Arithmetic in Early Modern England

Jay Hancock reviews Jessica Marie Otis, By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England (Oxford UP 2024) (H/t Tyler Cowen). The opening paragraphs of the review:

Steam engine entrepreneur James Watt, as responsible as anybody for upgrading the world from poor to rich, left a notebook of his work. Squiggly symbols such as “5” and “2” mark the pages. Without these little glyphs, borrowed by Europeans from medieval Arabs, Watt would not have been able to determine cylinder volumes, pressure forces, and heat-transfer rates. Isaac Newton would’ve struggled to find that gravity is inversely proportional to the square of a planet’s distance from the sun. Calculations for Antoine Lavoisier’s chemistry, Abraham de Moivre’s probability tables, and the Bank of England’s bookkeeping would have been difficult or impossible.

But before Hindu-Arabic numerals could fuel the Enlightenment and the Industrial Revolution, society had to start to think quantitatively. Jessica Marie Otis’ By the Numbers is about scribes starting to write 7 instead of VII, parish clerks counting plague deaths rather than guessing, and gamblers calculating instead of hoping and praying.

Why did some countries become wealthy after 1800? Historians argue about the relative influences of religion, climate, geography, slavery, colonialism, legal systems, and natural resources. But the key, famously shown by economist Robert Solow, who died in December, is technological innovation enabling more and more goods and services to be produced per worker and unit of capital. Innovation needs research, development, and engineering. All those require numbers and numeracy.

This is consistent with the argument that David Hays and I made in The Evolution of Cognition (1990):

The role which speech plays in Rank 1 thought, and writing plays in Rank 2 thought, is taken by calculation in Rank 3 thought (cf. Havelock 1982: 341 ff.). Writing appears in Rank 1 cultures and proves to be a medium for Rank 2 thinking. Calculation in a strict sense appears in Rank 2 and proves to be a medium for Rank 3 thinking. Rank 2 thinkers developed a perspicuous notation and algorithms. It remained for Rank 3 thinkers to exploit calculational algorithms effectively. An algorithm is a procedure for computation which is explicit in the sense that all of its steps are specified and effective in the sense that the procedure will produce the correct answer. The procedures of arithmetic calculation which we teach in elementary school are algorithms.

The algorithms of arithmetic were collected by Abu Ja'far Mohammed ibn Musa al-Khowarizm around 825 AD in his treatise Kitab al jabr w'al-muqabala (Penrose 1989). They received an effective European exposition in Leonardo Fibonacci's 1202 work, Algebra et almuchabala (Ball 1908). It is easy enough to see that algorithms were important in the eventual emergence of science, with all the calculations so required. But they are important on another score. For algorithms are the first purely informatic procedures which had been fully codified. Writing focused attention on language, but it never fully revealed the processes of language (we’re still working on that). A thinker contemplating an algorithm can see the complete computational process, fully revealed.

To be clear, what Hays and I argued goes against a widely held view of the matter, perhaps the standard view, which credits the invention of the printing press with the catalytic role. While the printing press was enormously important, its importance was in facilitating the spread of ideas. As instruments of thought, mechanically printed books offered no affordances that hand-copied books didn't have. But arithmetic, that's a cognitive technology and, as such, can have a direct influence on thought.

Hays and I then go on to discuss the effect of "crossing" algorithmic calculation with the development of mechanisms:

The world of classical antiquity was altogether static. The glories of Greece were Platonic ideals and Euclidean geometry, Phidias's sculptures and marble temples. Although Mediterranean antiquity knew the wheel, it did not know mechanism. Water mills were tried, but not much used. Hero of Alexandria invented toys large and small with moving parts, but nothing practical came of them. Historians generally assert that the ancients did not need mechanism because they had surplus labor, but it seems to us more credible to say that they did not exploit mechanisms because their culture did not tolerate the idea. With the little Renaissance, the first machine with two co-ordinated motions, a sawmill that both pushed the log and turned the saw blade, turned up (White 1978: 80). Was it something in Germanic culture, or the effect of bringing together the cultures of Greece and Rome, of Islam and the East, that brought a sense of mechanism? We hope to learn more about this question, but for the moment we have to leave it unanswered.

What we can see is that generalizations of the idea of mechanism would be fruitful for technology (and they were), but that it would take an abstraction to produce a new view of nature. The algorithm can be understood in just this way. If its originators in India disregarded mechanism, and the north European developers of mechanism lacked the abstraction, it would only be the accidental propinquity of the two that generated a result. Put the abstract version together in one culture with a host of concrete examples, and by metaphor lay out the idea of the universe as a great machine. What is characteristic of machines is their temporality; a static machine is not a machine at all. And, with that, further add the co-ordination of motions as in the sawmill. Galileo discovered that force alters acceleration, not velocity (a discovery about temporality) and during the next few centuries mechanical clocks were made successfully. The notion of a clockwork universe spread across Europe (note that the Chinese had clockworks in the 11th Century, but never developed the notion of a clockwork universe, cf. Needham 1981). For any machine, it is possible to make functional diagrams and describe the relative motions of the parts; and the theories of classical science can be understood as functional diagrams of nature, with descriptions of the relative motions of the parts.