“The only way you could formulate a complete rule (in premodern sensibility): you had to foresee the exceptions, it is both specific and supple. The habit doesn’t simply enforce the rule, it embodies it, just like this spearbearer statue embodies the canon of male beauty. More than that the habit’s discretion is not supplementary to the rule, it is part of the rule. It is the leaden ruler that adjusts the straight iron ruler to the curves of the individual case.”
In all of these cases, what to our modern eyes looks like supplementary materials, models exceptions examples experience is not only integral to the pre-modern rule it supplies the means by which universal are applied to particulars. So how do we get from there to here? From the prototypical rule as model to the prototypical rule as algorithm? Or to be more precise algorithms so definite so exact so finite that they can be executed by machine? How did rules become mechanical? A full answer would require another lecture in its own right time but I ll spare you, but the short answer is that rules became mechanical before they could be executed by machines. In the late eighteen and early nineteen centuries the size of the most heavy duty calculation mainly astronomical observatories began to devise ways of dividing up the task of thses massive calculations into smaller simpler steps that could be executed by assistants. So first these assistants were schoolboys and later they were women who usually had only elementary mathematical training sometimes only addition and substraction and could be paid very low wages. The most famous and by no mean the only one of these factory-like calculation projects was the one organised by the French engineer Gaspard Riche De Prony in 1791t create new logarithm tables using base ten in order to complement the new metric system which had been introduced by the French revolution. These tables by the way were never published in their entirety, there is still the manuscript in the Observatoire de Paris. Because it was a prestige project it was going to prove not only the superiority of the metric system but also the superiority of the French revolutionary government to the world. The project took absolutely grandiose proportions. They would calculate ten thousands signed values to twenty five decimal places. And 200 000 logarithms to fourteen decimal places. And Prony had read Adam Smith wealth of nations. He read that description of the pin factory. Adam Smith in turn had read the Encyclopédie on pin manufacture. So he got the idea that you could divide the mathematical ever so finely that you have seventy to eighty workers doing only addition and subtraction who could perform about a thousands of these arithmetic operations per day. By method that Prony then described as purely mechanical. So in the midst of the political and economical torment of Terror during seventy nineties Prony’s factory of algorithms managed actually to complete this task in a few years. They filled seventeen folio volumes with manuscript tables which are as I said still preserved at Observatoire de Paris. Th eperson who had the published was none other that Charles Babbage. And when Charles Babbage came in eighteen nineteen to Paris to try and get these manuscript tables published it became for him the inspiration for his own analytical engine. He reasoned that if these uneducated workers could perform these operations, a machine could do it. Now no machine more complicated than a quill pen was used to compute these tables. What made it mechanical in the eyes of Prony and Babbage was rather the nature of the labour. The mechanisation of calculation was a much more gradual process than we like to think. It began in the late eighteen century with the application of the principles of the division of labour to human calculators and it by no means ended in the eighteen fifties when reliable mass manufacturers produced calculating machines like this one [Thomas de Colmar Arithmometer] for the first time became available. There had been calculating machines invented before for example Pascal xxx seventeenth century but no one could get them to work. You always had to check the results by hand. For almost a century between the appearance of the arithmometer and the middle of the nineteenth century until really I think the nineteenth seventies or the sixties in astronomical observatories and insurance offices census bureaus and war time weapon projects like the Manhttan project humans and machines worked in tandem to apply algorithms. Only in the final quarter of the 20th century was a near full automation of the execution of computational algorithms achieved by pre-programmed electronic devices. Mechanisation was first a metaphor and only much later a reality. The mechanization of algorithms in the 19th and 20th century created a new prototype of a rule that requires no interpretation, no examples, no context. It was a genuinely free standing rule. This was the dream of artificial intelligence and indeed much of a cognitive sciences to simulate and perhaps even surpass human mental processes by algorithms that could be used to program a computer. It is a great irony that after decades of frustration in trying to teach computers how to recognize patterns, translate languages and other non-calculational tasks AI has been long abandoned for machine learning in which computers are fed just like commercial arithmetic texts of the fifteenth century thousands and thousands of examples rather than algorithms. In one of the most famous passages of 20th century philosophy Ludwig Wittgenstein contrasted mechanical rule-following with rule-following as a practice taught by example rather than by precept within a community of users. Wittgenstein concludes: to obey a rule to make a report to give an order to play a game of chess, these are customs, uses, institutions. Although Wittgenstein probably didn’t realise it he thereby brought the history of rules full circle. Even the most mechanical rules turn out to be practices taught by experience. But Wittgenstein was still philosophising about how humans follow rules not machines and did his main point was that humans think and learn differently than machines do. With the recent successes of machine learning it may well turn out that the most modern and literally mechanical forms of rule following show that the pre-modern sense of rule as model is alive and well.