Probability Theory: the synthesis of commerce and ethics

This post presents details of the argument that mathematical probability emerged between 1564 and 1713 out of the ethical analysis of commercial practice.  This development of probability in the early modern period was built on foundations laid in the high Middle Ages.  This piece is taken from a longer article Reciprocity as a Foundation of Financial Economics .

From 1000 C.E. until about 1300 C.E. there was a rapid development of the economy in Western Europe as it evolved from an agriculturally based feudal society towards a commercially based bourgeois society, initially in Italy then, in the twelfth century, in North Western Europe. One physical manifestation of this change was the volume of coin circulating in the European economy, as the population doubled over the three hundred years, the amount of coin per person tripled. ([29, Chapter 3 & 4], [22, pp 15–16], [25, p 72])

Practice

Medieval European merchants, unlike their contemporaries in the Middle East, India or China, had to contend simultaneously with prohibitions on usury and the heterogeneity of currency. Muslim merchants had usury prohibitions but homogeneous currency, Indian and Chinese merchants had to (sometimes) deal with heterogeneous currencies but without the centralised religious prohibitions on usury.

Usury derives from the Latin usus meaning ‘use’, and referred to the charging of a fee for the use of money. Interest comes from the Latin interesse and originated in the Roman legal codes as the compensation paid if a contract was broken [19, p 73]. Shortly after 1200 the theologian, Peter the Chanter, argued that “a buyer or seller may be excused from usury if he exposes himself to the risk of receiving more or less” [13, pp 263–264] and this idea that usury was absent in the presence of risk became firmly established in the thirteenth century.

The basic financial instrument at this time was the census that originated when ninth century monasteries guaranteed a fixed regular income in exchange for a donation of land. Censii developed to be written on the back of a diverse range of assets, including a craftsman’s labour, resembling modern day securitisation. In time ‘structured’ contracts emerged such that a borrower would receive a lump sum secured against the future cash-flow from an asset, rente à prix d’argent, without necessarily relinquishing ownership of the asset ([19, pp 75–76], [28, pp 31–33]).

Modern structured finance was anticipated in the triple, or German, contract (contractus trinus), developed to fund long distance trade. It involved a loan to fund the venture (the first contract); the transformation of the variable return of the venture into fixed cash-flow (the second contract); and an insurance contract to guarantee the fixed payment (the third contract). In terms of contemporary finance this third contract is a Credit Default Swap and the whole contract has the same structure of a Special Purpose Vehicle. This contract was declared illicit by the Catholic Church in 1586 on the basis that the lender received a risk-less return. [26, pp 209–220]

The heterogeneity of currency was a consequence of feudalism and the desire of magnates to assert their authority by issuing coin. The Italian peninsula had over twenty currencies, the Kingdom of France three, and each prince of the Holy Roman Empire would mint their own coin. Alfred Crosby describes the activities of a Tuscan merchant in supplying cloth to Venice from Mallorcan wool that involved at least five currencies [8, p 201]. William Goetzman explains that as a consequence of the multitude of currencies, European medieval merchants “operated in a world of complete relativism” [14] while Crosby remarks that there was an “abstraction of Western merchants’ scale of value” and “no people were more obsessed with counting and counting and counting”[8, p 72, 74].

A solution to the problem of the complexity of Medieval commerce came in Fibonacci’s Liber Abaci first published in 1202, the initiant of Financial Economics ([8, 43–47], [32, Introduction]). It was an immediate success and a second edition was produced in 1228, a remarkable feat in an age when books were hand copied [32, p 4]. The text introduces Arabic/Hindu numerals and explains basic arithmetic over seven chapters. It then presents four chapters applying the theory by presenting cases on practical commercial problems. The text finishes with a more theoretical section on iterating to a solution of a problem. ([32], [14])

Before the Liber Abaci, European merchants, like their contemporaries across the globe, would have used an abacus to perform arithmetic calculations, and once a calculation had been made, it was recorded. The technologies described in the Liber Abaci, particularly Hindu numbers, meant that merchants could write down their calculation method, the algorithm, which could be copied and modified by others. Knowledge, in the form of best practice, could be created, distributed and improved.

