Encoding mathematics: A concept for a data model for representing early modern and modern mathematical notations in Digital Editions

Authors: Rinner, Elisabeth

Date: Wednesday, 6 September 2023, 4:15pm to 5:45pm

Location: Main Campus, L 1.202 <campus:note>


In my paper, I will present a concept for encoding historical mathematical texts. As this work is part of an investigation of possibilities to expand the use of digital methods in the Leibniz Edition, it is based on passages from mathematical writings of G.W. Leibniz ([LSB1923ff]).

In 17th centuries’ European mathematics, a notation system for writing down mathematical content developed. Nevertheless, specific notations and spatial arrangements are also used in procedures such as long division, or they provide means that aim on establishing new knowledge. By this, notations are an essential part of “doing mathematics”. As notations and their specific characteristics vary between scholars and over time, they are object of investigation themselves, and preserving them in text editions is crucial for research in the history of mathematics ([Unguru1975]).

However, [TEI P5] proposes to use markup languages which do not “tag” the text according to the actual structures of historical notations―if they can represent them at all. Since an adequate encoding is missing, digital editions of such texts are rare, and existing specimen can scarcely be evaluated digitally in a way that is meaningful for historians of mathematics ([Briefportal], [Görmar2023], [Newton]).

The encoding is indeed challenging: the data model needs to be capable to represent deviations from linear text flow as well as complex and multiple spatial relationships between text elements. In complex spatial arrangements, even the meaning of a single letter can depend on its context if it is part of several interwoven text elements. This illustrates that mathematical notations exceed the limits of today’s manifestation of TEI in [TEI P5] (cf. e.g. [Sahle2013] p. 374). In addition, some passages prove to be multi-modal, in that in one case it is the text strings, and in others rather the spatial arrangements from which meaning is derived ([Rinner2023], section “Résumé und Ausblick”).

The core idea of the proposed approach is to regard mathematical notations as a sign system in the sense of Ch.S. Peirce’s semiotics (so the signs are characterized, among others, by their meaning and their visual representation), and to individuate their structure elements accordingly. A documentation of the signs can be a resource for research on its own. In order to connect several digital editions, a collaboratively established “Dictionary of mathematical notations” seems to be a desideratum for the history of mathematics.

Representing the semiotic signs as nodes in a graph-based approach leads to an appropriate encoding of the complexities of mathematical notation. By separating the “bits” of mathematical notations such as short text strings (often, they will comprise only single characters) and additional graphical elements from the encoding of structure elements, re-interpretations as they can be observed within and across historical mathematical texts as well as additional research-specific perspectives of mathematical notation can be added to an already existing digital edition. By capturing different modes such as textual or diagrammatic aspects the text is represented in full depth, and options for searching it can be expanded.

A useful and well-adjusted set of components for an implementation is still to be established. Due to the character of this approach, this should be preceded by discussions in the history of mathematics.


[Briefportal] [Leibniz-Forschungsstelle Hannover der Akademie der Wissenschaften zu Göttingen beim Leibniz-Archiv der Gottfried Wilhelm Leibniz Bibliothek – Niedersächsische Landesbibliothek]: Briefportal Leibniz. Ausgewählte Briefe in HTML. https://leibniz-briefportal.adw-goe.de/.

[Görmar2023] Maximilian Görmar: Briefportal Leibniz. Ausgewählte Briefe in HTML, Niedersächsische Akademie der Wissenschaften zu Göttingen in Kooperation mit der SUB Göttingen (ed.), 2016-2019. (Last Accessed: 29.08.2022). Reviewed by Maximilian Görmar (Herzog August Bibliothek Wolfenbüttel). In: RIDE – A review journal for digital editions and resources, Issue 16: Scholarly Editions (FAIR criteria), ed. by Tessa Gengnagel, Frederike Neuber, Daniela Schulz and Ulrike Henny-Krahmer, Frederike Neuber, Martina Scholger as managing editors, 2023. DOI: 10.18716/ride.a.16.3.

[LSB1923ff] Gottfried Wilhelm Leibniz: Sämtliche Schriften und Briefe, ed. by Preußische Akademie der Wissenschaften et al., Darmstadt et al. 1923ff. A substantial part of volumes as well as additional material to the Leibniz Edition such as pre-print versions of upcoming volumes and comprehensive registers can be accessed online via https://leibnizedition.de/, https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/ and https://rep.adw-goe.de/handle/11858/00-001S-0000-0006-AB16-C.

[Newton] The Newton Project, https://www.newtonproject.ox.ac.uk.

[Rinner2023] Elisabeth Rinner: Die Zerfällungsschemata der „Regula discerptionum et triscerptionum universalis“ von Gottfried Wilhelm Leibniz. In: [Tagungsband der gemeinsamen Jahrestagung der Fachsektion „Geschichte der Mathematik“ der DMV und des Arbeitskreises „Mathematikgeschichte und Unterricht“ der GDM vom 15.03.2023 bis 19.03.2023 in Naunhof bei Leipzig (preliminary title)] (submitted).

[Sahle2013] Patrick Sahle: Digitale Editionsformen. Zum Umgang mit der Überlieferung unter den Bedingungen des Medienwandels. Teil 3: Textbegriffe und Recodierung (Schriften des Instituts für Dokumentologie und Editorik 9). BoD, Norderstedt, 2013.

[TEI P5] The TEI Consortium: TEI P5: Guidelines for Electronic Text Encoding and Interchange. Originally edited by C.M. Sperberg-McQueen and Lou Burnard for the ACH-ALLC-ACL Text Encoding Initiative now entirely revised and expanded under the supervision of the Technical Council of the TEI Consortium. Version 4.5.0. Last updated on 25th October 2022, revision 3e98e619e. Text Encoding Initiative Consortium, 2022. https://tei-c.org/release/doc/tei-p5-doc/en/Guidelines.pdf (accessed on 04/04/2023).

[Unguru1975] Sabetai Unguru: On the Need to Rewrite the History of Greek Mathematics. In: Archive for History of Exact Sciences. Vol. 15, No. 1 (1975), pp. 67-114. DOI: 10.1007/bf00327233

About the Author

Dr. Elisabeth Rinner, DH person and editor of Leibniz’s mathematical writings in the project Leibniz Edition at the Leibniz-Archiv (= Leibniz-Forschungsstelle Hannover der Niedersächsischen Akademie der Wissenschaften zu Göttingen beim Leibniz-Archiv der Gottfried Wilhelm Leibniz Bibliothek – Niedersächsische Landesbibliothek (GWLB/NAdWGö) in Hanover/Germany, https://www.gwlb.de/leibniz/leibniz-archiv). Research interests include data models, digital publishing, methodology of DH, and history of mathematics (especially mathematical notations).

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