinion backed by authority, probability slowly became a quantitative measure of certainty. At the core of this change is the intersection of a new rationality which was leaving behind the traditional idea of certainty with the desire of applying mathematics to new areas of experience. ( Gigerenzer, Swijtink, Porter, Daston, Beaty, & Kruger, 1989, p. 1). Without the entanglement between both, none of them alone would have been enough to advance probabilities as we know them today (Hacking,1990).

In the classical period of probabilities, from *circa *1660 to 1840, there were three undiscerning ways to which mathematicians referred to as probabilities.

The first was about equal possibilities based on physical symmetry which was applied to gambling scenarios. For instance, the work of Blaise Pascal in understanding the relationship between risks and uncertain pay-off is an example of this new rationality where it was “not enough to consider how good or bad an outcome is in itself, but also the likelihood that it will come to pass (Arnauld and Nicole, [1662] 1965, pp. 352-3 in Gigerenzer et al., 1989 p. 5).

The second way referred to probabilities as observed frequencies of events, through analysing different kinds of data collected since the sixteenth century. The permeation of the probabilities outside the realm of gambling can be seen by the hand of the Dutch Christiaan Huygens who continued Pascal’s work and applied probabilities to the study of equitable contracts. Chance and risk as random events were at the heart of the dealings in the Amsterdam Stock Exchange where financing marine ventures required the need to to determine what needed to be paid in exchange contracts, in order to guarantee equity and justice. Another of its applications was in calculating the price of annuities, which evolved to become life insurance (Gigerenzer et al., 1989 p. 6).

Finally, the third concept of probabilities is connected to the notion of sufficient knowledge and it arose from analysing procedures in courtrooms and miracles. If order in nature was universal, invariable and eternal and a sign of divine providence, then the question on how to interpret miracles was yet to be answered. One of the works which contributed to the development of this idea was David Hume’s essay on miracles. He defended that a miracle was less probable than a naturalistic explanation, raising the notion of intrinsic evidence; and rejected the evidence arising from testimony, considered as extrinsic evidence on past events as an inference to predict future ones. It is the connection of intrinsic and extrinsic evidence that characterises probabilities as models of causation and degrees of subjective certainty or belief (Gigerenzer et al., 1989 p. 9, p.28).

These three interpretations regarding probabilities led to a distinction *a posteriori *between objective and subjective probabilities. Still, this distinction shouldn’t be seen as a binary opposition though but rather as a complex web weaved by classical probabilists who also used both meanings randomly. For example, Pierre-Simon Laplace was an astronomer who used both objective and subjective elements in his theories. Moreover, Laplace was also one of the founders of the classical theory of probability and an avid supporter of determinism to allow the control and domination of nature which would be possible under the condition we have sufficient knowledge. This conception of a predetermined universe carved the way for probability to rise as a new mathematical program and a new rationality where probability was then to be understood a state of mind relative to human knowledge rather than a state of the physical world.

*Circa *1840 A.A. Cournot wrote “the double sense of probability, which at once refers to a certain measure of our knowledge”, this is subjective probabilities, “and also to a measure of the possibility of things independently of the knowledge we have of them", this is the equal possibilities based on physician symmetry and frequencies. (Daston, 2007). Hacking distinguishes probabilities as modelling and as inference , when referring to objective and subjective probabilities respectively (Hacking, 1990 p. 98).

Though it seemed mathematics could then be applied to other domains of experience, there was a catch in the program: the variability found in data represented a problem. Classical probabilists faced huge criticisms and the program was discredited (Wissenschaftskolleg zu Berlin, 2011, January 6). What initially survived was the objective conception of probability applied in gambling and actuarial problems. To successfully push the program forward and connect both the objective and subjective senses of probability two more elements were needed: the first element required probability as a measure of knowledge or state of mind had to be shown in practice, meaning it needed to attract people who were motivated to find in society the equivalent of Newton’s gravitational universal law; and the second element was evidence needed to be quantified for which the availability of printed numbers was key.

**The printed number and the rise of statistics **

The idea of gathering numbers about populations can be traced back to the eighteenth century when government officials such as Leibniz, the “philosophical godfather of Prussian official statistics'' (Hacking, 1990 p. 18), who saw the potential of a central office estimating population and their behaviour to serve different administrations such as taxation and military service. Nevertheless, a universal statistical law of human nature didn’t emerge in Prussia but in England and France in government offices.

This vision was shaped in the beginning of the nineteenth century. By this time, there was a solid printing network in place and an increase of the capacity to print propelled by the Industrial Revolution. This made the appearance of an avalanche of printed numbers technically possible, developed and implemented by the rising modern states and their new bureaux, which aimed at creating stability and avoiding unbalances to peace after the turbulence of the Napoleonic era.

The new bureaucratic state institutions were instrumental to gather and diffuse the avalanche of numbers and it engendered the need of creating categories where individuals could fall into. The idea was to show that in practice society would follow into the same universal umbrella of predictability and control. In order to do so, everything was enumerated, from birth, deaths, average date of marriage, voting predisposition and, in particular, wrongdoings, such as suicides (Hacking, 1990, p. 64). It is in the wrongdoings that statistics finds its social mission, the one of improving society by applying reason.

