The intellectual void at the heart of the Occupy movement

The Bayesian algorithm at Amazon recommended David Graeber’s book, Debt: The First 5,000 Years with the product description

Economic history states that money replaced a bartering system, yet there isn’t any evidence to support this axiom. Anthropologist Graeber presents a stunning reversal of this conventional wisdom. For more than 5,000 years humans have used elaborate credit systems to buy and sell goods. Since the beginning of the agrarian empires, humans have been divided into debtors and creditors. Through time, virtual credit money was replaced by gold and the system as a whole went into decline. This fascinating history is told for the first time.

I tell my students, almost exclusively looking towards careers in banking or insurance, that they should take some interest in the “nature of money”, given this is the central topic of their current studies and future careers. Practising what I preach I thought I would get Graeber’s book.

The claim that “This fascinating history is told for the first time.” is publicist’s hyperbole. Within economics there has always been a debate as to the nature of money. Nicole Oresme wrote a Treatise on the Origin, Nature, Law, and Alterations of Money in the mid fourteenth century and then Copernicus wrote on On the Minting of Coin long before he wrote on the planets. In the first decades of the nineteenth century, the Bullionist Debates dominated British economics while the 1900 book The Wonderful Wizard of Oz is sometimes seen an allegory in favour of the State Theory of Money, that money is created by governments, as opposed to the Commodity Theory, that money is the most convenient commodity to facilitate exchange.

In the twentieth century it was anthropologists, like Malinowski and Mauss, who challenged the standard economic argument that money emerged out of barter.  More orthodox was  John Maynard Keynes’s A Treatise on Money , motivated by the conundrum, in the historical record, that credit existed before money. In 1985 the anthropologist Caroline Humphrey summarised the situation saying that

Barter is at once a cornerstone of modern economic theory and an ancient subject of debate about political justice, from Plato and Aristotle onwards. In both discourses, which are distinct though related, barter provides the imagined preconditions for the emergence of money …[however] No example of a barter economy, pure and simple, has ever been described, let alone the emergence from it of money; all available ethnography suggests that there never has been such a thing.

A modern narrative of this story, though not novel, would be interesting, and Graeber’s book is full scholarship, summarising the work of other social scientists who have addressed the basic question “what is money”. It presents a strong case that the standard economic argument, that first there was barter and then money emerged as a commodity to facilitate exchange, is myth unfounded in fact. This is important since so much of contemporary monetary policy has been based on the assumption of money as a commodity.

The fact that Graeber castigates modern orthodox economics is important since it sets the benchmark against which he should be measured.

In addition, there is  a note referring to the novelist Margaret Atwood

[Atwood] then proceeds to explore the nature of our sense of economic morality …Despite the brilliance of many of its arguments, the result is a rather sad testimony to how difficult it is for the scions of the North Atlantic professional classes not to see their own characteristic ways of imagining the world as simple human nature.

As a (middle class) novelist, Atwood is falling into the trap of subjective analysis, something the scientist, even political scientists and certainly anthropologists, should avoid.

The problem is, that when Graeber begins to consider exchange in the context of financial markets, he relies onsimilar economic and subjective assumptions.

In the case of …commercial exchange, when both parties in the transaction are only interested in the value of the goods transacted, they may well – as economists insist they should – try to seek the maximum material advantage

Given that up to this point the book has given a series of examples of when observed behaviour does not mimic economic theory, it is striking that this one point is made based on that same economic theory without further comment. Graeber goes on to say

What marks commercial exchange is that it’s “impersonal”: who it is that is selling something to us, or buying something from us, should in principle be entirely irrelevant.

The fact is, that when sociologists and anthropologists observe the actions of financiers, they realise that the economic theory is not, in fact, put into practice. While the market is capable of automating trade, so that decisions are based only on value, the markets emerged, evolved and exist on the basis of trust between participants, an emotions centred on personal relationships.

The sociologist Donald MacKenzie highlights this when discussing how Leo Melamed, the chairman of the Chicago Mercantile Exchange negotiated with the academic, Milton Friedman, about sponsoring a paper advocating the introduction of currency futures in the early 1970s. The speculators at the Merc, and the CBOT, were acting together in, what they believed, was the common good, to create the market, and not in their personal interest, maximising their wealth by focussing on competitive trading. This leads to the “delightful paradox”, that it seemed

the very markets in which Homo economicus, the rational egoist, appears to thrive cannot be created (if they require the solution of collective action problems, as in Chicago) by Homo economicus.

The belief that markets are about maximising wealth, rather than creating networks, comes out of political philosophy, emerging in early Victorian Britain with the liberal philosopher John Stuart Mill arguing that economics

is concerned with [man] solely as a being who desires to possess wealth, and who is capable of judging the comparative efficacy of means for obtaining that end.

Around the same time, the poet-, Alfred, Lord Tennyson, wrote about nature “red in tooth and claw”. In 1859 Darwin published the Origin of the Species which explained evolution in terms of natural selection. In the popular perception, nature became seen as being driven by a bitter struggle for survival, un-regulated by the ethics of divine architect. It was the leading economist at Cambridge University of the time, Alfred Marshall, who would synthesise Mill’s approach to economics with Darwinian metaphors in the late nineteenth century. In the twentieth, Friedman defined ‘positive economics’ as being disconnected from morality and his views have been summarised as

Any deviation from that single–minded pursuit of profit–maximisation by the admission of some other social responsibility is “fundamentally subversive”, “pure and unadulterated socialism”, something which could “thoroughly undermine the very foundations of our free society.” Businessmen subjected to “a social responsibility other than making maximum profits for stockholders” cannot know what interests to serve.

While this is the view from broader society, it is not natural in finance. This point is captured in a key case in the English courts in 1950, Buttle v Saunders. Saunders managed a trust for Buttle, and had agreed to sell a piece of land owned by the trust for £6,142 to a Mrs Simpson. Before the transaction had become legally binding, another person offered the trust £6,400 for the land. However, Saunders believed “my word is my bond” and declined the higher offer in favour of the original agreement with Mrs Simpson. The beneficiary of the trust, Buttle, took Saunders to court, and the court ruled in favour of Buttle

The only consideration which was present to [the trustees] minds was that they had gone so far in the negotiations with Mrs Simpson that they could not properly, from the point of view of commercial morality, resile from those negotiations.

‘Commercial morality’ was not a valid consideration and the sale to Mrs Simpson was declared null and English home-buyers could never again be certain that a purchase would be completed, ‘gazumping’ had arrived, not at the instigation of the financial adviser but on the insistence of a judge.

The scientist Graeber has fallen into the trap that he criticises the novelist Atwood of having succumbed to; he is imagining financiers rather than studying them. This is significant because it means his whole thesis is based on an assumption of what finance is about, this assumption is based on what academic economists imagine what it is about, rather than what the actual behaviour of bankers tells us.

Googling Graeber I discover that he is a central figure in the “Occupy” movement and is regarded as providing an intellectual justification for the physical manifestation of the movement.  However, I view this justification as being, itself, based on a false axiom/assumption, and that is the assumption about the nature of finance.

While we can acknowledge and accept the point the FCIC makes in concluding that

there was a systemic breakdown in accountability and ethics. The integrity of our financial markets and the public’s trust in those markets are essential to the economic well–being of our nation. The soundness and the sustained prosperity of the financial system and our economy rely on the notions of fair dealing, responsibility, and transparency.

The critical question is whether the lack of ethics the FCIC report is endogenous to the markets, and possibly incompatible with them, as an Occupy protester may argue, or whether the natural ethics of the market have been expunged by a series of commentators external to the markets, from Mill to Friedman. The point is, coming to this question with anarcho-communist preconceptions is not going to help a serious analysis.

That said, Graeber does acknowledge that finance has not always been as it is now. He remarks that the Medieval understanding of finace was concerned with maintaining social relations. This leads me to suggest that there is a blind-spot for many of the commentators on finance, and that is in regard to the problem of randomness.