Abaco or rekoning schools sprang up throughout Europe teaching apprentice merchants the techniques originating the Liber Abaci. The impact of these abaco schools was enormous, algebra became an important tool used by the large and influential community of Europeans and would provide the reservoir of mathematicians on which the scientific developments of the seventeenth century were built. The unique circumstances of medieval European commercial practice offer a solution to Needham’s question that asks why European technological development accelerated so much faster than Chinese after 1600. ([16, Chapter 1], [32, Introduction], [18] )

Theory

The societal changes before 1200 led to a need to revitalise the Catholic Church, particularly to combat unorthodoxy such as Catharism. The Dominican and Franciscan orders were established to engage with the emerging bourgeoisie and would come to dominate Scholasticism, the intellectual movement that integrated Greek philosophy and Christian theology in Europe’s universities until the Reformation.

The science that emerged in Western Europe in the seventeenth century is distinctive in its use of mathematics to describe the laws of nature. The Greeks, and their Muslim successors, generally regarded ‘pure’ mathematics as being irrelevant to the sensible world while Chinese scientists used mathematics to calculate but not to describe ([8, p 16], [11, p 164], [12, p 53]). Richard Hadden, Alfred Crosby and Joel Kaye have all argued that the ‘mathematisation’ of European science began with the synthesis of commercial practice and Scholastic ethics in the thirteenth and fourteenth centuries ([16], [8], [22]).

A key component of this synthesis was Aristotle’s Nicomachean Ethics that addresses how an individual can live as part of a community and it discusses economics in Book V in the context of the virtue of Justice. Aristotle saw reciprocity in exchange as being important in binding society together, and Aristotle believed exchange was performed to correct for inequalities in endowment and to establish a social equilibrium, not in order to generate a profit ([22, p 51], [5, 1133a15–30]).

Aristotle distinguishes economic justice into two classes, distributive and directive (or corrective, restorative). Distributive justice is concerned with the distribution of common goods by a central authority in proportion to the recipients’ worth and is determined by equating Geometric Proportions. Directive justice applies in cases where the parties are considered to be equal, for example in commerce, in which case justice is determined by equating Arithmetic Proportion and is based on reciprocity ([22, p 41–43], [5, 1130b30–31a5]).

What is most striking in Aristotle’s treatment of economic exchange is that he approached it as a mathematical problem. This is remarkable in itself because Aristotle rarely applied mathematics to the sensible world elsewhere ([16, p 75], [8, p 13], [5, 1094b15–28]). Aristotle realised that if there was to be equality and Justice then

everything that is exchanged must be somehow comparable. This is the role that is fulfilled by currency [nomisma], so that it becomes, in a way, an intermediate [5, 1133a19–20]

These lines are significant for two reasons. Firstly the word nomisma for currency/money is related to the concepts of custom and law, not to ‘labour and expenses’. Secondly, ‘intermediate’ is in the sense of a mediator between two objects, rather than simply as a token, which is a more modern interpretation. Furthermore, Aristotle defined the quality that money measured by the word chreia, which was initially translated to opus (work), but was later corrected to indigentia (need) [22, pp 68–70]. This is important because it demonstrates that Aristotle and the Scholastics viewed money as a social construction binding society by allowing an exchange based on need, rather than as a simple commodity facilitating the exchange of sensible quantities, such as labour and expenses.

The significance of the Scholastic analysis to the development of science was that when Aristotle discussed measurement in the context of physics he argued that the measure shared the ‘substance’ of the measured; this meant that wine was incommensurable with cloth, time incommensurable with space. The Scholastics realised that money was a very special measure; it applied to all goods in a market, and only occasionally shared the substance of the goods. This insight enabled them to revolutionise the concept of measurement, in a way that contemporary Muslim scholars did not, and allowed Jean Buridan to identify the concept of inertia. ([4, p 263-268], [8, p 67–74], [22, pp 65–70])

Out of Aristotle’s discussion of market exchange, Scholastics developed the concept of the ‘Just Price’, which has been the subject of considerable modern debate. For example, Raymond de Roover [10], argues against viewing the Just Price in a Marxist, labour theory of value, sense but rather as the market price, in a neo-classical, liberal sense. However, neither of these modern positions corresponds to how the Scholastics viewed the concept. The interpretation of the Just Price we shall employ, based on the Scholastic attitudes to Aristotle’s description of exchange, is the one discussed by Fabio Monsalve [24, pp 6–7]. The Just Price represents an “intellectual construct: an ideal price that guarantees equality in exchange” and that it represents a mathematical ‘medium’ or a ‘mean’.

