Conference on Diagrammatizing Mathematical Work: A Historical Approach

This hybrid conference is organised by Karine Chemla, Zehra Bilgin, Alex Garnick, Agathe Keller and Xiaohan Zhou in the context of context of the British Academy’s Global Professorships Programme 2023.

Please email your abstract to the conference organisers by November 15 2026.

In our exploration of the historiography of mathematical symbolism during the first phase of the project, it emerged that several previous historians (e.g., Nesselmann 1842, Neugebauer 1932-1933, Burnett 2002) had emphasized a key feature of symbolism that might serve as a promising point of departure for our work. Namely, mathematical symbolisms are primarily written notations, not transcriptions of spoken language, and may even be unrelated to speech altogether.

To explore this property of symbolisms, the seminar this year has focused on the diagrammatic dimensions of various mathematical notations and inscriptions, along with their corresponding practices. It may be helpful, first, to recall what we have meant by such a focus.

Why did our theme specify both “notations” and “inscriptions”? As far as the ancient and medieval sources are concerned, mathematical notations that are distinct from linguistic signs appear occasionally in the documents themselves, though at times only in the margins. Sometimes, they are merely described in the discursive parts of texts, the notations themselves rather used on ephemeral media only: It is these ephemeral kinds of notations that we denote with the term “inscription.” Although most often such inscriptions must be reconstructed indirectly (from the evidence that our sources contain about them), we must keep them in our focus. This is, in large part, because some of the notations that eventually feature in later sources present strong continuities with ephemeral inscriptions that are alluded to in earlier sources.

Moreover, why did we concentrate on the “diagrammatic dimensions” of notations and inscriptions? Our hypothesis was that concentrating on textual entities that broke with the continuous writing of a discursive text line by line—or column by column, for Chinese texts—would naturally lead us to focus on what we called “written practices.” By these practices, we meant ones relying on notations and inscriptions that, like symbolisms, did not merely transcribe spoken utterances. Indeed, such entities appeared promising, since they presupposed a distance between oral language, on the one hand, and mathematical notations and inscriptions, on the other. We thus thought that such an approach would help us better understand what the diagrammatic dimensions of 16th- and 17th-century algebraic symbolisms owed to previous practices that involved similar dimensions. This was a way to inscribe these symbolisms within a deeper history of mathematical practices, from yet another perspective.

The various contributions to the seminar this past year dealt with sources in Arabic, Chinese, Sanskrit, and several European languages. Together, they illustrated that, geometrical diagrams aside, these sources attest to a great variety of mathematical practices—far greater than what we had first envisioned—which bring into play devices with diagrammatic features: cells, rows, columns, and tables; horizontal, oblique, and vertical lines; dotted and continuous lines; positions, and so on. It has thus appeared to us that the diagrammatic features of the notations and inscriptions on which we had first concentrated belong to a broader set of graphic devices employed in mathematical practices. Examining the nature, purpose, and use of the diagrammatic features of such devices in general, we believe, will help us understand the specificity of these features within symbolic practices in particular and thus advance our historical endeavor. This is what we propose to address in the conference.

Here are some of the issues that appear to us worthy of attention:

What and where? 

Which kinds of devices with diagrammatic features do our sources contain, or to which ones do they refer? Which diagrammatic features do these devices put into play? Where do these devices occur, in the context of their use as well as in that of their representation (the distinction is necessary since devices used on material media can also be represented in a document)? Are they placed in the main text, or in the margins, or do they feature on material media only? Moreover, in relation to what specific mathematical topics and tasks are they used? Are they tied essentially to practices of computation, or do we encounter them in other types of mathematical practices?

Why? 

What are the purposes and functions of these devices? What status and meaning are attributed to them, and to the practices involving them? Are they used to express something (e.g., a proof) or as a support for an activity (e.g., the execution of a computation)? Are they schematic ways of prescribing actions, or diagrams on the basis of which to carry out computation or reasoning? Are some of these devices specific to a kind of activity, or can the practices involving them combine various uses? What properties do these diagrammatic features impart to the mathematical practices relying on such notations and inscriptions?

How? - 1 

How are the diagrammatic features brought into play in relation to the purposes and functions of the devices? How do actors navigate through the devices when performing a task? Which feature is used for which operation?

How? - 2 

What types of signs (e.g., words, symbols) and configurations thereof are used within the devices, and how can the choice of signs be correlated with the use of the device How are these signs used in particular?

How? - 3 

Are these devices used in and of themselves, or are they used in relation to discursive developments in the corresponding sources? If the latter, how do the devices with diagrammatic features used by actors relate to the discursive parts of the sources in which they occur or that refer to them? 

Actors’ categories and views

Can we find actors’ categories referring to such devices or statements describing how actors view their epistemic function? 

One goal

Against the backdrop of this broader category of graphic devices, how can we situate the diagrammatic features of notations and inscriptions that use place-value notations as well as algebraic symbolisms of the Viètean kind? What do practices involving such notations and inscriptions owe to earlier practices shaped by actors to carry out their cognitive work with these devices (such as spatial navigation)? How do they differ from other types of diagrammatic practices? In particular, what relationship can be established between the deployment of notations and inscriptions with diagrammatic features and the use of place-value notations?

Giaquinto (2007) has explored the visual aspects of “calculation” and “symbol manipulation,” analyzing how (in our terms) diagrammatic dimensions of notations and inscriptions were used in such mathematical work. The program he has formulated to carry out a fine-grained analysis of this type of work (identifying the key elementary operations that such work requires) raises a question that is fundamental for our endeavor: What is the history of these diagrammatic features and of the practices that bring them into play?

In the context of the conference, we welcome contributions dealing with sources from as wide a range as possible. 

These may include the tabular types of computation that we find in, for instance, Sanskrit sources (resp. Keller, Montelle, and Koolakudlu forthcoming). They may also include Chinese sources such as Qin Jiushao’s 秦九韶 Mathematical Writings in Nine Chapters (數書九章, 1247) or Arabic sources such as Ibn al-Yāsamīn’s Grafting of Opinions of the Work on Dust Figures, composed in the Maghreb in the twelfth century (Oaks 2007). Indeed, the latter two cases appear as interesting inasmuch as their authors viewed as “figures” the graphical devices employed to compute using place-value decimal notations and to determine “the” root of equations. Accordingly, mathematical work with these devices put into play similar diagrammatic features.

For our historical survey of the diagrammatic features of symbolic notations and of the practices with them, we also hope to include contributions that will deal with modern notations and practices with them. These features 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). 

In conclusion, 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. The aim of this conference is to examine how ancient and mediaeval practices bearing on devices with diagrammatic dimensions played a part in the history of symbolic work in mathematics.