Mill, Marx, Darwin and Dickens were all contemporaries living at a time when science was attempting to relegate randomness, chance, to history. In the 1830s English law criminalised most forms of gambling, the consequence was the difficulty Melamed had in creating a foreign exchange futures contract in 1970. Marxism proved popular not with the industrial proletariat, but in agrarian economies of Russia, China and Cuba, societies exposed to random climatic changes. Richard Dawkins is often heard stating that “evolution is not a random process”. Novels of the nineteenth century repeatedly use the device of a character being ruined by reckless speculation or feckless gambling.

The economic theory that developed at this time followed the dominant cultural direction. Laplace had advised that mechanics should be confined to the physical science, while the social sciences should employ probability to manage uncertainty, however, economists ignored this advice and built their science on deterministic mechanics.

The point is, earlier generations had accepted chance as an integral part of life, in particular economic life. Scholastic analysis of finance, which Graeber commends, was based on a distinction between what would happen with certainty and what was subject to chance.  The pioneers of mathematical probability, Pascal, Fermat, Huygens and J. Bernoulli all came to the topic trying to understand finance. 

Further back in time, gambling is an almost universal feature of primitive society. For the Greeks, the brothers Zeus, Poseidon and Hades cast lots to divide up the universe, Zeus winning the sky, Poseidon the sea and Hades the underworld. Hindus believe the world was a game of dice played between Shiva and his wife, while at the heart of the epic tale of the Mahabharata is an, unfair, dice game between the Kauravas and the Pandavas.

Contemporary anthropologists recognise that gambling plays a fundamental role in contemporary neolithic communities. Consider a case observed in an Australian aboriginal group, the Momega in a remote area of Arnhem Land around 1980. The community had access to social security payments and there was often a surplus left over after essentials had been bought. As the anthropologist, Jon Altman, studying the group observed

this surplus was not equally bestowed. …This variability in bestowal was extremely arbitrary and it resulted in inter–household variability in access to cash.

This variability can seen as subjective discrimination of the community by the Australian government. Gambling, according to Altman, “acted effectively to both redistribute cash …[it] provided a means for people with no cash income to gain cash” and from a small stake a larger cash reserve could be generated. The random distribution created by gambling, while not uniform, some would lose a lot, some win a lot, was none the less objective and most people ended up with a fair share of the cash this was important in a non-hierarchical community because it meant that the arbitrary bestowal of money was not corrected by another subjective distribution, such as redistribution by a chief.

Another anthropologist, William Mitchell considered the role that gambling plays in disrupting hierarchical social structures, such as the Indian caste system, by studying the Wape in New Guinea around the same time

An important task of Dumont’s classic study of Indian caste was to demonstrate how inequality is maintained. My task is the obverse, that is, to reveal how the Wape defeat the formidable principle of hierarchy to maintain male equality. How do the Wape, who, as individuals, desire wealth and who, since the 1930s, have been directly tied to a world capitalist market system, prevent wealth from being successfully manipulated by a few men to raise themselves above others? The paradoxical answer is deceptively simple: through gambling.

This is an explanation for the pervasive nature of gambling in neolithic communities, appearing in the Vedic scriptures, potlach ceremonies of North America, and in aboriginal Australia and New Guinea and the Hazda; it is an objective, fair, mechanism for the redistribution of wealth.

We can criticise Graeber’s methodology and his reliance on unsupportable assumptions, but the real failure of the Occupy movement is their inability to appreciate the positive aspects of gambling and speculation, central to the markets.Perhaps they should read more anthropology and less Dickens.

Notes

${}^1$Humphrey (1985, p 48)

${}^2$Graeber (2011, p 404, n5)

${}^3$Graeber (2011, p 103)

${}^4$Graeber (2011, p 103)

${}^5$MacKenzie (2008, p 151)

${}^6$Persky (1995, quoting Mill, p 223)

${}^7$ Backhouse (1985, 10.1), Thomas (1991)

${}^8$Watchman (2001, p 27)

${}^9$FCIC (2011, p xxii)

${}^{10}$Katz (1993, p 685)

${}^{11}$Ménard (1987)

${}^{12 }$Sahlins (2003, p 27), Brenner and Brenner (1990, p 1–5)

${}^{13}$Altman (1985, p 56)

${}^{14}$Altman (1985, pp 60-61)

${}^{15}$Mitchell (1988)

${}^{16}$Sahlins (2003, p 27)

References

    Altman, J. (1985). Gambling as a mode of redistributing and accumulating cash among aborigines: a case study from Arnhem Land. In Caldwell, G., Dickerson, M., Haig, B., and Sylvan, L., editors, Gambling in Australia, pages 50–67. Croom Helm.

    Backhouse, R. (1985). A History of Modern Economic Analysis. Blackwell.

    Brenner, R. and Brenner, G. A. (1990). Gambling and Speculation: A theory, a history and a future of some human decisions. Cambridge University Press.

    FCIC (2011). The Financial Crisis Inquiry Report. Technical report, The National Commission on the Causes of the Financial and Economic Crisis in the United States.

    Graeber, D. (2011). Debt: The first 5,000 years. Melville House.

    Humphrey, C. (1985). Barter and economic disintegration. Man, 20(1).

    Katz, V. J. (1993). A History of Mathematics: an Introduction. Harper Collins.

    MacKenzie, D. (2008). An Engine, Not a Camera: How Financial Models Shape Markets. The MIT Press.

    Ménard, C. (1987). Why was there no Probabilistic Revolution in economic thought? In Kruger, L., Gigerenzer, G., and Morgan, M. S., editors, The Probabilistic Revolution: Volume 2: Ideas in the Sciences. MIT Press.

    Mitchell, W. E. (1988). The defeat of hierarchy: Gambling as exchange in a Sepik society. American Ethnologist, 15(4).

    Persky, J. (1995). Retrospectives: The ethology of Homo economicus. The Journal of Economic Perspectives, 9(2):221–231.

    Sahlins, M. (1972 (2003)). Stone Age Economics. (Routledge).

    Thomas, B. (1991). Alfred Marshall on economic biology. Journal of Financial Intermediation, 3(1):1–14.

    Watchman, P. (2001). A legal framework for the integration of environmental, social and governance issues into institutional investment. Technical report, UNEP Finance Initiative/Freshfields Bruckhaus Deringer. .

Why don't more mathematicians see the potential of economics

The question is, how did economics change its attitude to mathematics in the forty years between Håvelmo’s The Probability Approach in Econometrics and his Nobel Prize in 1989, when he was pessimistic about the impact the development of econometrics had had on the practice of economics. Coinciding with Håvelmo’s pessimism, many economists were reacting strongly against the ‘mathematisation’ of economics, evidenced by the fact that before 1925, only around 5% of economics research papers were based on mathematics, but by 1944, the year of Havelmo and von Neumann-Morgenstern’s contributions, this had quintupled to 25%. While the proportion of economics papers being based on maths has not continued this trajectory, the influence of mathematical economics has and the person most closely associated with this change in economic practice was Paul Samuelson.

Samuelson is widely regarded as the most influential economist to come out of the United States and is possibly the most influential post-war economist in the world. He was the first U.S. citizen to be awarded the Nobel Prize in Economics in 1970 because “more than any other contemporary economist, he has contributed to raising the general analytical and methodological level in economic science”. He studied at the University of Chicago and then Harvard, were he obtained his doctorate in 1941. In 1940 he was appointed to the economics department of M.I.T., in the final years of the war he worked in Wiener’s group looking at gun control problems, where he would remain for the rest of his life. Samuelson would comment that “I was vaccinated early to understand that economics and physics could share the same formal mathematical theorems”.

In 1947 Samuelson published Foundations of Economic Analysis, which laid out the mathematics Samuelson felt was needed to understand economics. It is said that von Neumann was invited to write a review Foundations in 1947 declined because “one would think the book about contemporary with Newton”. Von Neumann, like many mathematicians who looked at economics, believed economics needed better maths than it was being offered. In 1948 Samuelson published the first edition of his most famous work, Economics: An Introductory Analysis, one of the most influential textbooks on economics ever published, it has run into nineteen editions and sold over four million copies.

There appears to be a contradiction, Håvelmo seems to think his introduction of mathematics into economics was a failure, while Samuelson’s status seems to suggest mathematics came to dominate economics. In the face of contradiction, science should look for distinction.