Monsalve points out that Scholastic analysis was conducted in a definite moral frame of reference, and so the Just Price “could not refer indiscriminately to whatever price might be obtained in the market” [24, p 8, quoting Langholm]. This aspect was discussed in detail by the Scholastics prompted by a question ‘Whether the seller is bound to state the defects of the thing sold?’ posed by the important Dominican Thomas Aquinas [1, II, ii, qu. 77, art. 3, ad. 4]. Specifically Aquinas addresses a problem originating in Stoic philosophy relating to the conduct of a merchant carrying a supply of food to a starving country. The merchant knows that they are the first of a number of merchants bringing food, the question is, should he sell the food at the high ‘market’ price or a lower price based on his knowledge.

Kaye makes the point that Aquinas separates the Just Price, determined by divine law, from the ‘market price’, established by men, and explains that if the Just Price equated with the market price then an “individual’s responsibility in economic activity is effectively eliminated” [22, p 98]. Despite realising this distinction, the answer from Aquinas is a little surprising. Aquinas observes that the merchant may believe that there are more grain shipments on the way, but does not know: the future is uncertain. On the basis that there is no certainty, and on the authority of Peter the Chanter, the merchant may charge the going market price, making an excessive but nevertheless legitimate profit, though it would be more virtuous to charge the lower price.

Aquinas’ argument was criticised by Pierre Jean Olivi, a leader of the ‘Spiritual Franciscans’. The Spiritual Franciscans argued that the vow of poverty meant monks should limit their use of property, usus pauper, not simply not own property. As a consequence of this extreme position Olivi was posthumously condemned as a heretic in 1326, hindering the subsequent transmission of his thought. The Franciscans, unlike the empirical rationalist Dominicans such as Thomas Aquinas, were fideists and this philosophical approach meant that Olivi argued that the metaphysical probability of more grain arriving had a certain reality, which Aquinas was ignoring [22, p 121]. Olivi said

The judgement of the value of a thing in exchange seldom or never can be made except through conjecture or probable opinion, and not so precisely, or as if understood and measured by one invisible point, but rather as a fitting latitude within which the diverse judgements of men will differ in estimation [22, p 124].

This distinction is essential in demarcating the Just Price, an imprecise abstraction, from the market price, which is observed at a fixed point [24, Section 3.2.1].

Olivi seems to have interacted with merchants and been a close observer of markets and considered a number of aspects of commerce including the problem of usury [13, p 265]. Based on the principle that a lender could charge a borrower compensation for a loss (interesse) Olivi recognised that borrowers should compensate lenders for the ‘probable profit’ they could earn by employing capital elsewhere. Fair exchange was a question of restoring ‘probable equivalence’, not of precise equality ([22, p 119], [13, pp 265–267]). As part of this argument Olivi commented that a valuation did not only depend on ‘need’ but also on a good’s scarcity, usefulness and desirability. Since both need and desirability are subjective, different people will value the same good differently and based on these ideas, Olivi was able also to explain the ‘value paradox’ ([30, pp 60–61], [22, pp 123–124]). Ultimately, according to James Franklin, Olivi thought of probability as a trade-able entity, and so could be quantified [13, pp 266–267].

The Science of Conjecture

The Science of Conjecture, or Probability, is the rational method for dealing with uncertainty. Aristotle classified events into three types: certain events determined by specific causes; probable events that usually happened; and unpredictable events, including games of chance, not amenable to science [17, p 30]. The development of Probability over the past five hundred years has been concerned principally with reducing the scope of those events ‘not amenable to science’.

While Olivi and merchants developed the idea of probability in relation to commercial exchange and jurists and theologians addressed questions of proof the concept of quantifying chance did not fully materialise until the mid-sixteenth century with Cardano’s Liber de Ludo Alea. Ian Hacking has remarked that the emergence of the concept of absolute chance was late; however, this identification of mathematical probability in the context of finance precedes both Descartes’ introduction of absolute space (Cartesian co-ordinates) and Newton’s of absolute time.

Up until the 1950s, and a re-assessment of his work by Øystein Ore [27], Cardano’s contribution to probability theory had been widely ignored. In the context frequentist interpretations of probability, that dominated the nineteenth and early twentieth centuries, it was seen as incoherent. More recently, David Bellhouse [2] has re-evaluated the Liber looking at it as a humanist philosophical text, not as a mathematical document, based on the fact that Cardano, himself, did not list it as one of his mathematical works. Bellhouse’s hypothesis is that in the Liber Cardano is trying to establish under what grounds gambling can be considered ethical in the context of Nicomachean Ethics.

