Monday, 22 August 2011

The origins of modern algebra and its notations

I am now working on the chapter of my book (on formal languages) which focuses on the historical development of formal languages and mathematical formalisms more generally. I wish I could go into these developments much more thoroughly than I will be able to in the book (given word limit and time constraints), as they are truly fascinating. (It is no secret to anyone having read my previous posts that I am obsessed with the topic of notations and symbolic systems!) So for now, let me share some bibliographical suggestions and a digest of my main findings.

Some of you may be thinking that this topic is not of obvious interest in the context of a blog on mathematical philosophy. But one of the hallmarks of mathematical philosophy is certainly the use of mathematical formalisms and special notational systems, and thus a better understanding of this methodology (including its history) does seem to fall within the remit of the blog. (At any rate, feel free to stop reading if you don't care much about history!) For those who are still with me, here we go.

A question that had puzzled me for years is: where does the explosive progress in mathematical notation of the 16th and 17th centuries come from? I knew enough about Latin medieval *academic* mathematics to know that nothing in the quadrivium curriculum seemed to anticipate the birth of modern algebra with Viete and Descartes. So it had to come from somewhere else, but where? Well, last year I attended a conference in Nancy, and the mystery stated to be dispelled with a talk by Albrecht Heeffer on the medieval abbaco tradition. Here is a passage from the paper corresponding to that talk:

By the end of the fifteenth century there existed two independent traditions of mathematical practice. On the one hand there was the Latin tradition as taught at the early universities and monastery schools in the quadrivium. Of these four disciplines arithmetic was the dominant one with De Institutione Arithmetica of Boethius as the authoritative text. Arithmetic developed into a theory of proportions as a kind of qualitative arithmetic rather than being of any practical use, which appealed to esthetic and intellectual aspirations. On the other hand, the south of Europe also knew a flourishing tradition of what Jens Hoyrup (1994) calls “sub-scientific mathematical practice”. Sons of merchants and artisans, including well-known names such as Dante Alighieri and Leonardo da Vinci, were taught the basics of reckoning and arithmetic in the so-called abbaco schools in the cities of North Italy, theProvence, and Catalonia. The teachers or maestri d’abbaco produced between 1300 and 1500 about 250 extant treatises on arithmetic, algebra, practical geometry and business problems in the vernacular. The mathematical practice of these abbaco schools had clear practical use and supported the growing commercialization of European cities. These two traditions, with their own methodological and epistemic principles, existed completely separately. (Heeffer, forthcoming)

Basically, the abbaco tradition was the missing link between Arabic algebra as consolidated in al-Khwārizmī’s “Book on restoration and opposition” (which in turn was inspired by other mathematical traditions, such as the Indian tradition) and the algebra of Viete and Descartes. The interesting thing is that Viete himself emphasizes his indebtedness to three Greek mathematicians (Pappus, Diophantus and Eudoxus -- see this paper by Danielle Macbeth), but makes no reference to either Arabic algebra or to the sub-scientific abbaco schools (this fits well the Renaissance ethos of the time, going back to the Classics!). But besides the 'canonical' Arabic tradition, the abbaco tradition was much inspired by the introduction of special symbols and techniques to operate with the symbolism emerging in the mathematical tradition of the Maghreb in the later Middle Ages (again, it makes sense, as the abbaco people were merchants and thus traveled a lot!). The Maghrebian tradition appears to be the historical place of birth for many of the notational conventions still widely used, such as the notation for fractions.

Thus, one lesson to be learned here is that focusing only on the official, 'academic' story is simply not enough to understand the emergence of modern mathematical symbolism; the sub-scientific tradition of the abbaco schools is a crucial piece in the puzzle.

Here are some additional references:

- A whole volume with the title Philosophical aspects of symbolic reasoning in Early Modern mathematics (and freely available!). In particular, the paper by Hoyrup tells the story of the development towards algebraic symbolization from circa 1300 to 1550, and the paper by Heeffer covers the ground immediately preceding Viete.

- Another paper by Hoyrup, a concise survey of proto-algebra and pre-modern algebra.

- Chapter 5 of Bellos' Alex's Adventures in Numberland, which I also mentioned in my previous post. In it (p.181), we discover for example that the reason why we use x as the main symbol for the unknown is because it is one of the least used letters in French! Descartes introduced the convention that letters towards the end of the alphabet would be used for unknown quantities (while those at the beginning would be used for known quantities), but as his La Geometrie was being printed, the printer was running out of letters (think of the old-fashioned printing method of using small lead letters to imprint the paper). He asked if it mattered whether x, y or z was used; since it did not matter, he opted for x simply because it is used less frequently in French. And here we are, still stuck with x's!

UPDATE: Kai von Fintel makes the excellent suggestion of adding some links to the (still) definitive account of the history of mathematical notations: Florian Cajori's A History of Mathematical Notations (1928).The first volume can be downloaded (for free) here, the second seems not to have been scanned yet. And here is a compilation (from Cajori's book) of earliest uses of various mathematical notations.


  1. Catarina, many thanks. Look forward to this book.
    I'm interested in various notational matters, and I'm pleased you've gathered some interesting material from the history of algebra.

    An example that bothers me is the best way to write sequences, and I've finally settled on

    $(a_i: i \in I)$,

    where $I$ is the index set.
    Although we see things like:

    $(a_0,\dots, a_n)$

    $\langle a_0, \dots, a_n \rangle$




    $(a_i)_{i \in I}$

    $\{a_i\}_{i \in I}$



    and no doubt others ... And sometimes a sequence starts with $a_0$ and sometimes with $a_1$.

  2. I had a question about your comments on why "x" is most frequently used as a variable, and the assertion that it is because it is the least frequently used letter in the French alphabet.

    I first came across this idea in a book by Art Johnson. I contacted him about it and asked if he could provide the original source of this bit of information. He said that he didn't have the reference. So far, every mention of this that I have ever seen in any book has listed Art Johnson as a source. (not that I have done an exhaustive search.) The earliest mention of anything connected with this idea that I was able to find was in an 1885 magazine article in Biblioteca Mathematica, but it still did not substantiate the claim or give other references.

    I was wondering if you have any references on this bit of information that you'd be willing to share. Citations are fine... I'll go find the originals. I'm not trying to discredit the claim, but rather, I'm interested in researching it and I'm not sure where to continue.

    Thanks so much,


  3. Hi Dave, legitimate question. My source for this piece of information was Alex Bellos' book, but right now I don't have it handy to check whether he gives any further references. But you are right, it may well be one of those 'urban legends' that everybody keeps repeating... It would be great to find a more definitive source for the story.

  4. Looks like urban legend to me. the 1637 text is online atéométrie_(éd._1637)
    and a cursory inspection shows not only that x, y, and z are all used as variables in the text, but also that 'z' occurs far less frequently in the body text than either 'x' (with uses similar to modern french) or 'y' (used in many cases where modern orthography would have an 'i'). Not to mention that a french printer, especially in that time, would want to have many instances of 'z' available for setting dialog. (« frère Jacques, dormez-vous ? »)

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