I think the clue is in Samuelson’s attachment to “formal mathematical theorems”, and that his conception of mathematics was very different from that of the earlier generation of mathematicians that included everyone from Newton and Poincaré to von Neumann, Wiener and Kolmogorov.

A potted history of the philosophy of mathematics is that the numerologist Plato came up with the Theory of Forms and then Euclid produced The Elements which was supposed to capture the indubitability, the certainty, and immutability, the permanence, of mathematics on the basis that mathematical objects where Real representations of Forms. This was used by St Augustine of Hippo as evidence for the indubitability and immutability of God, embedding into western European culture the indubitability and immutability of mathematics. The identification of non-Euclidean geometries in the nineteenth century destroyed this edifice and the reaction was the attempt to lay the Foundations of Mathematics, not on the basis of geometry but on the logic of the natural numbers. Frege’s logicist attempt collapsed with Russell’s paradox and attention turned to Hilbert’s formalism to provide a non-Platonic foundation for mathematics. The key idea behind Formalism is that, unlike Platonic Realism, mathematical objects have no meaning outside mathematics, the discipline is a game played with symbols that have no relevance to human experience.

The Platonist, Kurt Gödel, according to von Neumann, has “shown that Hilbert’s program is essentially hopeless” and

The very concept of “absolute” mathematical rigour is not immutable. The variability of the concept of rigour shows that something else besides mathematical abstraction must enter into the makeup of mathematics

Mathematics split into two broad streams. Applied mathematics, practised by the likes of von Neumann and Turing, responded by focussing on real-world ‘special cases’, such as modelling the brain. Pure mathematics took the opposite approach, emphasising the generalisation of special cases, as practised by Bourbaki and Hilbert’s heirs.

Formalism began to dominate mathematics in the 1940s-1950s. Mathematics was about ‘rigorous’, whatever that means, deduction from axioms and definitions to theorems. Explanatory, natural,  language and, possibly worse, pictures, were to be removed from mathematics. The “new math” program of the 1960s was a consequence of this Formalist-Bourbaki dominance of mathematics.

It is difficult to give a definitive explanation for why Formalism became dominant, but it is often associated with the emergence of logical–positivism, a somewhat incoherent synthesis of Mach’s desire to base science only on phenomena (which rejected the atom), mathematical deduction and Comte’s views on the unity of the physical and social sciences. Logical-positivism dominated western science after the Second World War, spreading out from its heart in central European physics, carried by refugees from Nazism.

The consequences of Formalism were felt most keenly in physics. Richard Feynman, the physicists’ favourite physicist, hated its abandonment of relevance. Murray Gell-Mann, another Noble Laureate physicist, commented in 1992 that the Formalist-Bourbaki era seemed to be over

abstract mathematics reached out in so many directions and became so seemingly abstruse that it appeared to have left physics far behind, so that among all the new structures being explored by mathematicians, the fraction that would even be of any interest to science would be so small as not to make it worth the time of a scientist to study them.

But all that has changed in the last decade or two. It has turned out that the apparent divergence of pure mathematics from science was partly an illusion produced by obscurantist, ultra-rigorous language used by mathematicians, especially those of a Bourbaki persuasion, and their reluctance to write up non–trivial examples in explicit detail. When demystified, large chunks of modern mathematics turn out to be connected with physics and other sciences, and these chunks are mostly in or near the most prestigious parts of mathematics, such as differential topology, where geometry, algebra and analysis come together. Pure mathematics and science are finally being reunited and mercifully, the Bourbaki plague is dying out.

Economics has always doubted its credentials. Laplace saw the physical sciences resting on calculus, while the social sciences would rest on probability, but classical economists, like Walras, Jevons and Menger, wanted their emerging discipline economics to have the same status as Newton’s physics, and so mimicked physics. Samuelson was looking to do essentially the same thing, economics would be indubitable and immutable if it looked like Formalist mathematics, and in this respect he has been successful, the status of economics has grown faster than the growth of maths in economics. However, while the general status of economics has exploded, its usefulness to most users of economics, such as those in the financial markets, has collapsed. Trading floors are recruiting engineers and physicists, who always looked for the relevance of mathematics, in preference to economists (or post-graduate mathematicians).

My answer to the question “why don’t more economists see the potential of mathematics” is both simple and complex. Economists have, in the main, been looking at a peculiar manifestation of mathematics - Formalist-Bourbaki mathematics - a type of mathematics that emerged in the 1920s in response to an intellectual crisis in the Foundations of Mathematics. Economists have either embraced it, as Samuelson did, or were repulsed by it, as Friedman was.

Why this type of mathematics, a type of maths that would have been alien to the great mathematicians of the twentieth century like Wiener, von Neumann, Kolmogorov and Turing, became dominant and was adopted by economics is more complex and possibly inexplicable. The question is, can academic mathematics return to its roots in relevance, or will it wither in its ivory towers?

Notes

${}^1$Mirowski (1991, pp 150–151)

${}^2$The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1970.

${}^3$MacKenzie (2008, p 63–64)

${}^4$Mirowski (1992, p 134)

${}^5$Mirowski (1992, p 122, quoting von Neumann)

${}^6$Mirowski (1992, p 122–124)

${}^7$Gell-Mann (1992, p 7)

${}^8$Katz (1993, p 685)

References

    Gell-Mann, M. (1992). Nature conformable to herself. Bulletin of the Santa Fe Institute, 7(1):7–8.

    Katz, V. J. (1993). A History of Mathematics: an Introduction. Haper Collins.

    MacKenzie, D. (2008). An Engine, Not a Camera: How Financial Models Shape Markets. The MIT Press.

    Mirowski, P. (1991). The when, the how and the why of mathematical expression in the history of economic analysis. Journal of Economic Perspectives, 5(1):145–157.

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

Why don't more economists see the potential of mathematics

A research student, working in econometrics has e-mailed me with the comment

I am a little confused why many economists do not see the potential of mathematics.

The discipline of econometrics was introduced in the 1940’s with the key monograph being Trygve Håvelmo’s The Probability Approach in Econometrics. Håvelmo’s motivation for writing the paper is eloquently stated in the preface

The method of econometric research aims, essentially, at a conjunction of economic theory and actual measurements, using the theory and technique of statistical inference as a bridge pier. But the bridge itself was never completely built. So far, the common procedure has been, first to construct an economic theory involving exact functional relationships, then to compare this theory with some actual measurements, and, finally, “to judge” whether the correspondence is “good” or “bad”. Tools of statistical inference have been introduced, in some degree, to support such judgements, e.g., the calculation of a few standard errors and multiple-correlation coefficients. The application of such simple “statistics” has been considered legitimate, while, at the same time, the adoption of definite probability models has been deemed a crime in economic research, a violation of the very nature of economic data. That is to say, it has been considered legitimate to use some of the tools developed in statistical theory without accepting the very foundation upon which statistical theory is built. For no tool developed in the theory of statistics has any meaning– except, perhaps, for descriptive purposes –without being referred to some stochastic scheme.

The reluctance among economists to accept probability models as a basis for economic research has, it seems, been founded upon a very narrow concept of probability and random variables. Probability schemes, it is held, apply only to such phenomena as lottery drawings, or, at best, to those series of observations where each observation may be considered as an independent drawing from one and the same “population”. From this point of view it has been argued, e.g., that most economic time series do not conform well to any probability model, “because the successive observations are not independent”. But it is not necessary that the observations should be independent and that they should all follow the same one–dimensional probability law. It is sufficient to assume that the whole set of, say $n$, observations may be considered as one observation of $n$ variables (or a “sample point”) following an $n$-dimensional joint probability law, the “existence” of which may be purely hypothetical. Then, one can test hypotheses regarding this joint probability law, and draw inference as to its possible form, by means of one sample point (in $n$ dimensions). Modern statistical theory has made considerable progress in solving such problems of statistical inference.

