(This is a speedrun post, where I set a one hour timer to see what I can find out about a subject. See the category tag for more examples.)

I’m currently reading Catarina Dutilh Novaes’s *Formal Languages in Logic*, and one part of the section on the historical development of mathematical notation jumped out at me as potentially interesting. Abbaco (‘abacus’) schools were a kind of practical school in medieval southern Europe that trained the sons of merchants and artisans in useful mathematics for bookkeeping and business. Apparently the mathematical culture associated with these schools actually went beyond the university education of the time in some respects, and helped push forward the development of algebra:

Indeed, modern algebra (and its notation) will ultimately emerge from the sub-scientific tradition of the abbaco schools, rather than the somewhat solidified academic tradition taught at the medieval universities.

I find these sort of semi-informal institutions on the edges of academia intriguing… I’m not sure how much I care about the details, but it seems worth an hour of investigation at least. There’s also a mention of Leonardo da Vinci and Danti Alighieri attending these schools, which could be interesting to follow up.

This speedrun session is also a bit different because we’re trying out a group speedrun event, and David MacIver and Eve Bigaj have also joined. Let’s see how it goes… As usual I typed this as I went and have done only minor tidying up afterwards, so there may be a bunch of typos and dodgy formatting.

There’s a wikipedia article, but it isn’t very long. Looks like there are a few other useful links though

Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra, among other subjects. These schools sprang after the publication of Fibonacci’s Book of the Abacus and his introduction of the Hindu-Arabic numeral system. In Fibonacci’s viewpoint, this system, originating in India around 400 BCE, and later adopted by the Arabs, was simpler and more practical than using the existing Roman numeric tradition. Italian merchants and traders quickly adopted the structure as a means of producing accountants, clerks, and so on, and subsequently abacus schools for students were established.

So, yep, practical education for merchants and traders.

Significant for a couple of reasons. First they got rid of Roman numerals.

The number of Roman characters a merchant needed to memorize to carry out financial transactions as opposed to Hindu-numerals made the switch practical. Commercialists were first introduced to this new system through Leonardo Fibonacci, who came from a business family and had studied Arabic math. Being convinced of its uses, abacus schools were therefore created and dominated by wealthy merchants, with some exceptions

Also they were instrumental in rising literacy levels.

Nothing about algebra here! Another thing on the search page mentioned Cardano though so hopefully there will be a link.

Then there’s a bunch of stuff about the school system.

Italian abacus school systems differed more in their establishment than in their curriculum during the Middle Ages. For example, institutions and appointed educators were set up in a number of ways, either through commune patronage or independent masters’ personal funds. Some abbaco teachers tutored privately in homes. All instructors, however, were contractually bound to their agreement which usually meant that they could supplement their salary with tuition fees or other rates.

Could be an overlap here with medieval guild funding of universities (e.g. in Bologna), another subject I’m considering speedrunning on.

Independent teachers could also be hired by the commune, but for lower wages.[19] Most times, free-lance masters were contracted by a group of parents in a similar fashion to that of communal agreements, thus establishing their own school if the number of students being tutored was significant in size.[20] Abbaco apprentices training to become masters could also tutor household children and pay for their studies simultaneously.

Last (short) section is on the curriculum.

Arithmetic, geometry, bookkeeping, reading and writing in the vernacular were the basic elementary and secondary subjects in the abbaco syllabus for most institutions, which began in the fall, Mondays through Saturdays.

… Mathematical problems dealt with the everyday exchange of different types of goods or monies of differing values, whether it was in demand or in good quality, and how much of it was being traded. Other problems dealt with distribution of profits, where each member invested a certain sum and may have later withdrawn a portion of that amount

Well that wasn’t a very informative article. There isn’t one in Italian either, just Arabic (same info as English) and Persiian (a stub where I’m not going to even bother to hit translate). So I need to leave wikipedia very early.

OK, this looks good and more what I was after. ‘Solving the Cubic with Cardano – Aspects of Abbaco Mathematics’ by William Branson.