Cardano latches on to the idea that Justice is equivalent to equality and argues that in dice games ‘equality’ was established by counting the ways a player could win and comparing that number to the ways a player would lose. On this basis the ‘chance’ of winning could be deduced, and if the stakes did not match the chances, the gamble was unjust. Summarising his findings he states, “a just gamble is one between willing and knowledgeable players”, making an explicit association between science and ethics. Almost immediately after coming to these ethical conclusions, Cardano observes that

These facts contribute a great deal to understanding but hardly anything to practical play [9, p 58 quoting from Chapter 9 of the Liber]

since they offer nothing to help forecast the outcome of the dice throw.

Throughout the sixteenth century there was a flow of mathematical thinking from the vernacular abaco schools into academic circles, not just in the universities that had declined in status during the Renaissance but also into the courts of autocrats. For example, Lucca Pacioli was trained in the abaco tradition, probably by Piero della Francesca, before working for a Venetian merchant. He then took Franciscan orders and entered the University of Perugia before ultimately joining the court of the Milanese autocrat, Lodovico Sforza, where he taught perspective to Leonardo da Vinci. Thomas Gresham was born into an important English commercial family around the time Pacioli died and spent much of his life manipulating the exchange rate in Antwerp on behalf of the English monarchy. When he died he created the first chair in mathematics in England with the establishment of Gresham College, which was the foundation of the Royal Society ([20], [6]). Simon Stevin also trained in the abaco tradition and worked as a merchant’s clerk then as a tax official before moving to the University of Leiden in 1583. Ten years later he became involved with the government of the Dutch Republic during the wars of independence against Spain. As part of the war effort, the Dutch Mathematical School was established to train military engineers with Stevin as Director. Stevin’s vocational syllabus attracted soldiers (such as Descartes) and merchants from across northern Europe and the fees they paid raised the status of its mathematicians, who when associated with the scholastic syllabus of abstract arithmetic and geometry, astronomy and music, were widely regarded as irrelevant. Stevin wrote a number of textbooks in French or Dutch, not in the exclusive Latin of the universities, which became read widely and emphasised the practical usefulness of mathematics in everyday life. His influence was profound and forced other institutions change their curricula; our use of decimal notation is due to Stevin who recognised its utility in commerce. ([31], [23, p 121], [28, pp 131–132], [11, p 104])

One problem Cardano considered was the so-called Problem of Points which appears in a text by Pacioli and is based on the following situation:

Two players, F and P, are playing a game based on a sequence of rounds, and each round consists of, for example, the tossing of a fair coin. The winner of the game is the player who is the first to win 7 rounds, and they will win 80 francs.

The Problem of Points is how the 80 francs should be split if the game is forced to end after P has won 5 rounds while F has won 4.

Edith Dudley Sylla notes that the Problem comes from the abaco tradition of using ‘stories’ to give examples of how to solve problems in commercial arithmetic. In this case the Problem of Points, the story represents the case of how the capital tied up in a business partnership should be divided if the venture has to finish prematurely [33].

Pacioli’s solution was statistical, the pot should be split 5:4. Cardano realised this was absurd since it would give a manifestly unfair result if the game ended after one round out of a hundred or when F had 99 wins to P’s 90. Cardano makes the point that the correct solution would be arrived at by considering what would happen in the future, it had to be forward-looking, in particular it had to account for what ‘paths’ the game would follow. Despite this insight, Cardano’s solution was still wrong, and the correct solution was provided by Pascal and Fermat in their correspondence of 1654.

The Pascal-Fermat solution to the Problem of Points is widely regarded as the starting point of mathematical probability. The pair (it is not known exactly who) realised that when Cardano calculated that P could win the pot if the game followed the path PP (i.e. P wins and P wins again) this actually represented four paths, PPPP, PPPF, PPFP, PPFF, for the game. It was the players’ ‘choice’ that the game ended after PP, not a feature of the game itself and this represents an early example of mathematicians disentangling behaviour from problem structure. Calculating the proportion of winning paths would come down to using the Arithmetic, or Pascal’s, Triangle – the Binomial distribution. Essentially, Pascal and Fermat established what would today be recognised as the Cox-Ross-Rubenstein formula [7] for pricing a digital call option.

The Pascal–Fermat correspondence was private, the first textbook on probability was written by Christiaan Huygens in 1656. Huygens had visited Paris in late 1655 and had been told of the Problem of Points, but not of its solution ([9, p 111], [17, p 67]), and on his return to the Netherlands he solved the problem for himself and produced the first treatise on mathematical probability, Van Rekeningh in Speelen van Geluck (‘On the Reckoning of Games of Chance’) in 1657.