In fact, if we consider actual economic research–even that carried on by people who oppose the use of probability schemes–we find that it rests, ultimately, upon some, perhaps very vague, notion of probability and random variables. For whenever we apply a theory to facts we do not–and we do not expect to–obtain exact agreement. Certain discrepancies are classified as “admissible”, others as “practically impossible” under the assumptions of the theory. And the principle of such classification is itself a theoretical scheme, namely one in which the vague expressions “practically impossible” or “almost certain” are replaced by “the probability is near to zero”, or “the probability is near to one”.
This is nothing but a convenient way of expressing opinions about real phenomena. But the probability concept has the advantage that it is “analytic”, we can derive new statements from it by the rules of logic.

Håvelmo’s argument can be split into four key points. If economics is to be regarded as ‘scientific’, it needs to take probability theory seriously. He then notes that economists have taken a naive approach to probability, and possibly mathematics in general, and introduces the Lagrangian idea of representing $n$ points in one dimensional space by one point in $n$-dimensional space. Finally he makes Poincaré’s point that probability is a convenient solution, it makes the scientist’s life easier, and finally he makes Feller’s point that it enables the creation of new knowledge, new statements.

Håvelmo then goes on to tackle the issue that goes back as far as Cicero, at least, “there is no foreknowledge of things that happen by chance” by making the critical observation, nature looks stable because we look at it in a particular way

“In the natural sciences we have stable laws”, means not much more and not much less than this: The natural sciences have chosen very fruitful ways of looking on physical reality.

Håvelmo is saying that if economists look at the world in a different way, if the right analytical tools are available to them, they may be able to identify stable laws.

At about the same time, Oskar Morgenstern was working with John von Nueumann on The Theory of Games and Economic Behavior, a “big book because they wrote it twice, once in symbols for mathematicians and once in prose for economists”. Morgenstern begins the book by describing the landscape. On the second page he, makes the case for using mathematics in economics, just as Håvelmo had, but with a more comprehensive argument. Morgenstern reviews the case as to why mathematics is inappropriate to economics, no doubt with von Neumann at his shoulder,

The arguments often heard that because of the human element, of psychological factors etc., or because there is – allegedly – no measurement of important factors, mathematics will find no application [in economics] [von Neumann and Morgenstern 1967 p 3]

However, Morgenstern points out that Aristotle had the same opinion of the use of mathematics in physics

Almost all these objections have been made, or might have been made, many centuries ago in fields fields where mathematics is now the chief instrument of analysis.

While measurement may appear difficult in economics, measurement appeared difficult before the time of Albert the Great, again before Newton fixed time and space, when objects were either ‘hot’ or ‘cold’ or before the idea of potential energy being released into kinetic energy emerged.

The reason why mathematics has not been more successful in economics must, consequently, be found elsewhere. The lack of real success is largely due to a combination of unfavourable circumstances, some of which can be removed gradually. To begin with economic problems were not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are. There is no point in using exact methods where there is no clarity in the concepts and the issues to which they are to be applied. Consequently the initial task is to clarify the knowledge of the matter by further careful description. But even in those parts of economics where the descriptive problem has been handled more satisfactorily, mathematical tools have seldom been used appropriately. They were either inadequately handled, as in the attempts to determine a general economic equilibrium …, or they led to mere translations from a literary form of expression into symbols, without any subsequent mathematical analysis. [von Neumann and Morgenstern, 1967, p 4]

Morgenstern makes the critical observation, that the ‘correct’ use of mathematics in science leads to the creation of new mathematics

The decisive phase of the application of mathematics to physics – Newton’s creation of a rational discipline of mechanics – brought about, and can hardly be separated from, the discovery of [calculus]. (There are several other examples, but none stronger than this.)

The importance of social phenomena, the wealth and multiplicity of their manifestations, and the complexity of their structure, are at least equal to those in physics. It is therefore expected – or feared – that the mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this field. [von Neumann and Morgenstern 1967 p 5]

In 1989 Håvelmo was awarded the Nobel Prize in Economics “for his clarification of the probability theory foundations of econometrics and his analyses of simultaneous economic structures”. In his speech, the economist Håvelmo reflected on the impact of his work,

To some extent my conclusions [are] in a way negative. I [draw] attention to the – in itself sad – result that the new and, as we had thought, more satisfactory methods of measuring interrelations in economic life had caused some concern among those who had tried the new methods in practical work. It was found that the economic theories which we had inherited and believed in, were in fact less stringent than one could have been led to think by previous more rudimentary methods of measurement. To my mind this conclusion is not in itself totally negative. If the improved methods could be believed to show the truth, it is certainly better to know it. Also for practical economic policy it is useful to know this, because it may be possible to take preventive measures to reduce uncertainty. I also mentioned another thing that perhaps could be blamed for results that were not as good as one might have hoped for, namely economic theory in itself. The basis of econometrics, the economic theories that we had been led to believe in by our forefathers, were perhaps not good enough. It is quite obvious that if the theories we build to simulate actual economic life are not sufficiently realistic, that is, if the data we get to work on in practice are not produced the way that economic theories suggest, then it is rather meaningless to confront actual observations with relations that describe something else. [ Prize Lecture Lecture to the memory of Alfred Nobel  ]

Håvelmo’s aim in the 1940s, along with that of John von Neuman, had been to improve economic methodology, the consequence was, in Håvelmo’s case, was that it highlighted deficiencies in economic theory. The question is, what happened in economics in the forty years between Håvelmo’s paper on econometrics and his Nobel Prize in 1989 to lead to such a negative reflection on the development of economics. I shall come back to this in my next post.

References

   J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Wiley, 3rd edition, 1967.

The Girl with the Dragon Tattoo, and why we can't solve the Euro crisis

I, like a lot of technically minded-people it seems, like a good crime thriller.  Having spent a year mesmerised by Sara Lund's sweaters I picked up The Girl with the Dragon Tattoo.  It's a good story, but a bit contrived with the plot being resolved by unexplained and unbelievable computer hacking.

So why write about it on a blog about science and finance?

The plot is set in the context of our hero, a journalist, Blomkvist and his relationship between two businessmen,  Wennerström and Vanger.  Blomkvist publishes a story claiming Wennerström was involved in a fraud, which he failed to corroborate and the novel begins with his conviction for libel.  This makes him vulnerable to an approach from Vanger to investigate the mysterious disappearance of his niece in the 1960s, which is the central mystery in the novel.

What I find interesting is how the different businessmen are presented.   Wennerström has become a billionaire from nowhere, building his wealth by participating in such unsavoury activities as options trading and currency speculation.  Vanger, on the other hand, is old-money - a traditional industrialist whose family had arrived in Sweden with Napoleon's puppet king, Jean-Baptiste Bernadotte, and was granted land holdings on which they developed paper mills that spawned a firm involved in manufacturing, technology and the media.  Within this setting the message is clear:  Vanger may be an unpalatable capitalist but Wennerström is beyond the pale.

The sub-text is physiocratic, the Vagner's are legitimate capitalists because they hold land.  Wennerström, who has made his own money through speculation, by judgement and foresight, is corrupt.  The irony that the hero Blomkvist becomes rich by speculating,  that Wennerström is a crook, is passed by.  A second irony is that if the events described in the book had actually occurred in England, they might have become of interest to the Leveson Inquiry, looking at actual cases of the relationship between journalists and private detectives involved in phone hacking. 

The theme of the corruption of finance is not new, but the novel ignores St Augustine's observation that if a merchant was a cheat, the fault was with the individual, not the profession. What is disturbing is the deep conservatism of the books' sub-text:  if some thug in history had not given your ancestors land, you do not deserve to be rich.

I finished reading this book the weekend that France and Germany looked to solve the Euro crisis and impose tighter regulations on financial services.  The French President, commenting on the UK's withdrawal from the process, went on to say that "a good part of the world's problems come from the deregulation of financial services". 

Both the UK's FSA's, in their recent report on the collapse of RBS, and the US's Financial Crisis Inquiry Commission (FCIC) observe that the regulations existed, but they were not effectively implemented.  The UK-US crisis was a failure of government agencies, at the behest of politicians, to enforce rules.  The Euro crisis has similar roots in a failure to enforce rules on sovereign debt ratios.   The crises seem to have less to do with bankers and more to do with the profligacy of politicians.

Much of the narrative of the ongoing financial crises has been about the imbalance, particularly in the UK, between manufacturing, represented by Vanger, and financial services, the Wennerströms.  Implicit is that manufacturing, creating physical objects, is something more real than financial services, moving money around, and moreover, more reputable.

