In the present, mathematics frames much of our policies and political discussion. Remember how during the COVID-19 pandemic the R-value took centre stage in the news cycle? Network science was a key tool informing our policy makers. Similarly, climate modeling shapes international agreements, and optimization tools inform the allocation of infrastructure investments. These are prime examples of the application of mathematics in making decisions, something that is expected of modern governance. But in past times governments were less well informed.
One obstacle preventing the application of mathematics is that policy decisions are made under uncertainty, and the necessary probability theory to deal with uncertainty is surprisingly nascent. The conceptualization of probability is only a few hundred years old, and its formalization as a branch of mathematics is a lot younger than fields like geometry or number theory. But from the moment probability showed up, its power for application was recognized not only by gamblers: for this story, we go to the 17th century, when the notions of probability were just starting to emerge. We see how Dutch statesman Johan de Witt was in the middle of this pivotal moment in mathematical history, and one of the first to apply these new ideas. De Witt, as a leading politician, had great interest in calculating the value of the debt securities the Dutch state was issuing. If they were undervalued, the state could go insolvent and burden future generations with paying them off. As we will see, these securities depended on life expectancy. For the first time in history, there was a method for this to be calculated.
The emergence of probability: different perspectives
In his book The Emergence of Probability1 philosopher Ian Hacking argues that no singular event or work can be pointed at as the start of probability. Instead, the concept of probability was practically absent before the 1660s, and then suddenly, over the course of a decade, many probabilistic ideas appeared independently and concurrently all over Europe. Why is that?
Probability is simply how likely something is to occur. Try giving a more specific definition, and we stumble onto a duality. On the one hand, probability means the relative frequency of some random but stable mechanism: the statistical chance. Think of the chance of getting heads when flipping a coin, or getting 6 when throwing a die. But probability can also mean the degree of belief that something will happen based on the available evidence, say, the diagnosis of a disease given the symptoms. These two perspectives are intimately related, but the connection was not always clear.
The first perspective came out of aleatory chance: calculations on bets and games. For thousands of years, humanity gambled like mad without knowing how to calculate their chances of winning. This started to change in the 17th century. First comes Gerolamo Cardano, an Italian polymath known for discovering the formula for solving cubic equations2. He paid the bills with gambling and wrote a book with calculations on games of chance in 1564, but it was not published and widely read until 1663. He is also known for discovering the formula for solving cubic equations (if you’re interested, read about it here or watch this video!). In 1654, Chevalier de Méré3 wondered how a pot should be divided if a game of dice is abandoned before it is completed. This inspired a famous letter exchange between Pascal and Fermat about this so-called Unfinished Game Problem4. The Dutch mathematician Christiaan Huygens heard about the correspondence while in Paris, and subsequently wrote De Ratiociniis in Aleae Ludo (‘Calculations in Games of Chance’). It is considered the first textbook on the calculations of probability, and was published in 1657. By the middle of the century, gaming had inspired the development of the first methods for statistical calculations.
Simultaneously, the philosophical leap to epistemological chance was unfolding. Before the 17th century, the word probable had a different meaning, disconnected to empirical observation or objective assessment of chance. Instead, a ‘probable opinion’ was one supported by authority, and it was distinct from demonstrable knowledge. A modern notion of evidence, that of information that objectively but not decisively supports an opinion, is lacking. Most closely related was the medieval concept of signs. Signs allow the testimony of the world to be read like any authority. This may sound like superstition, but not all signs were without merit: swarms of mice were a sign that forebode disease5, not far-fetched in times of plague. Halfway through the 17th century, evidence as we know it appears. Hacking argues that it originates in the ‘low sciences’ like alchemy and medicine. Unlike astronomy or mechanics, which could rely on demonstration, the low sciences have to grapple with uncertainty, and this caused the rigid divide between opinion and knowledge to be blurred. For example, a stomach ache is not a demonstrable proof for appendicitis, but it is objectively correlated with appendicitis, making it evidence partially supporting the diagnosis. The book Port-Royal Logic, published in 1662, first makes this explicit distinction between evidence and testimony relying on authority. It is also around this time Pascal formulates his Wager, which—although often criticized for its false dichotomy—is a first in decision making under uncertainty.
The emergence of probability can be seen as making the connection between aleatory and epistemological chance. There is no single event or person marking it, but there is a witness. In 1665, the nineteen years young Gottfried Leibniz was writing a thesis called On Conditionals for his Bachelor of Law. Not familiar with the Huygens work on games, Leibniz suggested that courts could formulate numerical degrees of belief based on available evidence. When years later he did encounter Huygens’ Alae Ludo in Paris, he summarized the pivotal decade:
“Chevalier de Méré [...] – a man of penetrating mind who was both a gambler and philosopher – gave the mathematicians a timely opening by putting some questions about betting in order to find out how much a stake in a game would be worth, if the game were interrupted at a given stage in the proceedings. He got his friend Pascal to look into these things. The problem became well known and led Huygens to write his monograph De Aleae. Other learned men took up the subject. Some axioms became fixed. Pensioner de Witt used them in a little book on annuities printed in Dutch”6
And there, in the midst of probability’s emergence, is Johan de Witt. How does this statesman find himself in the middle of a discussion so at the forefront of mathematics? To answer this question, we need to take a little detour. What was happening in The Netherlands at that time? And what role does Johan de Witt play?