To understand the abbaco mathematics used by Cardano, we have to step back and look at the medieval tradition of abbaco schools and their masters. Though the subject is a fascinating and deep one, there is one particular aspect of this tradition that is crucial in the following account: abbaco masters thought in terms of canonical problems, and one particular canonical problem, the “Problem of Ten,” arises in the solution of the cubic that we will examine.

Quick summary of what they were, similar to wikipedia.

Abbaco mathematics was rhetorical—in Cardano’s time, most of the algebraic symbols with which we are so familiar were either recently invented, concurrent with the Ars Magna, or were well in the future. For example, ‘(+)’ and ‘(–)’ were first recorded in the 1480s, and were not in common use in 1545, when the Ars Magna was published. Robert Recorde would not invent the equals sign until 1557, and the use of letters and exponential notation would have to await Francois Viete in the 1590s and the Geometrie of Rene Descartes of 1637 [Note 2]. What Descartes would write as (x^3=ax+b,) Cardano wrote as “cubus aequalis rebus & numero” [Cardano 1662, Chapter 12, p. 251].

OK this is similar to what Dutilh Novaes was saying, people were solving problems that were algebraic in nature with unknowns to solve for, but the notation was still very wordy.

Rhetorical formulas can be difficult to remember, so algebraic rules were presented with canonical examples, which encoded the rules as algorithms within the examples. Thus, the mind of the abbaco master was a storehouse of such canonical examples, to which he compared the new problems that he came across in his work. When he recognized a parallel structure between the new problem and a canonical problem, he could solve the new problem by making appropriate substitutions into the canonical example.

So these ‘wordy’ forms still had some kind of canonical structures, it wasn’t just free text but was a kind of notation.

Such canonical examples occurred even in the foundational texts of abbaco mathematics, including the Algebra of al-Khwarizmi. An important example for us, one that occurs implicitly in Cardano’s solution to the cubic, is the “problem of ten” [Note 3]. Most abbaco texts had such problems, and one from Robert of Chester’s 1215 translation of al-Khwarizmi’s Algebra into Latin [al-Khwarizmi, p. 111] ran as follows:

Denarium numerum sic in duo diuido, vt vna parte cum altera multiplicata, productum multiplicationis in 21 terminetur. Iam ergo vnam partem, rem proponimus quam cum 10 sine re, quae alteram partem habent, multiplicamus…

In his translation of this passage into English, Louis Karpinski used (x) for ‘rem’ (thing), and so I offer my own translation, without symbols [Note 4]:

Ten numbers in two parts I divide in such a way, in order that one part with the other multiplied has the product of the multiplication conclude with 21. Now therefore one part we declare the thing, and then, with 10 without the thing, which the other part is, we multiply…

My god I can’t even be bothered to read all of that that… very glad we don’t do maths like that now…

The structure of the “problem of ten” was that of a number (a) broken into two parts (x) and (y,) with a condition on the parts; symbolically: [x+y=a\,\,{\rm and}\,\,f(x,y)=b] for some function (f(x,y)) and number (b.) The usual method of solution was to express the two parts as “thing” and “number minus thing” and then to substitute into the condition, as al-Khwarizmi did above. The “problem of ten” was canonical for quadratic problems, and served as a way to remember the rules for solving such problems.

This was used in Cardano’s solution to the cubic, apparently, but there’s no more detail on this page, it just ends there. Looks like a book extract or something.

There’s another MAA page on abbaco schools, though, so I’ll read that next. This is ‘Background: The Abbaco Tradition’ by Randy K. Schwarz.

Bit more detail on where these schools were:

They arose first in northern Italy, whose economy was the most vibrant in Europe during this period (Spiesser 2003, pp. 34-35). A banker and official in Florence, Italy, reported that in 1345 at least 1,000 boys in that city alone were receiving instruction in abbaco and algorismo (Biggs 2009, p. 73). Such schools also began to appear in neighboring southern France, and a few in Catalonia (the area around Barcelona, Spain) and coastal North Africa. These four regions of the western Mediterranean had extensive trade and cultural ties with one another at the time, so it isn’t surprising that they shared methods of practical mathematics and its instruction (Høyrup 2006).