This bias seems to be reflected in UK government science policy.  In the recently published Innovation and Research Strategy, the government identifies the following, key, technology based sectors: life sciences, high-value manufacturing, nanotechnology and digital technology.  Financial services, while contributing around 10% of UK's GDP, is not mentioned.  This approach ignores the advice from the Royal Society in their 2009 report Hidden Wealth.

I would argue that the recent financial crises are about a general lack of understanding of modern finance that enabled institutions to behave irresponsibly.  It is hardly surprising that this situation has arisen given the blindness to policy-makers to investing in fundamental research into finance and its underpinning technologies, whether they be the contracts traded, the mathematical models and their computational implementations.  This aversion to understanding finance will persist as long as popular culture is dominated by the Vanger=good, Wennerström=bad model.

... but what about the South Sea Bubble...

David K Waltz has made a comment on my last post, I realised my response was going to be a bit longer than a "comment".

David raises the spectre of the South Sea Bubble and Tuplipmania.  I am not an expert on the Tulip Bubble, but I am aware that there is a question as to whether it was a significant as popular imagination suggests (i.e Dumas's The Black Tulip).

The South Sea Bubble (1720) can tell us a lot.  In the aftermath the British constitution was completely overhauled - the introduction of the position of Prime Minister (held by Walpole,still the longest serving Prime Minister) was the most obvious impact.  In addition, British public finance was put on a firm footing that laid the foundation of successive military victories and the Empire.  France's response to the Mississippi Bubble (1719) was less dramatic (they blamed the financier Law) and the consequence was a succession of military defeats.

Also, the popular response included Defoe's Complete English Tradesman where he writes (1726)

A tradesman's books are his repeating clock, which upon all occasions are to tell him how he goes on, and how things stand with him in the world: there he will know when it is time to go on, or when it is time to give over; and upon his regular keeping, and fully acquainting himself with his books, depends at least the comfort of his trade, if not the very trade itself. If they are not duly posted, and if every thing is not carefully entered in them, the debtor's accounts kept even, the cash constantly balanced, and the credits all stated, the tradesman is like a ship at sea, steered without a helm; he is all in confusion, and knows not what he does, or where he is; he may be a rich man, or a bankrupt-for, in a word, he can give no account of himself to himself, much less to any body else.

This contrasts with  the Rape of Lady Credit by stock-jobbers that Defoe described in 1709

The first Violence they committed was downright Rape ... these new-fashion'd thieves seiz'd upon her, took her Prisoner, toss'd her in a Blanket, ravish'd her, and in short us'd her barbarously, and had almost murther'd her

Much of the political and public rhetoric in recent years has been focused on criticism of financiers (but, unlike Defoe, not before 2006).  However, given the performance of Europe's politicians since 2008 is terrifying (for a European citizen) in that they seem to refuse to take responsibility (which countries were the first to break Euro rule, was it Germany and France?) and prefer to see the crisis as an opportunity to score political points (the Conservative party in the UK).

The British, between around 1688 and 1728, seem to have recognised the democratic nature of markets. Something that Aristotle commented on, and was discussed by scholastics that influenced the development of science through the likes of Bradwardine and Oresme.  Napoleon was scornful as Britain as a nation of shopkeepers, but that is precisely where its strength lay. And it is not simply a financial strength, how different is Defoe's description of a Tradesman from Hume's empiricism?

Its not just science that can learn from the markets.

Does finance need a scientific revolution or science need a revolution inspired by finance

Nature, the British version of Science, rarely makes forays into exploring issues in finance. They invited me to talk at a meeting in September 2009 and I get the sense that they would like to do more but are a bit perplexed by the subtleties of finance. The mind is willing but the body is weak.

In October 2008 Nature published a piece, Economics needs a scientific revolution, by the French econo-pysicist Jean-Philippe Bouchaud, which argued that the fault was with the academic discipline of economics: "Classical economics is built on very strong assumptions that quickly become axioms" where as "Physicists, on the other hand, have learned to be suspicious of axioms. If empirical observation is incompatible with a model, the model must be trashed or amended". More recently they have published a broader article, reflecting the diffusion of the financial crisis into society at large, Science's attitudes mustreflect a world in crisis, by the science writer Colin Macilwain.

Physicists are never slow to criticise the short-comings of other disciplines, but the truth is physics, with its never changing laws, is simple. While physicists are paddling in the play pool, economists are hanging on to a rubber ring in the middle of a North Atlantic gale. It is not surprising they don't look like good swimmers. That said, economists are well aware of the limitations of their discipline. From the mathematician Cournot's criticism of the economists' use of mathematics in the 1830s to the contemporary economist's criticism of the wholesale adoption of the ergodic hypothesis by Paul Davidson, economics could do better.

Since the Second World War, economics has transformed itself from a relatively minor discipline to a dominant field. It is ironic that in the 2008 "Research Assessment Exercise", UK economists judged their work to be "excellent", just as the tsunami of the credit crisis was crashing on the shore. This dominance was built, substantially, on "positive" economics, adopting the ideals of the logical positivists that emerged in central European maths and physics in the aftermaths of the First World War. The authority that these scientist had rested on the fact that they had defeated Germany and Japan.

It was not square-jawed commandos that defeated the Axis, but mathematicians and physicists (like  Turing, von Neumann, Kolmogorov, Weiner, Shannon, ... the list is not short) working in Operations Research, the Manhattan Project and code-breaking, that won the war. In particular the code breakers were able to transform streams of random letters into meaningful messages. This analogue seems to have driven economic research through the fifties and sixties: given the right algorithm the randomness of asset prices can be read. Then, just as the idealised economic order of Bretton Woods collapsed under the reality of people and politics, Black-Scholes-Merton and financial mathematics (apparently) emerged to save the world from random chaos.

The financial crisis of 2007-2009 was a demonstration of limitations of the science that emerged out of logical-positivism and in particular the ergodic hypothesis placed at the core of positive economics. It was not a failure of maths and science in general, but a particular type of science that emerged in the 1920s and dominated society between 1940 and the present.

Macilwain is concerned with he fact that many science policy gurus, generally wedded to the tenets of  positivism,  are "clearly more comfortable discussing the planet's ecological crises than the economic ones currently alarming the general population" and that there is "great danger is that scarce funding will consolidate around single-discipline research". More Big Physics like the LHC, or repeating Tony Blair's "betting the house"on genetics - a bet that has not paid out.

Financial markets are manifestations of random, non-ergodic phenomena, as such they can hardly be amenable to deterministic techniques. Specifically, a single approach to understanding them, whether rooted in analogues from the Olympian disciplines of mathematics, physics or biology, will fail. Scientific revolutions have historically been based on the inter-disciplinary interaction, such as between financiers, lawyers (Bacon, Descartes, Fermat, Huygens) and mathematicians. If society stops to think about the current financial turmoil and what it can tell us, we would surely be at the dawn of another scientific revolution.

De Coding Da Vinci

It is well known that Leonardo Da Vinci became interested in the "golden ratio" or "divine proportion".  It is somewhat less well known is he learnt about the number from Lucca Pacioli, the Francisican friar and grandfather of accounting.  What is virtually unknown is that Pacioli probably learnt his mathematics from the financial mathematician and artist, Piero della Francesca  (as an artist).

I have written an article, Decoding Da Vinci: Finance, functions and art, on this for plus! an on-line magazine aimed at youngster. The piece explains why the ratio was considered divine because of its form as a continued fraction, and how another financial mathematician laid the foundations for functional analysis by popularising decimal notation.

St Thomas Aquinas comes to the defence of the vampire squid?

On April 16 2010, the SEC filed fraud charges against Goldman Sachs, the “great vampire squid wrapped around the face of humanity”. The case in court is that the bank claimed the assets in the fund ABACUS 2007-AC1 were selected by an independent advisor, when in fact they were selected by John Paulson to enable his funds to short the assets, and in making the claim, the bank deceived investors. However the case is usually presented as the bank being taken to court for acting immorally structuring an asset designed to fail and selling this on to unsuspecting clients (for example The New York Times’ report and their prediction in December 2009 or in the UK Jeremy Warner of The Telegraph). The vampire squid was sucking the life out of innocent investors. 