One of the most relevant statistical realisations is the law of large numbers. It is a part of a theorem refined and presented by Siméon Denis Poisson in 1837 which expresses the stability of mass phenomena. It acknowledged there were regularities in the societies and this became a fact which “didn’t need” to be empirically checked because it was how things were, the collected numbers showed a frequency in the many events such as the number of deaths and births. Though there were many sceptics to Poisson’s theorem, the proposition about the law of large numbers had the soil to flourish for some decades once the stability of relative frequencies was associated with a sign of divinity. Statistical law was now admitted in social affairs (Hacking, 1990, p. 104).

Nevertheless, the regularities in wrongdoings such as crime and suicide were difficult to understand in a deterministic universe, so the law of large numbers was later discarded as

explanation. The question of what could be the equivalent of the laws of regularity which govern the universe to the ones that govern individuals remained opened.

Maxwell, a scientist in the field of mathematics physics and Quetelet, an astronomer and statistician, had a different vision regarding what determinism was and defended that many of the irregularities found by bureaucrats were also found in physics such as in the statistical laws of thermodynamics. Quetelet thought it wasn’t necessary to know everything about all the properties in absolute detail to address the puzzle of societal wrongdoings and favoured society and not the individual as the basis of social statistics. He introduced the curve of normal distribution at a societal level, which was already used in physics. The curve of normal distribution, described observational errors in astronomy and is composed by the mean and measure of dispersion. The dispersion is very relevant because it determines if an average is or not reliable. When it is, it is called a normal distribution and is represented by what we know today as a Bell Curve. If the universe also had its own errors, then it was only normal society had theirs too, determinism was not at stake.

Underlying this notion is the concept of “normality”. Quetelet, by accepting deviations and errors present in the universe and society, gave birth to the notion of the average man (Gigerenzer et al., 1989 p. 53). The average man became an objective notion of “a” man against whom all others can be compared to and judge if they are average or a deviation. The Bell curve or normal distribution is still used in social matters. For instance, the intelligence quotient (IQ) remains one of the most controversial products of statistical reasoning expressed through a Bell curve. Used to measure intelligence, it conceives it in a universal and supposedly neutral way.

Moreover, the law of large numbers and normal distributions are present in the students’ exams case portrayed in the introduction. Ofqual’s algorithm was applied because it was considered there was sufficient data which provided evidence about the stability and regularity in centres’ performance in the past years and it also had a way to compare students to each other through distribution of their grades. Ofqual also expected some students whose grades would fall out of the patterns or the normal distribution curve, but considered it wouldn’t compromise the large scale average and predictability.

On that account, the variations found in astronomers' observations had consequences: if some variability was found in the universe, then some chance and randomness needed to be readmitted both in physics and society. Though variability didn’t put an end to determinism it led to an erosion, what Ian Hacking defines as taming chance. Nevertheless, the laws of nature differ from those of society: while elements of the universe don’t have an ambition of being considered normal or constant, humans do look forward to being seen as such, meaning we will try to adapt to what the rules are, which will consequently affect what is considered to be “the average man”. Despite the initial failure of the probabilistic program, we became statistical subjects thriving to be considered normal people. This situation was only possible due to the avalanche of numbers which helped create the statistical laws of societies. Yet, without the belief that there was an equivalent of a newtonian-laplacian universe in society, the statistical laws might have never been inferred in those numbers (Hacking, 1990).

**Conclusion **

We have seen how probabilities and statistics are connected with determinism and how they changed through the course of the classical period of probabilities.

The classical probabilities set in motion a new mathematical program and a new conception of how to think about certainty, trying to exclude chance. The regularity of the universe was desired on earth, to allow insights, predictions and control in all spheres of human life. Though the program had many sceptics, probabilities and statistics carved its way into the core of our rationality and are engraved in social sciences and in our rationality. Probabilities were a huge success at many levels. For instance, they are an epistemological success because when we think about evidence, data, experiments and credibility we do it in probabilistic terms; it is a success in logic due to inference theory. As the history of probabilities and statistics is an ongoing endeavour, further considerations may be added for future research. I suggest, for example, if a plausible comparison can be made between the relevance of printed number in the nineteenth century and the relevance of data points in contemporary society for the purpose of determining our behaviour; and understanding if algorithms are the most recent techno-deterministic flavour of our society, governing all spheres of our life from whom we may vote, whom we may be attracted to or what are we going to buy. These questions are valid because they seem supported by the fact that despite the evidence of some level of indeterminism in the universe brought in by other theories in physics, such as quantum theory, determinism remained mostly unshaken.

Note 1:

GSE, AL are the UK equivalent to school-leaving or university entrance examinations in different subjects. GSE stands for General Certificate of Secondary Education, AS for Advanced Subsidiary level and A is a combination of advanced levels.

**References **

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A-levels and GCSEs: How did the exam algorithm work? (2020, August 20). BBC. Retrieved from https://www.bbc.com/news/explainers-53807730

A-levels and GCSEs: U-turn as teacher estimates to be used for exam results. (2020, August 17). BBC. Retrieved from https://www.bbc.com/news/uk-53810655

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