Oedipus and the difficult relationship between maths and economics

Noah Smith recently wrote a “rant” on maths in (macro)economics, which elicited a swift response from Paul Krugman, which in turn prompted Bryan Caplan to argue that “Economath fails the cost benefit analysis”. As a mathematician (i.e. someone who is employed by a maths department of a university) I think all the commentators make valuable points but, maybe somewhat surprisingly, my strongest affinity is with Bryan Caplan’s position that maths is failing economics. What I intend to do in this post is present an argument that, historically, `physics maths' has been driven by economic intuition and the contemporary problems in economics are that it is adopting maths from the physical science rather than generating a more insightful mathematics of its own. In the context of Noah’s original article, econmath is not enjoyable in the same way that ill fitting clothes are not enjoyable, or playing football in tennis shoes can be painful Metaphorically, maths is Oedipus and economics is King Laius: maths does not recognise its father, father does not recognise child, and tragedy follows.  

In a recent post Caplan challenges Krugman to “Name three important economic insights you think we owe to economath.”, I will support my case by naming three critical developments in “physics math” that come out of economic intuition.

The mathematisation of physics

Aristotle, and his successors such as the medieval Islamic scientists, did not think that maths had anything to offer physics, probably similar to Caplan’s view that maths has little to offer economics. Maths had nothing to say about causes, what was important in understanding nature were the qualities of objects, whether they were heavy or light, hot or cold, wet or dry, hard or soft. Since mathematics is concerned with precision, numbers and proportions, it could not give any insight into these qualities; “boulders were never perfectly spherical nor were pyramids perfectly pyramidal, so what was the use of treating them as such” [7, p 16]. The critical distinction for Greek science and its descendants was that mathematicians dealt with pure abstract objects while natural philosophers and engineers work with the concrete, messy, qualities of physical objects [9, p 65]. This approach was not just Greek, other cultures, such as the Chinese, took a similar line on the usefulness of maths to science [11, p 53].

Approaching physics through mathematics was uniquely European and was first recorded by the ‘Merton Calculator’ Thomas Bradwardine in

[Mathematics] is the revealer of genuine truth, for it knows every hidden secret and bears the key to every subtlety of letters. Whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start that he will never make his entry through the portals of wisdom [13, p 176]

Bradwardine had entered Merton College in 1323 and twelve years later went on work for the Bishop of Durham, who was the Treasurer and Chancellor of England, was eventually appointed Archbishop of Canterbury shortly before his death in 1349.

Arguing that physics was inseparable from mathematics was revolutionary because it completely broke with Aristotle’s approach to physics. From this point ‘Latin’ science would take a very different path to Islamic science, which was based on the same Hellenistic tradition. Bradwardine put this theory into practice by looking for a mathematical description of Aristotle’s laws of motions in his 1328 book De proportione velocitatum in motibus (‘Concerning the ratio of speeds in movement’), he achieved this, but being based on Aristotle’s incorrect principles, it was wrong. The ideas were developed by subsequent Calculators and the French philosophers Jean Buridan, who identified the concept of inertia, and Nicolas Oresme, who identified the mean speed theorem.

Key to unlocking the laws of motion was the realisation that speed was the ratio of distance divided by time. This seems obvious today but was not to Aristotle who had investigated measurement in Physics and Metaphysics, parts of the Organon where he claimed that a measure is always the minimum unit of the thing being measured. The measure shares the same substance as the subject of measurement. For example, numbers are measured by the smallest number, ‘1’, distances are measured by the smallest length of distance, say an inch, while liquids are measure in pints and so forth. This is not conceptually unnatural, the Chinese language still incorporates ‘measure words’ when quantifying objects. When Aristotle measured physical objects, as Richard Hadden has noted, “he [was] careful not to mix quantities differing in kind in the same expression” [12, p 75], he, and scientists in the Greek tradition could not get their heads around the very concept of ‘metres per second’ or ‘foot—pound’.

So why did Bradwardine ‘get it’? Joel Kaye has argued that the 45 Fellows at Merton were not just occupied in academic activities but were involved in the daily management of the College’s extensive resources

The three bursars who had collective responsibility for the college’s monies, the warden who headed the yearly audit and visited the far-flung manors at harvest time to assess the year’s taxation, in money or in kind, [others] oversaw the books and calculated the profits of the college’s many properties [17, p 33]

The point is Bradwardine would have developed his mathematical skills through economic application. But this is insignificant on its own, what was critical was the change in the conception of measurement that had occurred when Albert the Great studied Aristotle’s Nicomachean Ethics, in the 1250s after it had been translated into Latin in 1248/9.