The SEC could not take Goldman Sachs to court for acting unethically, the courts are about legality not morality. However, it would come as a surprise to many that, according to Catholic doctrine at least, there was probably not much morally wrong with Goldman’s actions. 

In the third quarter of the thirteenth century, Thomas Aquinas worked on his Summa Theologica, integrating Aristotelian philosophy with Catholic doctrine and established himself as a “Doctor of the Church”. In the “Second part of the Second part” of the Summa, Aquinas addressed the issue of individual morality, including the concept of justice. One question the theologian considered was “Whether it is lawful to sell something for more than it is worth?” and examined a case presented by Stoic philosophers

A grain merchant from Alexandria arrives at Rhodes, which is gripped by famine. The merchant knows that other merchants are following him with plentiful supplies of grain, though the town’s inhabitants do not know this. How should the merchant price the grain he has? ( De Officiis Book 1, XII)

The Roman jurist Cicero (Tully) was typical in arguing that the merchant should not take advantage of the misfortune of the starving and charge a lower price based on the knowledge of the coming relief.

Aquinas disagrees

in the case cited, the goods are expected to be of less value at a future time, on account of the arrival of other merchants, which was not foreseen by the buyers. Wherefore the seller, since he sells his goods at the price actually offered him, does not seem to act contrary to justice through not stating what is going to happen. If however he were to do so, or if he lowered his price, it would be exceedingly virtuous on his part: although he does not seem to be bound to do this as a debt of justice. (Summa II.II Q77(A3) Reply to Obj. 4)

This is an almost shocking conclusion from a saint. To Aquinas, the merchant, having arrived at Rhodes, may think there are more grain shipments on the way, but does not know. Aquinas argues that

because the just price of things is not fixed with mathematical precision, but depends on a kind of estimate, so that a slight addition or subtraction would not seem to destroy the equality of justice. (Summa II.II Q77(A1) Reply to Obj 1, para 2)

It is not unjust to charge the higher price, since whether of not this price is ‘just’, or not, is uncertain.

Aquinas’ concept of the “equality of justice” comes from Book 5 of Aristotle’s Nicomachean Ethics which had been translated into Latin in 1250. In Ethics, Aristotle considered the morality of economic exchange and argues that market exchange is not performed in order to generate a profit, for gain, but to correct for inequalities and to establish a social equilibrium. Using the example of a builder and a shoemaker, Aristotle argues that

The builder, then, must get from the shoemaker the latter’s work, and must himself give him in return his own. (Ethics p 79)

For justice to exist in the exchange, there needed to be an equality between the shoes the shoemaker produced and the results of the builders’ work.  For such an equality to exist, there needed to be a measure of the value of the goods produced by the builder and the shoemaker to enable a just exchange . 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. (Ethics p 79)

The relationship at the heart of the exchange between the shoemaker and the builder that Aristotle went on to establish was explained mathematically, as a geometric relationship. This was significant since Aristotle never used mathematics in addressing questions in physics. 

However Aquinas’ use of an analogy with mathematics was not simply following Aristotle, he was re-interpreting in the context of contemporary events. The twelfth century had seen a “renaissance” in western Europe, a population explosion that brought with it Gothic architecture and a commercial revolution. 

In the aftermath of the collapse of the Roman Empire in the west, the concept of turpe lucrum, or ‘shameful gain’ emerged and merchants were restricted in what prices they could charge. As Europe emerged out of the ‘castellan’ society, characterised by isolated local lords held together by feudal service relations, into one of integrated trade networks, turpe lucrum was no longer a feasible basis on which to run the economy. It was in this environment that Leonardo of Pisa, Fibonacci, published the Liber Abaci. 

One aspect of the complexity of the transactions was to disguise usurious contracts. There is a subtle difference between usury, which is associated with charging for the use of money, and interest which is a compensation for loss. For example a farmer could lend a cow for a year and expect to be re-paid with a cow and a calf, since in the normal course of events a cow would give birth to a calf over the year. Interest could be charged on ‘productive’ assets, with the consequence that all, legitimate, medieval securities would be ‘asset backed’. Gold, being inorganic, was not productive and so charging for its use was unjust.

However, this attitude to money stifled innovation and so more complex structures emerged to disguise the charging of interest on a money loan. For example, the ‘triple contract’ enabled an entrepreneur to raise money to invest in a trading venture. At the heart of the triple contract was a partnership between the entrepreneur and investors, this was the first contract. The second contract would be an insurance contract taken out by the entrepreneur to insure against the loss the investors’ capital. The third contract was another ‘insurance’ contract given to the investors by the entrepreneur, where by the investors surrendered their rights to a share of the profit in exchanged for a fixed payment from the entrepreneur, this payment was guaranteed by the second contract. Not quite a Credit Default Swap but definitely credit insurance. 

Eventually, in 1236, the canon (church) jurist, Alanus Anglicus, determined that turpe lucrum did not exist if the future price of the good was uncertain in the mind of the merchant and this ruling was embedded in Catholic doctrine some ten years later. Aquinas, in the Summa was integrating the theories of the newly translated Aristotle with emerging Catholic doctrine and existing commercial practice to argue that a profit was ‘just’ so long as it was uncertain and might lead to a loss, it entailed a ‘risk’. 

Islamic societies had equally stringent prohibitions on usury, and so this alone cannot explain why Fibonacci would have such an influence in Europe. The key difference between the environment for medieval European merchants and their colleagues in the Middle East, India or China was the range of currencies being used, a consequence of the castellan society of the ‘Dark Ages’. For example there were 28 different currencies in Italy at one time or another and as Groetzman has observed

Reading Liber Abaci one has the sense that Italian merchants of the 13^th century operated in a world of complete relativism. With no central government, no dominant currency, and even competing faiths and heresies, value is expressed quite abstractly only in a set of relative relations to other items. (NBER Working Paper No. 10352, 2004)

It was in this environment that ‘arbitrage’, the process by which one commodity ‘arbitrates’ between the value of two other commodities, just as money ‘mediates’ between the value of two commodities, emerges and is discussed in the Liber. 

Aristotle’s use of mathematics in examining the nature of exchange was first noted by the medieval proto-scientist, and Aquinas’ teacher, Albert the Great. This observation was important philosophically since it separated the measure from the measured, money does not share the same ‘nature’ as shoes. This seems insignificant today, but at the time it was an important conceptual development that spawned a ‘mania’ for measuring and mathematics such as that undertaken by the ‘Merton Calculators’. 

The historian Joel Kaye has argued that the line of thought initiated by Albert and Aquinas developed by Oresme and the Calculators, was based on

the transformation of the conceptual model of the natural world ,…, [which] was strongly influenced by the rapid monetisation of European society taking place [between 1260–1380] (Kaye, p 1)

The scholars took their approach to nature having observed the operation of the markets that had emerged in the century before and in response

[were] more intent on examining how the system of exchange actually functioned than how it ought to function. (Kaye, pp 219-220)

Kaye, and a fellow historian, Alfred Crosby, believe this paradigm shift played a pivotal role in the development of European science. The process of this conceptual change is clear. Innovations in finance led to the development of vernacular mathematics encapsulated in the Liber Abaci and disseminated through abbaco schools. University based scholars then began to try and make sense of what was actually happening and tried to identify the essence of the markets, with Aquinas concluding that a profit was ‘fair’ provided that it was uncertain and came with the risk of a loss. In modern terms, a market was viable if it did not admit arbitrages, so long as there is ‘No free lunch with vanishing risk’. 

In much of the public discourse on modern finance, financial institutions are presented as immoral and rapacious beasts, initiating ‘unnatural’ financial instruments in order to take advantage of the innocent. These opinions belie a more complex reality, that finance is integral to society, complex financial products have been with us for at least 900 years, that when people, like Aquinas, employ reason and morality to examine the markets there are some surprising conclusions, and that the study of markets leads to profound insights for science in general. Criticism of finance is as likely to reflect a lack of rationality and morality in society, in general, than a specific lack of ethics and reason, in the markets.