Johan de Witt and the Stadtholderless Period
In the year 1671, the Dutch Republic was in the evening of its Golden Age. Lucrative trade from the Baltic Sea to Western Europe7, rich Flemish immigrants, and, notoriously, the East India Company's monopoly on the spice trade and the West India Company’s role in the transatlantic slave trade had made the small country a formidable international power. Its relatively open attitude towards different religions and freedom of thought had allowed art and science to bloom. This was the period of the Dutch masters, such as Rembrandt and Vermeer and thinkers like Spinoza, Huygens and Van Leeuwenhoek.
Johan de Witt held the office of Grand Pensionary, a position comparable to today’s prime minister. It was a special time, the First Stadtholderless Period, in which the federal assembly ruled and did not share power with the Stadtholder, a position otherwise held by the Prince of Orange. Johan de Witt called it De Ware Vrijheid (‘True Freedom’), a state of affairs in which prominent citizens ruled and did not share power with the nobility. In this, the Dutch Republic was an exception in Europe of the time. By 1671, Johan de Witt had held office for 18 years, during which he strengthened the Dutch navy, promoted trade, and twice successfully secured peace with the English.
It is here that Johan de Witt, our protagonist, wrote a public letter to the assembly of Holland. It consists of a short treatise titled Waerdye van lyf-rente naer proportie van los-renten (Value of annuities in proportion to redeemable annuities)8. It proposes a novel method for valuing life insurance, using probability to calculate life expectancy. De Witt already had a reputation in mathematics for his 1646 work on conic intersection, and the Waerdye shows that he was closely involved with the probabilistic revolution.
De Waerdye
The main mathematical contribution in the Waerdye is the calculation of life expectation using a mortality table. The notion of expectation was still so green that De Witt had to define it from scratch. Life expectancy is of great relevance to the Dutch state because it raised public funds using life annuities. Life annuities were a type of government bond, structured like reverse life insurance. Imagine a wealthy individual looking to see some return on their savings. They decide to buy a life annuity from the Dutch government, paying a lump sum—the purchase price—and designating a life: typically a young, healthy citizen. For every year that the life is alive, the government pays an installment. When the life dies, the government stops paying the installments. Part of the reason annuities were preferred over loans was because of the stigma against usury (making money through interest), which was seen as exploiting the less fortunate. But there is another reason: if because of a plague or a war a significant part of the population is wiped out, so is the public tax burden.
Life annuities were used to fund public expenditures as early as ancient Rome9. But just as long is the history of state defaults. Governments simply did not know how to price life annuities. For example, the underlying lives were often chosen to be young girls. Without mortality tables and life expectancy, the issuing administrators did not realize that this stratum of the population has the highest life expectancy. Girls of that age have survived child mortality, and were not exposed to the work and war related risks that their male counterparts faced. Their long life expectancy meant that the governments were stuck paying annual installments for many years.
It was this issue that De Witt investigated in the Waerdye. Specifically, he wondered at what years’ purchase (i.e. at what multiple of the annual installment) life annuities (lijf-renten) were competitive with loans (los-renten).
Together with his friend, mayor of Amsterdam, and mathematical collaborator Johannes Hudde, he took a close look at the annuities the state had issued in the past 80 years. From the age of the life at the start of the contract and the number of paid installments, he could construct mortality tables for the typical annuity life.
To go from mortality tables to life expectancy, De Witt first needed to define the concept of expectation. He took the first presupposition straight from Huygens’ Ludo Alae. The ‘fair price’ (rechte waerdye) of two events with equal chance, one giving nothing and one giving 20 florins, is exactly 10 florins. He generalizes this in a series of propositions, starting with equally likely events:
Chances
Each chance of
florins
______________ +
Fair price
And then extending the method to proportional chances:
Chances
Each chance of
is
___________
_______
Fair price
His second presupposition states that for a healthy human, the chance of dying in the first half or second half of a given year is equivalent to the chance of a coin falling heads or tails. This nuance is relevant, as the Dutch annuities of the time were paid in semi-annual installments. He continues:
“it depends entirely upon chance as to the side on which it shall turn, and this to so high a degree that the penny may fall head 10, 20, or more times following, without once falling tail; and vice versa.”
De Witt seemed to anticipate criticism of the fact that his mortality data shows, by chance, large differences in the number of deaths between two consecutive half-years.
The third presupposition states De Witt’s assumptions about life expectancy that he based on the data of the mortality tables. It states that for a man having lived past his vigorous years (around 53 or 54), in the next 10 years the chance of death in a given half-year is 3/2 that of a man in his vigorous years. This ratio increases to 2-to-1 for men between 63 and 73 and 3-to-1 for men between 73 and 80. In De Witt’s calculations, no one lives past 80. Not a surprising conclusion, given that De Witt’s data on annuities went back only 80 years. With these assumptions, he calculated life expectancy to be just under 37 years.