Book V of Nicomachean Ethics considers the justice of economic exchange, and as a philosophical text it is disjointed and hard to follow. However, what is clear is that Aristotle sees reciprocal, fair, exchange in the market as being fundamental to a well functioning society since it binds individuals together [17, p 51]. Since Nicomachean Ethics was concerned with justice, Aristotle needed to identify under what conditions economic exchange was ‘just’ and he did this by insisting that there needed to be an equality between the goods being exchanged [17, p 45]. Aristotle identified that for equality to be established there needed to be a measure of the value of the goods, and this measure, the price, was provided by money.

all things that are exchanged must somehow be comparable. It is for this end that money has been introduced, and it becomes in a sense an intermediate; for it measures all things, and therefore the excess and the defect — how many shoes are equal to a house or to a given amount of food [17, p 47 quoting Ethics]

In his translation, Grosseteste described money as the medium of exchange. Today we interpret this in the sense that money is a physical token, however this is a modern interpretation. For the medieval scholar the Latin word medium was more commonly used in the sense of a mediator or intermediary, money is a neutral agent that links two distinctive commodities, such as shoes and houses.

Albert realised that if Aristotle was right about money being a measure he could not be right about a measure sharing the substance of the measured. This insight enabled Albert, and his successors, to revolutionise the concept of measurement, in a way that contemporary Muslim scholars did not. In particular students of Albert were able to reject Aristotle’s theories of measurement and consider concepts like metres per second or kilograms—metres per second (momentum). [17, p 67]

The mathematisation of Western Science is a consequence of the ethical assessment of an en economic activity.

The development of calculus

Bertrand Russell (amongst others) stresses the fundamental role probability (and statistics) has in science [21, page 301], it is less well known that probability emerges out of the ethical assessment of commercial contracts following Aristotelian concepts of Justice. Even less well known is the role financial practice and theory had in the genesis of Newton’s calculus on which his physics is based.

Many people appreciate that Arabic (Hindu) numbers were introduced into Europe through Fibonacci’s Liber Abaci, a financial text book. But Fibonacci employed fractions, not decimals. The significance of the decimal notation was highlighted by the Flemish mathematician Simon Stevin in his 1585 text , De Thiende (‘The tenths’). Stevin had not invented decimal fractions, they had been used by the Arabs and Chinese and first appear in Europe in a German text on algebra of 1525, but the audience for De Thiende was ‘practical men’ and Stevin pointed out that ‘all computations that are met in business may be performed’ using his notation. Stevin’s notation was in fact a bit cumbersome, decimal fractions as we know them appear in English in 1616. [2, p 316—317] Its also worth pointing out that the ‘=’ sign was introduced in The Whetstone of Witte written by Robert Recorde who controlled the English Mint).

The significance of decimal fractions to calculus is in how Newton tackled the problem of motion, building on Oresme’s work. While in Lincolnshire avoiding the 1666 Plague, Newton thought about how a point turned into a line by moving in an infinitesimal moment, for example how a pencil-line is drawn on a piece of paper. Newton called the resulting curve a fluent and he called the velocity of how the fluent grew in a moment its fluxion. For example a fluent could be the distance of a cannon ball from a cannon, the fluxion its velocity, or a force (the product of mass and acceleration) on cannon ball was the fluxion and its momentum (the product of mass and velocity) the fluent. Despite coming up with these ideas before 1667, Newton only circulated them in 1671 in Tractatus de methodis serierum et fluxionum (‘A Treatise on the Methods of Series and Fluxions’) [16, p 462].

Following Descartes’ Newton was comfortable in describing the distance travelled by an object with a function, and following Buridan and Oresme he understood the fluxion was the rate of change of the fluent, but in order to solve problems in general he needed to develop a way of deriving a function for a fluxion from a function for a fluent.

The key to unlocking this puzzle lay in decimal notation. before Stevin, numbers were generally written as composed or continued fractions. For example, as a composed fraction the sum of money five pounds, seven shillings and nine pence £5 and 149 pence, or 5would be written as a composed fraction,

as a continued fraction,

while in decimal notation it is

Newton realised this was the same as writing

and was interested in whether a similar approach could be taken with functions.