Mentions the Fibonacci book again as a common ancestor. Ah so this is why Fibonacci knew this stuff:

He was only a boy, he reports, when his father, a customs official representing Pisan merchants at their trading enclave of Bugia, in what is now Algeria, brought him to the customs house there to be taught Hindu-Arabic numerals and arithmetic (Sigler 2002, pp. 3, 15)

This article is part of a series on something called the Pamiers manuscript, which translated some of this into French maybe? or some language in modern France anyway. look up later if time.

Nice picture of teaching in an abbaco school here.

In general, the abbaco texts offered practical, simplified treatments in which mathematical techniques were distilled into easy-to-remember rules and algorithms. The focus was on how to carry these out rather than on justifying the theory behind them. At the same time, the books were often innovative in their solutions to particular problems and especially in their pedagogical approach: their presentation was popular, and they introduced the use of illustrations and vernacular languages to the history of mathematics textbooks.

Reference here to something called Swetz 1987, ‘Capitalism and Arithmetic: The New Math of the 15th Century’.

OK this article finishes here too… and I still have 34 minutes, this might be a difficult speedrun for finding information. I may as well skim the intro page and find out what the Pamiers manuscript is while I’m here.

Pamiers is in the far south of France, south of Toulouse near the Pyrenees. Written in the Languedocian language.

One of the striking features of the Pamiers manuscript is the fact that it includes the world’s earliest known instance in which a negative number was accepted as the answer to a problem for purely mathematical reasons. The fact that this occurred in the context of a commercial arithmetic, rather than a more scholastic or theoretical work, is a surprise.

Ah, nice, this is the sort of thing I was hoping for, new ideas coming up in the context of practical problems.

Back to wikipedia for now, what else can I find?

I found a pdf by Albrecht Heeffer which is very short but does mention one interesting book.

The abbaco or abbacus tradition (spelled with double b to distinguish it from the material calculating device called ‘abacus’) has the typical characteristics of a so-called ‘sub-scientific’ tradition of mathematical practice (coined by Jens Høyrup). It is supported by lay culture, e.g. merchants, artisans and surveyors. Knowledge is disseminated through master-apprentice relationships, often within family relations. Texts, as far as they are extant, are written in the vernacular. The tradition is open to foreign influences, including cross-cultural practices. Typically, the tradition is underrepresented in the history of mathematics.

Dutilh Novaes also mentioned the Høyrup book so maybe that is what I should really be reading. It’s this ‘sub-scientific’ angle that I’m interested in.

Abbaco masters made subtle but important contributions to the development of early symbolism. Their two centuries of algebraic practice paved the road for the development of symbolic algebra during the sixteenth century. They introduced mathematical techniques such as complete induction which is believed to have emerged a century later

Yeah, ok, so this *is* an interesting subject but I probably need to be reading books to find the good bits, rather than skimming the internet. Similar to Vygotsky speedrun maybe.

Let’s find out what this Høyrup book is called. Ah it must be this book mentioned on his wikipedia page: ‘Jacopo da Firenze’s Tractatus algorismi and early italian abacus culture.’ Yes I’m definitely going to buy these chapters off Springer for 25.95 euros each, sounds like a great idea.

Ah here’s a copy of a pdf by Høyrup! It’s 34 pages so I don’t have time to go into the details, but I can skim it. Hm also it looks like it’s mainly arguing about the centrality of Fibonacci in the tradition, I’m not interested in that, I’m interested in the sub-scientific thing.

First though I’d like to chase up that thing about Dante and da Vinci.

20 minutes left.

Search ‘da Vinci abbacco school’, oh god the results are full of random schools named after him and references to The Da Vinci Code. Must include: abbaco.

I have found another vaguely useful paper though, ‘The Market for Luca Pacioli’s Summa Arithmatica’ by Alan Sangster and others. Something here about the two-track nature of education in Renaissance Italy, with these schools at the practical end.

The curriculum of the vernacular schools emerged from the merchant culture and was designed to prepare sons of merchants and craftsmen for their future working lives [Grendler, 1990]. There was another parallel set of schools, the Latin (either scholastic or humanist) schools, where the sons of the privileged were taught in Latin.