Armed with these presuppositions and a workable notion of expectation, a tedious calculation follows, which De Witt had checked by two accountants and Johannes Hudde. The full table sums up the total pay-out for survival for each of the 153 half-years. With a semi-annual installment of 10 million florins, discounted by the compounded 4% annual interest a loan would pay, the total table looks something like this, recalculated by me and Excel10.
Chances
Of (in florin)
The Life to survive Half-years
...
...
...
_________________
Sum=
Once=
...
...
...
_________________
Sum=
Two-thirds=
....
...
....
_________________
Sum=
Half=
...
...
...
years
_________________
Sum=
One-third=
___________
________________
Total
Fair price = Total = florins or florins
Thus, for the state to have a good deal, a 1,000,000 per annum life annuity should be sold for at least 15,971,302 florins. Put differently, De Witt concludes an annuity should be sold for at least 16 years’ purchase. This is starkly more expensive than the 9 years’ purchase which was common at the time. Clearly, his expectations showed that a life at four would be expected to live for more than 9 years.
De Witt’s legacy
Although the Waerdye was not mass published, we know that a translation reached Leibniz and interested Bernoulli, who intended to borrow it from the former. Leibniz never managed to find it between his papers, but told Bernoulli from memory what it contained. He had read it with great attention, and shared with Bernoulli his criticism of De Witt’s assumption of uniform mortality between 4 and 54. For a while, the Waerdye was considered lost. Nowadays, a Dutch original can be viewed at the Royal Library in The Hague11, where I took the pictures attached. De Witt is considered as the founding father of assurance science, and has been studied in that capacity as early as 185212. Several translations of the Waerdye in English are available online13 and it is considered the first application of probability in economics.
And what happened to Johan de Witt himself? The good times were not to last. Not for the Dutch Republic, nor for him. The year after writing the Waerdye, the Republic was attacked from three sides: by England, France, and Cologne-Münster. A political transition took place. The Orangists, supporters of the Prince of Orange, incited the populace to a feverish paranoia against the republican regents. They spread baseless rumors of treason by Johan de Witt and his brother. With the new information technology of mass-printed pamphlets14, the people's anger reached a boiling point, culminating in a furious mob led by Orangists lynching the brothers.
And what happened to Johan de Witt himself? The good times were not to last. Not for the Dutch Republic, nor for him. The year after writing the Waerdye, the Republic was attacked from three sides: by England, France, and Cologne-Münster. A political transition took place. The Orangists, supporters of the Prince of Orange, incited the populace to a feverish paranoia against the republican regents. They spread baseless rumors of treason by Johan de Witt and his brother. With the new information technology of mass-printed pamphlets, the people's anger reached a boiling point, culminating in a furious mob led by Orangists lynching the brothers.
This year of 1672 is called the Rampjaar (‘Disaster Year’) and marked the end of the Ware Vrijheid and the start of a period of decline for the Dutch Republic. Consequently, the country would not produce mathematicians of such high standing for at least 200 years. And will we ever see a prime minister so at the forefront of mathematics again?15 Looking back at Johan de Witt's story and his demise, I cannot help but see some parallels to today. Fake news on social media is not so different from mass-printed pamphlets. We see increasing harassment of scientists and free thinkers, not unlike De Witt in his time. Maybe we can take some comfort in the fact that his contributions outlasted the tumult.
Hacking, I. (2006). The emergence of probability: A philosophical study of early ideas about probability, induction and statistical inference. Cambridge University Press, ↩︎
Fracastoro, G. (1546). De sympathia et antipathia rerum tiber unus: De contagione et contagiosis morbis et eorum curatione, libri iii. Venice. (Translated by C. Wright as Fracastoro’s De contagione, New York and London, 1930.), ↩︎
Die philosophischen Schriften von G. W. Leibniz, ed. C. I. Gerhardt, Berlin, 1875–90,7 p 447. ↩︎
Lak, M. (2007). De Moedernegotie. Handel met Oostzeegebied bracht meer rijkdom dan de VOC. Historisch Nieuwsblad, 6, 26-29, ↩︎
Van Daele, G. (2020). De lijfrente verzekeringstechnisch en fiscaal. BEST Seminars, Abstracts. Presented at the De lijfrente verzekeringstechnisch en fiscaal, webinar. Translation here, ↩︎
Ciecka, J. E. (2012). Ulpians table and the value of life annuities and usufructs. J. Legal Econ., 19, 7. ↩︎
The original, unsurprisingly, contains some miscalculations. Neither was I able to reproduce the English translation from 1852, which, although almost 200 years smarter, still did not have the computational machines of today, ↩︎
Hendriks, F. (1852). Contributions to the History of Insurance, and of the Theory of Life Contingencies, with a Restoration of the Grand Pensionary De Wit’s Treatise on Life Annuities (Concluded from No. VI). The Assurance Magazine, 2(03), 222–258 ↩︎
Reinders, M. (2012). Printed Pandemonium: Popular Print and Politics in The Netherlands 1650-72 (Vol. 17). Brill, ↩︎
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florins
years
= 
florins or 
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