Rethinking the history of mathematical symbolism: A cooperative project

General histories of mathematics commonly take up the view that mathematical symbolism was an innovation introduced in Europe at the end of the sixteenth century. This project aims to explore the hypothesis that what was practiced at the time was rather, more precisely, a new type of mathematical symbolism drawing on previous ones. In other words, the project will suggest rethinking what we mean by ‘symbolism’. In particular, we will pursue the contention that mathematical symbolism has, in fact, a much more global history than has so far been assumed.

In its first year, the project began with a historical exploration of the historiographies of mathematical symbolism through a series of seminars and a conference. Our aim was to explore the historical shaping of the view that mathematical symbolism originated with sixteenth- and seventeenth-century European actors. The historical importance assigned to their work in this respect has typically derived from their use of literal computation, that is, computations on letters with unspecified values rather than particular numerals.  We wanted, in particular, to highlight the facets of symbolism that have been neglected, or that have been treated as mere consequents of other symbolic features perceived to be primary. Our aim was also to examine properties and virtues of mathematical symbolism that some actors have foregrounded in their historical analysis and that may constitute different starting points for a history of mathematical symbolism. 

This reflection has brought to light that several historians (e.g., Nesselmann 1842, Neugebauer 1932-1933, Burnett 2002) had emphasized a key feature of symbolism that might serve as a promising starting point for the project, namely, that mathematical symbolisms were a primarily written practice and thus independent from spoken language. This viewpoint has led both Neugebauer and Burnett to broaden the range of notations to be included in a history of mathematical symbolism, albeit each in completely different ways.

The second year will explore this property of symbolisms—their independence from spoken language—by focusing on the diagrammatic dimensions of various mathematical notations and inscriptions, along with their corresponding practices. Indeed, symbolisms often involve non-linear notations, and working with them has required the creation of specific mathematical practices to navigate through them.

Why do we mention both notations and inscriptions? In the ancient and medieval sources, notations of these kinds often appear in margins or are even simply described in texts, the notations being themselves used only on ephemeral supports—these are the notations for which we use rather the term “inscription”. In all these cases, what is the status and meaning of these notations and/or inscriptions with diagrammatic features and of the practices with them? Are they schematic ways of prescribing or are they diagrams on the basis of which to carry out computation or reasoning? Who drew them and for which purpose? These questions must be addressed, notably because some of the notations that will later feature within the works themselves present a strong continuity with these ephemeral and marginal notations and/or inscriptions (Chemla 2014). In any event, we will consider the assumption that practices that diagrammatize prescriptions, computations or reasonings played a part in the history of symbolic work in mathematics.

Such a focus on the diagrammatic dimensions of mathematical notations and/or inscriptions as well as of practices with them appears promising for developing the hypotheses of the project inasmuch these dimensions presuppose a distance between oral speech and mathematical notations or inscriptions. Moreover, attending to these dimensions will also allow us to highlight the fact that actors shaped practices of spatial navigation through such notations and inscriptions in the course of cognitive work with them (whether this be in for reasoning or calculating), something that we argue might be a key dimension of symbolic work. 

Giaquinto (2007) has explored the diagrammatic dimensions of “calculation” and “symbol manipulation,” analysing how visual aspects are used in such mathematical work. What is the history of these diagrammatic features and of the practices that bring them into play? Are they related to the tabular types of computation that we find in, for instance, Sanskrit sources (resp. Keller, Montelle, and Koolakudlu forthcoming)? These are questions that will be addressed on the basis of sources in, e.g., cuneiform, Chinese, Sanskrit, Arabic, and Latin, as well of modern mathematical sources, which all abound in notations and computations with such diagrammatic features. Indeed, whether we pick up at random a Chinese source such as Qin Jiushao’s 秦九韶 Mathematical Writings in Nine Chapters (數書九章, 1247) or an Arabic source such as Ibn al-Yāsamīn’s Grafting of Opinions of the Work on Dust Figures, composed in Maghreb in the twelfth century (Oaks 2007), the notation of numbers and computations using place-value decimal notations and notations for equations and for determining their roots were all viewed as “figures.” And accordingly, mathematical work with them put into play similar diagrammatic features. These remarks offer a perspective from which to examine, in particular, the relationship between these features as practiced with numbers written with a place-value notation and as practiced with other types of notations and/or inscriptions (Chemla, 1996). 

In our historical survey of the diagrammatic features of symbolic notations and of the practices with them, modern notations and practices with them will also be of interest for us. They were long recognized by mathematicians like David Hilbert, when he wrote in his famous 1900 address to the Paris congress: “The arithmetical symbols are written diagrams, and the geometrical figures are graphic formulas” (p. 259). We are interested in exploring the history of the shaping of such practices.

The research devoted to the diagrammatic features of mathematical notations and inscriptions will examine whether such features are essentially tied to practices of computation or whether we encounter them in other types of mathematical practice. If they do prove to be attached to computation, we will need to spell out the properties that these diagrammatic features impart to the practices of computation that rely on such notations and/or inscriptions. In particular, a key question will be to understand the relationship that can be established between the deployment of notations and inscriptions with diagrammatic features and the use of place-value notations. The symbolic work enabled by diagrammatic features of specific mathematical practices—the use of navigation they enable, and how this relates to prescription, computation and reasoning—will be central to this year’s work.