The two sets of schools taught very different subjects. The Latin schools sought to teach the future leaders of society and those that aided them, e.g., secretaries and lawyers [Grendler,1989, p. 311]. They specialized in the trivium of grammar, rhetoric, and logic… On the rare occasions when mathematics was taught in these schools, it took the form of “classical or medieval Latin mathematics” [Grendler, 1989, p. 309]. In contrast to the vernacular schools, boys leaving the humanist schools often went to university.

Hang on, why don’t I just look on da Vinci’s wikipedia page? It just says the following:

Despite his family history, Leonardo only received a basic and informal education in (vernacular) writing, reading and math, possibly because his artistic talents were recognized early.

which would at least be consistent with going to one of these schools. And Dante Alighieri:

Not much is known about Dante’s education; he presumably studied at home or in a chapter school attached to a church or monastery in Florence.

Hm, so what did Dutilh Novaes say? Ah, it’s a quote from Heeffer 2007, ‘Humanist Repudiation of Eastern Influences in Early Modern Mathematics’. Pdf is here. Should have looked this up to start with!

Actually I’m confused because, although this is very relevant looking, it doesn’t have the quote in it at all. Ah well, I may as well read it for the rest of the time anyway (only 5 minutes left!). The thing about Dante and da Vinci isn’t really important.

Here’s some more on the sub-scientific idea:

Jens Høyrup coined the term sub-scientific mathematics for a long tradition of practice which has been neglected by historians. As a scholar working on a wide period of mathematical practice, from Babylonian algebra to the seventeenth century, Høyrup has always paid much attention to the more informal transmission of mathematical knowledge which he calls sub-scientific structures.

This is pretty complicated to skim quickly.

The sub-scientific tradition was a cross-cultural amalgam of several traditions. Merchant type arithmetic and recreational problems show a strong similarity with Indian sources. Algebra descended from the Arabs. By the time Regiomontanus learned algebra in Italy it was practiced by abbaco masters for more than 250 years. The tradition of surveying and mensuration within practical geometry goes back to Babylonian times.

Some stuff on ‘proto-algebraic rules’.

Our main hypothesis is that many recipes or precepts for arithmetical problem solving, in abbaco texts and arithmetic books before the second half of the sixteenth century, are based on proto-algebraic rules. We call these rules proto-algebraic because they are, or could be based originally on algebraic derivations. Yet their explanation, communication and application do not involve algebra at all. Proto-algebraic rules are disseminated together with the problems to which they can be applied. The problem functions as a vehicle for the transmission of this sub-scientific structure. Little attention has yet been given to sub-scientific mathematics or proto-algebraic rules.

Ding! Time’s up.

Hm, that was kind of annoying to do a speedrun on, because the Wikipedia article was so short and I had to jump quickly to a bunch of other sources which all either had very limited detail or way too much detail. I never did get to the bottom of the Dante and da Vinci thing.

I’m also still not that clear on the details of exactly what new techniques they introduced, but looks like they were relevant to Cardano’s solution of the cubic, and also to the use of negative numbers in problems. They also introduced a bunch of schematic templates for solving problems, which later developed into modern algebraic notation.

The idea of ‘sub-scientific’ traditions sounds interesting more generally too, maybe I should look up the Høyrup book. Overall this looks like a topic where I’m better off reading books and papers than skimming random web pages.

andersJune 5, 2021 / 4:04 amTemplates sound more enjoyable to use than algebra is

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Lucy KeerJune 5, 2021 / 7:15 amHaha I had the opposite reaction, so many words I couldn’t be bothered to even start reading! But I suppose you’d build up familiarity with the templates over time.

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Lucy KeerJune 5, 2021 / 7:23 amGenerally I like meaning to be concentrated I think – I have a related rant half-formed about how single-letter variable names in maths are Good, Actually (sometimes programmers talk about how they hate them). E.g. θ has built up really concentrated ‘angle-ness’ energy over time, it’s a sort of talisman that’s deeply charged with meaning. Compare to like ‘angleBetweenLines’ or some wordy thing in a program.

Long variable names *are* great when you’re dealing with a one-off thing that doesn’t come with any standard meaning though, or you just need to name some boring intermediate stage that doesn’t mean much. Which is often the case in programming.

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