Who was the first Quant?

Scott Patterson, in his book The Quants, describes Ed Thorp as the 'godfather' of Quants. Without doubt, Thorp heralded the modern age of quantitative finance, but  does this mean he was the first Quant?

Many might point to Louis Bachelier as being the original Quant. Bachelier had been  born in Le Harve, in Normandy, in 1870. His father was a wine-dealer while his mother came from a banking family. In the six months after graduating from school, both of Bachelier's parents died and he was forced to take over the management of the family firm, Bachelier fils, to provide for his younger siblings. After doing his military service in 1891, Bachelier moved to Paris and became involved in the Paris Stock Exchange. Simultaneously, he enrolled on the mathematics degree at the University of Sorbonne, where Poincare was a professor. He was not a brilliant student but in 1895 he embarked on a doctorate that synthesised the activities being undertaken at the Paris Stock Exchange and the theories of heat and of probability, at the cutting edge of physics at the time.  Bachelier then left finance and embarked on an academic career that lasted the following forty years, a career that appears to have been hindered by his early association with the markets.


This, however,  is not the career path of a Quant. The likes of Ed Thorp and James Simons had a training in science that they then applied to finance, Bachelier had started working in finance and used this experience as the basis of an academic career: he was a 'reverse-quant'.


The history of reverse-quants is probably more significant than that of Quants. Leonardo Bonacci, better known as Fibonacci, could claim to be the first reverse-quant, taking ideas from contemporary finance to change European culture, by changing not just how merchants undertook their business calculations, but how western science wrote mathematics.


Leonardo's father, Guglielmo Bonacci, was a merchant looking after the Pisan interests in the Algerian port of Bejaia. While we might not imagine that medieval finance was very sophisticated, we would be wrong. The historian Alfred Crosby describes a series of transactions undertaken by an Italian merchant, Datini, which, although they took place two hundred years later, would have been similar to the types of transactions Guglielmo Bonacci would have been involved in. In November 1394 Datini bought wool forward from Mallorca. The wool was sent to Pisa, via Barcelona, in summer 1395, arriving in Italy the following January. Datini sold some wool to a colleague in Florence and processed the the rest into cloth, which he sent to Venice for shipping back to Mallorca for sale in July 1396. However the market in the Balearics was weak and so the cloth was transported to Valencia and North Africa. The last piece of cloth was sold three and a half years after the wool was originally been contracted from the shepherds. Datini would have engaged in forward contracts, loan agreements and transactions in at least five currencies (Arogonese, Pisan, Florentine, Venetian, North African). To make a profit, he needed to be an expert at 'commercial arithmetic', or financial mathematics.


Fibonacci was born in Pisa around 1170 and educated, not only in Bejaia but, as far afield as, Egypt, Syria, Constantinople and Provence. He would write a number of books on mathematics, but his first and most influential was the Liber Abaci ('Book of Calculation'), which appeared in 1202. The Liber was heavily influenced by the Arabic book 'The Comprehensive Book on Calculation by Completion and Balancing'  written around 825 CE by al-Khwarizmi, who was himself motivated to write the book because

men constantly require in cases of inheritance, legacies, partition, law-suites and trade [a number]

and his book provided the easiest way of arriving at that number, using al-gabr ('restoration') and al-muqabala ('balancing'). Fibonacci collated these Arabic techniques into a single  textbook for merchants, such as Datini,  facing the increasingly complex financial instruments and transactions emerging at the time.


The impact of the Liber Abaci was enormous. Fibonacci became an adviser to the most powerful monarch of the time, Frederick II, Holy Roman Emperor and King of Sicily. More significant, Abaco or rekoning schools sprang up throughout Europe teaching apprentice merchants how to perform the various complex calculations needed to conduct their business. Pacioli, who taught Leonardo da Vinci maths, was a well known graduate. Less well known is the fact that Copernicus came from a merchant family and in 1526, seventeen years before his more famous, "epoch-making" 'On the Revolutions of the Heavenly Spheres', he wrote 'On the Minting of Coin' about finance.


The practical usefulness of the reckoning schools was that, by using positional numbers and algebra, merchants could execute complex financial calculations that would typically include an illicit interest charge, hidden from the mathematically unsophisticated, university based, Church scholars. The merchant bankers were using mathematics to keep one step ahead of the regulator and the effectiveness of the non-university mathematics would not have been lost on the sharper scholastics, observing market practice.


The most influential single Abaco graduate has to be the Dutchman, Simon Stevin. Stevin, who was born in 1548 in Bruges, had originally worked as a merchant's clerk in Antwerp then as a tax official back back in Bruges, where he wrote his first book Tafelen van Interest ('Tables of interest') which he published in 1582, before moving to the University of Leiden in 1583. About this time, he was appointed as adviser to Prince Mauritz of Nassau, who was leading the Dutch revolt against the Spanish, and eventually became the Dutch Republic's Finance Minister.


As well as being active in government, Stevin carried out scientific experiments, and it is believed his bookeeping inspired his physics. His most famous experiment showed that heavy and light objects fell to the earth, in the absence of air resistance, at the same speed, an experiment that disproved a belief of Aristotle and is usually attributed to Galileo dropping things from the Tower at Pisa some years later.
One of Stevin's most important posts was as the director of the Dutch Mathematical School, established in 1600 by Mauritz to train military engineers. In this capacity, in 1605, he published a textbook for the School, the 'Mathematical Tradition', which was a comprehensive overview of mathematics and included a whole section on 'Accounting for Princes in the Italian manner'. In a very short period, the Dutch Mathematical School became the centre for merchants' training in north western Europe. This success, in turn, forced the authorities at the  University of Leiden, which provided the School with its facilities, to take practical sciences, in particular maths, a bit more seriously. The Dutch Mathematical School would inspire the soldier Descartes to study maths and would train Huygens and a whole generation of European scientists.  In addition, it was Stevin's promotion of the use of decimals, to aid accounting, that inspired Newton to think of functions as power-series, giving birth to the discipline of Analysis.


So much for 'reverse-quants'. If a Quant is defined as a someone who moves from an academic career into finance, the most famous Quant is Isaac Newton. Newton essentially finished his work in physics with the publication of Principia in 1687, his last significant work, Optiks, published in English in 1704, was  based substantially on research undertaken in the early 1670s. After almost a decade of troubles, Newton moved into finance in April 1696 when he was appointed Warden of the Royal Mint. This was a largely ceremonial post, but Newton took to it so much that he became the Mint's operational manager, its Master, in 1699 (see Isaac Newton: Financial Regulator).


Being Master gave Newton an average income some 16 times what he would have had as an academic. Today a 'starting' salary for a jobbing professor in England is around £60,000, and we could expect the  Lucasian Professor at Cambridge to earn somewhat more than this. On this basis, the equivalent salary for Newton as Master of the Mint is in excess of £1 million, which pretty well places him in the bulge bracket. Newton, no longer needing the income from his position in Cambridge, resigned his Professorship at the end of 1701. Newton did become President of the Royal Society in 1703, an institution whose foundations were laid by the banker Thomas Gresham, and Newton followed in the footsteps of his patron, the the English Chancellor of the Exchequer, Charles Montagu. Gresham and Montagu, central to the establishment of the oldest national academy of sciences, are just two more examples of   'reverse-quants'.


Newton might be the most famous Quant, but he was not unique or the first. Huygens identified conditional expectation while investigating the pricing of life annuities, and J. Bernoulli identified the number e when investigating interest payments. Generations before Huygens, Bernoulli and Newton, we have Galileo, who was born in Pisa around 1564. He was appointed to the Professorship of Mathematics the prestigious University of Padua in 1592, where he made his famous astronomical observations. Galileo had been short of money all his adult life, as a mathematician he was poorly paid and was expected to supplement his salary by taking private students, but this interfered with his research. In 1613 the Duke of Tuscany, Cosimo II de Medici, offered Galileo the position of 'First and Extraordinary Mathematician of the University of Pisa and Mathematician to his Serenest Highness (i.e. Cosimio)' with a large salary and no duties, an ideal post for Galileo. It was some time in the next decade or so that Galileo wrote 'Upon the Discoveries of Dice', which he was almost certainly asked to write by Cosimo, who may have been trying to solve a practical problem in gambling. Galileo did not leave Padua to work in finance, but he did become the mathematical adviser to the the head of one of Italy's most important banking families.  Ironically, had Galileo stayed at Padua, under the protection of the religiously ambivalent Venetians, he would never had faced the Inquisition, and possibly would not have become as famous as he is today.