Newton’s approach, to write a function as a series

was not unheard of. Independently Indian mathematicians had considered writing functions as power series as early as the fourteenth century while in 1688 Nicolas Mercator considered power series for logarithms. (Mercator was German and originally called Niklaus Kauffman, he Latinised his name to Mercator; both Mercator and Kaufmann mean ‘merchant’). If a decimal number, written as a continued fraction, never ends it is an ‘irrational’ number rather than a ‘rational’ one. If a function when written as a power series/polynomial has a finite number of terms it is ‘algebraic’, if it has an infinite number of terms it transcends algebraic functions, it is transcendental, and usually given a name, such as sin, log, exp, etc.

Having realised that he could write a function as a power-series, working out how the function changed could be resolved if he could work out how a simple power changes, by establishing

What follows are two supreme achievements of Western science, Newtonian mechanics and Mathematical Analysis.

At this point it is worth observing that the diagrams in Noah’s original blog post are both directly related to these first two examples.

The Bachelier myth

Paul Samuelson tells the story of how

In the early 1950s I was able to locate by chance this unknown book by a French graduate student in 1900 rotting in the library of the University of Paris and when I opened it up it was as if a whole new world was laid out before me. [4, 13:00]

The book Samuelson refers to is Louis Bachelier’s thesis Théorie de la spéculation. The story that Samuelson tells, and has been disseminated, associates Bachelier’s work with Einstein’s work on Brownian motion rather than the pre-existing theories around finance. The alignment of Bachelier’s work with theoretical physics, rather than highlighting how the practice of applying mathematics to economics informs the development of mathematics, is an example of the tendency of academics to elevate theory over practice [10, Chapter IV].

Bachelier’s work had nothing to do with the physical process of Brownian motion, which Einstein was interested in, but was part of a long (French) tradition of employing the Binomial random walk model in finance. The canonical origins of mathematical probability are in the Pascal-Fermat solution to the Problem of Points, which introduced the model in 1654, a decade before calculus appears. Vernacular research into the subject includes French actuary Emmanuel-Etienne Duvillard’s Rechererches sur les rentes, les emprunts et les remboursements (‘Researches on annuities, loans and refunding’) [23] and Jules Regnault’s Calcul des Chances et Philosophie de la Bourse (‘Probability and the science of the markets’) of 1863 [15, .] In 1870, Henri Lefèvre de Chateaudun, an actuary who had been the private secretary of Baron de Rothschild [14] published Traité des valeurs mobilières et des opérations de Bourse: Placement et spéculation (‘Treatise of financial securities and stock exchange operations’)” [20]. In 1875 the Danish actuary and astronomer Thorvald Thiele introduced the concept of the random walk to model observational errors, incorporating a dynamic component to Gauss’ earlier work [18]. In 1894 Poincaré, France’s greatest mathematician of the time, chose to teach probability over all other subjects. At the time it was traditional for lecture courses by prominent mathematicians to be edited and published. In the case of Poincaré’s Calcul des Probabilitités, the editing of the lectures was carried out Albert Quiquet, an actuary working for the La Nationale insurance company, and not a university student, as was usual [3, p 279], highlighting the active interest of practising financiers in theoretical probability.

It was into this tradition that Bachelier was drawn. On account of his parents’ early deaths, Bachelier had been unable to enter university after school, and had worked initially for the family’s firm in Normandy and then moved to Paris where he traded rentes (i.e. consols) while studying mathematics at the University of Sorbonne, eventually taking his doctorate nominally under Poincaré ([6], [22]). Bachelier’s thesis is known for extending the discrete time Binomial random walk model into a continuous time model (i.e. it is a descendant of Newton’s work on calculus), and this is the aspect Samuelson picked up on.

Bachelier was not a succesful academic. It is rather obscure how Bachelier earned his living in the following decade, he obtained a few scholarships and in 1909 became a ‘free’ (unpaid) lecturer at the Sorbonne, lecturing on probability theory applied to finance. [6] In 1912 he published Calcul des Probabilitiés (‘Probability Calculus’) and then in 1914, Le Jeu, la Chance et le Hasard (‘Game, Chance and Randomness’). That same year, on the verge of being permanently appointed to the University of Paris, he, along with every other fit young Frenchman, was conscripted, as a private, into the army. He survived the cataclysm of the war, finishing it as a lieutenant, and in 1919 he took up a temporary post at the University of Besançon. He had further temporary positions at Dijon, between 1922 and 1925 and Rennes between 1925 and 1927.

