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%${}^{1}$.
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”${}^{2}$.
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${}^{3}$,
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${}^{4}$. 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${}^{5}$

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${}^{6}$.
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.${}^{7}$

Economics has always doubted its credentials. Laplace saw the physical
sciences resting on calculus, while the social sciences would rest on
probability${}^{8}$,
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?

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

### References

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.