European science did not start in the Renaissance, it existed in the High Middle Ages. The 'renaissance' of the 'long twelfth century' resulted in what the historian Joel Kaye describes as

the transformation of the conceptual model of the natural world ,..., [which] was strongly influenced by the rapid monetisation of European society taking place [between 1260-1380].

and played a pivotal role in the development of European science. Thirteenth century scholars

[were] more intent on examining how the system of exchange actually functioned than how it ought to function..

It seems that Fibonacci did not just influence medieval merchants, those scholars keeping an eye on merchant's dubious dealings, also, became obsessed with mathematics. This included the great scholastic scientist, Albert the Great, and the mathematically minded theologian, Thomas Aquinas. But the most famous scholars to turn to mathematics were the 'Merton Calculators'.


The first of the Calculators was Thomas Bradwardine, who entered Merton College in Oxford in 1323. While studying the quadrivium, mathematics, astronomy, music and geometry, Bradwardine observed that

[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.

This was a highly significant point in the history of science since it is the first time that scholars in the Hellenistic tradition, which included Jewish, Christian and Islamic philosophers, innovated in physics by using mathematics. In the footsteps of Bradwardine there was, amongst others, William Heytesbury, who identified the mean speed theorem, and Nicolas Oresme, who introduced the idea of the graph and advised the French king on money supply.


After twelve years at Merton, Bradwardine left the Oxford University to work for the   Bishop of Durham, who was the Treasurer and Chancellor of England. So Bradwardine was not only the first Physicist, as we would understand the term today, he was also the first person who left an academic career to take up one in finance. The first Quant was the first Physicist.


Looking at the relationship between scientists and finance reveals some important facts. Firstly, the  migration from academic careers in science to finance appear to be embedded, it is not a modern  phenomena. However, possibly more significant is the less well-appreciated role of the 'reverse-quants' in the development of science. The influence is captured by events in France in 1304-1305 when economic instability and a market failure led the French King, Philip the Fair, to issue decrees fixing the price of bread. His decrees failed spectacularly, and this was seen by contemporary observers as evidence that 'nature' ruled, and not the authority of the King, and that market prices where an objective, 'scientific' measure. This enabled the likes of Bradwardine to re-assess the role of mathematics in science. Later, people trained in commercial arithmetic - financial mathematics - such as Copernicus and Stevin, were able to challenge the authority of Aristotelian science, and argue that the Earth revolved around the Sun and that heavy and light objects fall at the same speed. Today, financial markets challenge assumptions about determinism and stability of systems, the question is, can science meet those challenges?

Maths and the markets

Dr Jack Stilgoe, a science policy wonk, has been thinking about Responsible innovation in financial services and asks the question, in relation to the Credit Crisis

Could mathematicians have done more to ensure that their models weren't abused, or is it not really about maths at all?

Jack's deceptively simple question is incredibly intricate.  There are many commentators who argue that the complexity of modern markets is such that they are mathematically intractable, and the best approach is analysis through discourse, as was popular in the Dark Ages and between the Black Death and Francis Bacon and Galileo.  My (biased) opinion is that these views are held principally by those educated in the ethos that developed before the collapse of Bretton-Woods, when a deterministic economy was managed by wise sages.  Unfortunately the world is not deterministic and the sages could not hope to manage the economy by agreeing treaties in luxury hotels.

However, mathematics itself cannot present a unified front.  We have Paul Wilmott and Nicolas Nassim Taleb arguing that the mathematical techniques that dominate the markets today, that of Ioannis Karatzas, Steven Shreve, Mark Davis (whom Wilmott has famously libelled in an ad hominen attack) and a Marek Musiela, to name a few, is the wrong sort of mathematics.  This is rather like someone claiming a Toyota Prius is not really a car in comparison to a Dodge Pickup, the fact that the Prius is unfamiliar does not mean it is not technically superior.

However, this does not mean that the academic discipline of financial mathematics does not have some issues to address.  The publication of the Heath-Jarrow-Morton framework created a demand for stochastic analysis skills in the markets, displacing the skills in the numerical solution of deterministic differential equations familiar to Wilmott, Taleb, physics and engineering.  This demand was met by the universities with a plethora of Financial Mathematics Masters degrees.  I feel that now the markets have moved on, but whether many of the MScs are keeping up, I am not so sure.

Part of the problem is that many academic mathematicians are more comfortable walking across campus to chat to their colleagues in the economics or finance departments than talk to mathematicians with direct experience of the markets, such as Claude Shannon, Edward Thorp and James Simons.  This means that the orthodoxy of Samuelson and his progeny dominates and ideas such as the Kelly Criterion, and those of stochastic control familiar to electrical engineering, have been missing from the rarefied curricula of some financial maths degrees.

But all this is a discussion of plumbing of the markets, a utility, and mathematics is not really a utility.  Mathematics is a science.

For Laplace, the roll of a dice is not random, given precise information of the position, orientation and velocity of a dice when it left a cup, the result of the roll was perfectly predictable.  At the heart of Laplace's determinism was knowledge, and `probability' was a measure of ignorance, and not of 'chance'. As a product of the Enlightenment, Bernoulli's God is replaced by 'an intellect', Laplace's demon.  The positions of Laplace and Bernoulli, however, differ significantly from Cicero who, in De Divinatione, distinguished between the predictable (eclipses), the foreseeable (the weather) and the random (finding of a treasure).  But between the Bernoulli's religious and Laplace's atheist conceptions of predestination, there is more than just a change in wording; there is a huge philosophical divide that was one of the key achievements of the Enlightenment.

A persistent problem with determinism is that it, logically, can lead to a collapse in moral responsibility. The syllogistic argument is:
Premise 1.        Actions are either pre-determined or random. 
Premise 2         If an action is pre-determined, the entity who performed the action is not morally responsible.  
Premise 3.        If an action is random, the entity that performed the action is not morally responsible.   
Conclusion.    No one is morally responsible for their actions.

An achievement of the Enlightenment was to realise that moral responsibility should not sit in the conclusion, but as a premise, and the argument became. 
Premise 1.        People should be held morally responsible for their actions.  
Premise 2.        If someone (i.e. a child) cannot foresee the consequences of their actions they cannot be held morally responsible for their actions.
Conclusion.    Moral responsibility requires that there be foresight.

In order to be 'morally responsible', people needed to have a degree of foresight, which can only be obtained through knowledge, or science.  This is the fundamental purpose of science, to enable people to take responsibility for their actions, whether related to the safety of industry or personal diet.  This was reflected in Humboldt's view that education should turn 'children into people', but very different from Bacon's opinion that 'knowledge is power'.

Society needs science to interact with the markets because science creates knowledge, knowledge enables foresight and foresight leads to responsibility.  If there is no science of finance, there can be no responsibility in the markets (if the Enlightenment was right).

Poincare dismissed the idea of 'science for science's sake', science is not a recreational pursuit.  Scientists need to ask the difficult questions at the extremities of knowledge and mathematics role is to tackle the questions that cannot be answered by experimentation.  This is why the the $3 billon investment in the Large Hadron Collider, in looking for the Higgs Boson, is seeking to prove a mathematical derivation.  Physical sciences are impotent in reaching out to the boundaries of knowledge without mathematics clearing the path.

The financial markets cannot be experimented on.  The very fact that they are complex means that the only tool science has in trying to understand them is mathematics.  The fact that the, predominantly, deterministic mathematics based on physical phenomena that most people are familiar with (even frequentist or objective probability is rooted in the 'physical' act of counting) is insufficient to understand the markets does not mean that mathematics will not provide the key to understanding the markets.  The point is, it will be "mathematics, but not as we know it", it needs to be created.

If society wants to understand the markets, and really wants them to act responsibility, it needs to fund financial mathematics on a par with the investment made into the physically very small or the very distant.