In 1926 a permanent position had become available at Dijon, which Bachelier applied for. His application was reviewed by a professor at Dijon who was not familiar with Bachelier’s work but believed an important article published in 1913 contained a ‘gross error’. The referee wrote to a young ‘doctoral brother’ (student of the same doctoral adviser) who was developing a reputation in probability, Paul Pierre Lévy, to comment on Bachelier’s work. Unlike Bachelier’s unconventional pathe into academia, Lévy was the son of an academic at the École Polytechnique, where he studied and published his first paper at the age of 19 in 1905 becoming a professor at the École Nationale Supérieure des Mines in 1913, and spent the war doing research for the French artillery. In 1920 he was appointed to the École Polytechnique and it was on the basis of his 1925 book Calcul des Probabilitiés that he was asked to report on Bachelier.

Lévy checked the page that contained the suspected error and agreed with the referee, Bachelier was blackballed. The issue that the referee and Lévy had with the paper was that it appeared to contradict a feature of Weiner’s 1921 formulation of Brownian Motion, the fact was it didn’t if you followed the approach Bachelier had been taking since his dissertation. The reviewer’s were unfamiliar with this work and so the 1913 paper was ambiguous and appeared wrong. Though the blackballing was painful to Bachelier, in 1927, at the age of 57, he finally secured a permanent position at Besançon where he would remain until his retirement in 1937. He wrote two more books on probability in retirement, and died in 1946 in Brittany.

While Bachelier was never appointed to one of the great French universities, perhaps because his vocational background did not conform to France’s Rational ideal, he had a successful academic career and to suggest his thesis was ‘rotting’ in a library ignores the significant contribution Bachelier made to mathematics. Notably his idea that probability was dynamic which was a significant conceptual leap at the time and was taken up by Kolmogorov when laying the foundations of modern probability.

In his thesis Bachelier discusses what he calls Rayonnement de la probibilité (‘Radiation of probability’) [1, p 46—47], [8, p 41], the idea that a probability density evolves in time. This was revolutionary, up until Bachelier had made his observation in his thesis it had been assumed that probabilities were static, even Bayesian approaches assumed the probability density was static but you learnt more about it in time. The idea emerged in physics following Einstein’s work in 1913 as the Fokker-Planck equation.

Kolmogorov became familiar with Bachelier’s work, and when Lévy this he realised also that if he had looked into the Bachelier’s work in 1927 when he refereed his application to Dijon, he might have come to Kolmogorov’s conclusions before the Russian and be famous today for laying the foundations of modern probability. Lévy apologised to Bachelier and publicly acknowledged Bachelier’s priority, not just over Wiener in the mathematical study of Brownian motion, but also over over Kolmogorov in linking Brownian motion to Fourier’s heat equation, and over himself in establishing certain properties of the Wiener process. [22, pp 20—21]

I think the Bachelier myth is incredibly important in demonstrating how academics, from Lévy in 1927 to Samuelson in 1999, denigrate the significance of economic phenomena in generating mathematical ideas that have profound impact in the physical sciences.

Conclusions

I sympathise with Bryan Caplan’s claim that modern mathematics obscures economic intuition. I think Noah Smith’s point that econmath is unappealing is because the mathematics developed in response to physical problems is not suited to economic problems.  

The observation that economists could do better with mathematics is nothing new, the mathematician Augustin Cournot did not think that economics was susceptible to precise quantification, in fact he was wary of attempts to ‘arithmetise’ economics

There are authors, like [Adam] Smith and Say, who, in writing on Political Economy, have preserved all the beauties of a purely literary style; but there are others, like [David] Ricardo, who, when treating the most abstract questions, or when seeking great accuracy, have not been able to avoid algebra, and have only disguised it under arithmetical calculations of tiresome length. Any one who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.

I propose to show in this essay that the solution of the general questions which arise from the theory of wealth, depends essentially not on elementary algebra, but on that branch of analysis which comprises arbitrary functions, which are merely restricted to satisfying certain conditions. [5, p 4—5]

More recently John von Neumann refused to write a review for Samuelson’s Foundations of Economic Analysis in 1947 because “one would think the book about contemporary with Newton”, like many mathematicians who look at economics, on Neumann believed economics needed better maths than it was being offered [19, p 134].

