## Tuesday, 17 January 2012

### 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?

### 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.

## Friday, 6 January 2012

### 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 Morgenstern1967, 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.