In Van Rekeningh Huygens starts with, what is essentially, an axiom,

I take as fundamental for such [fair] games that the chance to gain something is worth so much that, if one had it, one could get the same in a fair game, that is a game in which nobody stands to lose. [17, p 69]

Probability is defined by equating future gain with present value in the context of ‘fair’ games.

In the 1670’s probability theory developed in the context of Louis XIV’s appartements du roi, thrice weekly gambling events that have been described as a ‘symbolic activity’ not unlike potlach ceremonies that bind primitive communities [21, pp 31-42]. This mathematical analysis of an important social activity stimulated the publication of books describing objective, or frequentist, probability. The Empirical frequentist approach began to dominate the mathematical treatment of probability following the claimed ‘defeat’, or ‘taming’, of chance by mathematics with the publication of Montmort’s Essay d’Analyse sur les Jeux de Hazard (‘Analytical Essay on Games of Chance’) of 1708 and De Moivre’s De Mensura Sortis (‘The Measurement of Chance’), of 1711 developed in The Doctrine of Chances of 1718 [3]. These texts were developed more in the context of gaming rather than in the analysis of commercial contracts and The Doctrine was the more influential, introducing the Central Limit Theorem, and by 1735 it was believed that there was no longer a class of events that were ‘unpredictable’ [3].

Around 1684 James Bernoulli had begun working on problems in probability and between 1700 and his death in 1705 he worked on Ars Conjectandi (‘The Art of Conjecturing’), a title that emphases the practical rather than theoretical nature of conjecture, which was published posthumously in 1713. The Ars is made up of four parts, a commentary on Huygens’ Van Rekeningh, original work on calculating permutations and combinations, applications of these ideas to games of chance and finally the application of the ideas to “civil, moral and economic affairs” [17, p 224].

While the first three sections of the Ars are un-controversial, the final section is both the most significant and has proved problematic. Bernoulli, having discussed objective probability at length introduces the epistemic, or subjective, definition of probability as “a degree of certainty”. Anders Hald notes that this is “revolutionary” because Bernoulli is applying mathematics to propositions, not just to events [17, p 225]. This section of the Ars is significant in that it introduces what would become known as the ‘Law of Large Numbers’, which can be summarised as collecting a large amount of data will improve the accuracy of an observation – providing the system was stationary [17, p 225]. The section is problematic because Bernoulli considered situations where the sum of probabilities could be greater than one [34, p 27]. This is impossible if probability is calculated as relative frequency.

Sylla compared Bernoulli’s work to that of Huygens’ and other contemporaries, de Witt and de Moivre, in the process of translating the Ars and concluded that

equity among associates or partners rather than probabilities in the sense of relative frequencies provided the foundation for the earliest mathematical probability theory. [34, p 13]

and that

While traditional histories of mathematical probability start with Pierre Fermat, Pascal and Huygens because they give what are from the modern point of view correct frequentist solutions to the problems of division and expectations in games of chance …the foundations of Huygens’ method (…) was not chance (frequentist probability), but rather sors (expectation) in so far as it was involved in implicit contracts and the just treatment of partners. [34, p 28]

In the sixteenth and seventeenth centuries the motivation for the development of probability was in the ethical analysis of commercial contracts where Justice, or balanced reciprocity, ‘fairness’ dominated. The later Empirical approach to probability, based on observing relative frequencies, emerged out of the simpler analysis of games of chance in the context of fixed odds.

The case that Huygens was working in the context of Virtue Ethics is enhanced by recognising the difficulty he had in translating Van Rekeningh into Latin [15, pp 93–94]. Huygens struggled to translate the Dutch word kans (‘chance’, ‘lot’), which would normally be translated as sors, and eventually he, or his editor van Schooten, chose expectatio, giving the English term ‘expectation’ (in the mathematical sense). However, Huygens had considered using the Latin word spes [15, p 95] which was the term for the virtue ‘Hope’. In French, espérance is used when referring to mathematical expectation, reflecting this debate. The Dutch, who following Stevin’s focus on teaching mathematics in the vernacular, use their own terms in mathematics, in this case the equivalent is verwachting: hope, promise, expectation, forecast, prognosis.

Sylla also observes that The Port Royal Logic, a significant influence on Pascal, notes that “because the house takes part of the stakes, lotteries are manifestly unfair” and seventeenth century mathematicians recognised a distinction between actual gambles, involving transaction costs, and idealised, frictionless, markets, suitable for the mathematical study by academics. [33, p 327]

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