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.