Crunching the numbers

Alongside literacy, numeracy is a key pillar of education and progress. Basic arithmetic skills are in essence a requirement for modern-day life, while more sophisticated techniques are fundamental to any number of scientific and technological advances we nowadays take for granted.

In a new working paper, Danna, Iori and Mina consider the contribution of mathematics to Europe’s economic development. Specifically, they examine how the adoption of Hindu-Arabic numerals — the numbers 0 to 9 as we know them today — influenced growth in pre-modern Europe.

While mathematics has many applications, the central feature of interest for the purposes of this study is its application to commerce. Starting in the thirteenth century, the Commercial Revolution was marked by significant advances in financial activities, business structures and trade operations. Basic features of accounting, insurance and foreign exchange have their origins in this period.

Danna, Iori and Mina contend that the innovations of the Commercial Revolution were enabled by the introduction of Hindu-Arabic numerals and fractions. The preceding system of Roman numerals was less suited to mathematical operations more complex than addition and substraction. For one thing, Roman numerals have no zero.

Time for change. Source: JuergenG / Wikimedia Commons.

The adoption of Hindu-Arabic numerals across Europe was not instantaneous. Rather, it took centuries before new mathematical methods became widespread. Relying on the publication of arithmetic manuals (a proxy for instruction in and adoption of the new mathematical techniques), Danna, Iori and Mina find the earliest evidence of imported Hindu-Arabic numerals in thirteenth-century Italy. It was not until the fifteenth century that such manuals could be found in France or Germany; for England, the sixteenth century.

From zero to hero

To examine how mathematical advances contributed to Europe’s development, Danna, Iori and Mina test the effect of manual publications on city size. The authors control for differences in the introduction of the printing press (which first emerged in the mid-fifteenth century), which might both have facilitated the production of manuals and also supported information dissemination by other means. Other city-level controls relate to cultural, political and geographic factors: whether the city had a university, was a state capital or was located close to trade-enhancing infrastructure (for example, proximity to waterways).

The authors apply different econometric approaches, as illustrated in the table below. The first two — pooled OLS with time and country dummies, and panel fixed effects — cannot of themselves be used to established causality. This point is addressed in different ways by the second two approaches: a two-stage least squares (2SLS) model, and matching of similar cities with and without arithmetic manuals.

Effect of arithmetic manuals on city growth

Dependent variable:
City growth (log)
Pooled OLSPanel fixed effects2SLS
(second stage)
Propensity score matching
Number of manuals (log)0.112***
(0.032)
0.061*
(0.033)
0.375***
(0.143)
0.076*
(0.033)
Printed books0.122*
(0.068)
0.027
(0.042)
-0.059
(0.057)
0.023
(0.064)
City has university0.091**
(0.033)
0.024
(0.064)
0.042
(0.072)
0.037
(0.068)
City is capital0.424***
(0.054)
0.293*
(0.114)
0.335***
(0.110)
0.372***
(0.109)
City has navigable river0.089***
(0.018)
0.512
(0.346)
0.602*
(0.293)
0.489
(0.312)
Population (log)-0.302***
(0.016)
-0.776**
(0.038)
-0.808***
(0.039)
-0.745**
(0.064)
City fixed effectsYYYY
Country fixed effectsYYYY
Observations183818381838662
R20.3090.7260.6960.625
Significance levels: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01. Robust standard errors in parentheses.
Pooled OLS results shown here omit coefficients for additional geographic controls relating to historic roads and ports. 2SLS results reported here only for the second stage (the first stage measures the effect on the number of arithmethic manuals in a given city of the distance between that city and the first source of arithmetic manuals in the same country or the closest source of such materials outside the country). Propensity score matching is performed by considering similarities between cities, determined by the controls included above as well as additional geographic controls relating to historic roads and ports.
Sources: Tables 2, 3, 5 and 6, Danna, Iori and Mina (2022)

(The results shown here compare results for only one model based on the number of manuals in a given city in the preceding century. Danna, Iori and Mina also test alternative measures of manuals, yielding broadly similar results.)

A second look at the second stage

The 2SLS model uses as an instrument the proximity of cities to their initial sources of manuals. The first stage considers how the number of manuals in a given city is influenced by the effect of distance to the first city in the same country where a manual was published. And where the given city is the first city in its country to have such manuals, the distance is instead to the closest city outside the country from which manuals could be sourced. The second stage then uses this effect of distance on manuals to estimate the exogenous effect of manuals on city growth.

Perhaps curiously, the 2SLS estimates are markedly greater in magnitude than either the pooled OLS or panel fixed-effects estimates. The authors suggest that the instrument is capable of “detect(ing) fine-grained differences in the adoption and spread of Hindu-Arabic numerals in commercial practices”. In particular, they argue that measuring the number of manuals is only a crude proxy which likely underestimates how widely the new mathematical techniques were applied once available. In contrast to the other empirical approaches in the paper, the 2SLS strategy specifically considers the role of knowledge diffusion — how and when the manuals came to different cities, rather than purely how many manuals there were.

That said, it is also possible that the constructed instrument is not valid. Unobserved characteristics may influence both how likely it is that cities will adopt arithmetic manuals, and the growth of those cities over time — this is the primary reason for employing an instrumental variable. But if these unobserved characteristics are also spatially correlated with distance to other cities that are likely to adopt the manuals, then the empirical strategy here will not succeed in isolating the effect of the manuals. Indeed, the estimates produced by such a model would tend to overstate the effect of interest. Danna, Iori and Mina use examples to illustrate why this should not be the case here: Regensburg was the first German city where publication of an arithmetic manual is identified, but distance to Regensburg does not appear to have any bearing on the growth of other German cities.

Still, the fact the 2SLS estimates are so out-of-step with the other approaches invites some scepticism about their plausibility. This scepticism is hardly reduced by the alternative approach offered by Danna, Iori and Mina (the final column in the table above). The authors perform a matching exercise, forming pairs of otherwise similar cities where one has manuals in a given time period and the other does not. The resulting estimates are slightly larger in magnitude than the fixed-effects estimates: a 1 per cent increase in the number of manuals in a given city is associated with a 0.076 per cent increase in the city’s growth rate during the following century.

Growth potential. Source: Ecole polytechnique Université Paris-Saclay / Wikimedia Commons.

Beyond Europe?

Danna, Iori and Mina make a plausible case for the contribution of mathematics to the Commercial Revolution and Europe’s development. However, it is worth recalling that the ‘innovation’ at the centre of this study was imported. Indeed, as the authors note, the origins of the techniques enabled by Hindu-Arabic numerals can be traced back to fifth-century India. How do the effects of mathematics in Europe compare to its effects in Asia and the Middle East?

This reflects a broader weakness in economic history: a tendency to focus on Europe and the West to the exclusion of other parts of the world. In part, this is a story of data availability (in particular, Germany and northern Europe have a rich history of detailed record keeping). But it is perhaps also a product of institutional biases. Those of us in and from Western countries are more likely to view things from a Western perspective — if nothing else due to our knowledge of European languages and Western source material. Although there are researchers who have a detailed understanding of other parts of the world, there is plainly an inequality in resources and effort. This compromises our collective ability to comment on the world’s long-term development.

This is not a criticism of Danna, Iori and Mina. Their research is both interesting and relevant, using a detailed examination of source material to offer new insights on the spread and application of fundamental knowledge. One can debate the size of the effect of Hindu-Arabic numerals on growth — different models produce different estimates, and it isn’t clear which one the authors consider most reliable. Nevertheless, the key finding is that there is a significant and positive effect. One might even say that the numbers don’t lie.

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