Paul Krugman is right in emphasising the usefulness of mathematics as a rhetorical device, the problem is that he does not recognise that it is difficult to describe the Spitsbergen in winter in Arabic or the pleasures of a desert oasis in Sami.

My plea is that economists stop using existing mathematics and start commissioning new mathematics. I can see no resolution of the problem of econmath until there is the sort of relationship between economics and mathematics that mathematicians have with physical scientists and mathematicians work with economists to create a mathematics that enables the clear discussion of economic intuition.

In the Oedipus myth, although Oedipus does not realise he has killed his father, the murderer of Laius has to be brought to justice to end famine and pestilence in the kingdom.  Ultimately it is mathematics that needs to correct the wrongs to economics.

References

[1] L. Bachelier. Théorie de la spéculation. In Annales scientifiques de l’É. N. S. 3e série, tome 17 (1900), pages 21—86. École normale supérieure, 1900.

[2] C. B. Boyer and U. T. Merzbach. A History of Mathematics. John Wiley and Sons, 1991.

[3] P. Cartier. Poincaré’s Calcul des Probabilitités. In E. Charpentier, E. Ghys, and A. Lesne, editors, The Scientific Legacy of Poincaré. American Mathematical Society / London Mathematical Society, 2010.

[4] M. Clark. The Midas Formula (the Trillion Dollar Bet). BBC Horizon, 1999.

[5] A. A. Cournot. Researches into the mathematical principles of the theory of wealth (trans. N. T. Bacon). Macmillian, 1897. www. archive. org/details/researchesintom00fishgoog.

[6] J-L. Courtault, Y. Kabanov, B. Bru, P. Crépel, I. Lebon, and A. Le Marchand. Louis Bachelier on the centenary of Théorie de la Spéculation. Mathematical Finance, 10(3):339—353, 2000.

[7] A. W. Crosby. The Measure of Reality. Cambridge University Press, 1997.

[8] M. H. A. Davis and A. Etheridge. Louis Bachelier’s Theory of Speculation. Princeton University Press, 2006.

[9] P. Dear. Revolutionizing the Sciences. Palgrave, 2001.

[10] J. Dewey. The Quest for Certainty: A Study of the Relation of Knowledge And Action. Kessinger Publishing, 2005.

[11] P. Fara. Science: a four thousand year history. OUP, 2009.

[12] R. W. Hadden. On the Shoulders of Merchants: Exchange and the Mathematical Conception of Nature in Early Modern Europe. State University of New York Press, 1994.

[13] J. Hannam. God’s Philosophers: How the medieval world laid the foundations of modern science. Icon Books, 2009.

[14] F. Jovanovic. Was there a “vernacular science of financial markets” in france during the 19th century? History of Political Economy, 38(3):531—545, 2006.

[15] F. Jovanovic and P. Le Gall. Does god practice a random walk? the ’financial physics’ of a nineteenth-century forerunner, Jules Regnault. The European Journal of the History of Economic Thought, 8(3):332—362, 2001.

[16] V. J. Katz. A History of Mathematics: An Introduction. Haper Collins, 1993.

[17] J. Kaye. Economy and Nature in the Fourteenth Century. Cambridge University Press, 1998.

[18] S. L. Lauritzen. Time series analysis in 1880: A discussion of contributions made by T. N. Thiele. International Statistical Review, 49(3):319—331, 1981.

[19] P. Mirowski. What were von Neumannn and Morgenstern trying to accomplish?. In E. R. Weintraub, editor, Toward a History of Game Theory, pages 113—150. Duke University Press, 1992.

[20] A. Preda. Informative prices, rational investors: The emergence of the Random Walk Hypothesis and the Nineteenth-century ‘science of financial investments’. History of Political Economy, 36(2):351—386, 2004.

[21] B. Russell. An Outline of Philosophy. George Allen & Unwin (Routledge), 1927 (2009).

[22] M. S. Taqqu. Bachelier and his times: A conversation with Bernard Bru. Finance and Stochastics, 5(1):3—32, 2001.

 [23] Y. Biondi (trans J. d’Avingnon and G. Poitras). Duvillard’s Recherches sur les rents (1787) and the modified internal rate of return: a comparitive analysis. In G. Poitras, editor, Pioneers of Financial Economics: contributions prior to Irving Fisher, pages 116—148. Edward Elgar, 2006.