Periodic Reporting for period 2 - PHILIUMM (The Philosophy of Leibniz in the Light of his Unpublished Mathematical Manuscripts)
Période du rapport: 2023-03-01 au 2024-08-31
Gottfried Wilhelm Leibniz (1646-1716) is one of the most prominent philosophers and mathematicians of the Early Modern period. As he himself emphasized on numerous occasions, these two facets of his work were closely related. In a letter to Malebranche dated 1699, he declared: “mathematicians have just as much need to be philosophers, as philosophers to be mathematicians”. However, the exact nature of this interconnection between mathematics and philosophy is not easy to characterize. To clarify this relationship forms the core of the PHILIUMM project, not only as a study of Leibniz, but as a model for an interdisciplinary dialogue which needs to be pursued and extended into contemporary philosophy.
The main difficulty in realizing this task lies in the access to Leibniz’s mathematical texts. As surprising as it may sound, although Leibniz’s philosophy has been at the centre of a great deal of attention for three centuries, almost half of his mathematical production was still unpublished when our project began. Leibniz published very little during his lifetime (in mathematics, less than 5% of what he wrote) and remained very elusive about some of his deepest insights (such as his great idea of an analysis situs, his work on binary arithmetic, on logical calculi or his elaboration of what we now call “determinants”). Soon after his death, scholars felt compelled to dive into the ocean of his unpublished manuscripts, but mathematics was not well served in this editorial enterprise. Nor was it the easiest part to deal with either. The number of manuscripts was enormous (several thousands of folios) and their deciphering necessitated interdisciplinary skills which were rarely found. As a result, about half of Leibniz’ mathematical manuscripts were still unpublished in 2021 – a dramatic situation which has no equivalent for other great thinkers from that period. A second obstacle is that, even for the published texts, about half of them were established at the end of the XIXth century without adhering to rigorous and scientific standards – some texts of this group being in fact mere artefacts, which it is not possible to work on with any confidence.
Although it would be an exaggeration to say that all the content of the unpublished manuscripts is original, preliminary exploration of the Nachlass have indicated that they provide the seeds for a deep and fruitful reassessment of Leibniz’ philosophy. The main aim of the PHILIUMM project is to complete this task. In order to fulfil this aim, we have divided the corpus into five main topics of research: Dyadica (binary arithmetic), Ars combinatoria, Foundations of differential calculus, Leibniz’ doctrine of mathematical abstraction and ‘Machines and thought’.
The project relies on a research group that has developed over the past ten years in France and is now unparalleled worldwide. It also builds on a close partnership with the Leibniz-Archiv in Hanover and benefits from recent advances in the digitization of Leibniz's mathematical manuscripts (available online since 2016). Preliminary results have already been obtained on specific sets of texts. We have formulated original scientific hypotheses to guide the study of nearly 17,000 folios of unpublished material.
One of our leading hypothesis is a radical reinterpretation of what Leibniz meant by reducing mathematical truths to "identities." This hypothesis resonates strongly with current philosophy of logic and mathematics and, we hope to shed new light on contemporary debates in philosophy of mathematics. For that, we study the place Leibniz gave to identifications (not necessarily based on an absolute notion of logical identity) in his mathematical practice. We explore the way in which this approach to mathematics has echoes in foundational issues in contemporary mathematics, based on the relativization of identity (or “internal identity”) in category theory and, more recently, within Homotopy Type Theory.
We also aim to use this project to make Leibniz's thought more accessible (especially to historians of mathematics, mathematics professors, and students) through the publication of online text editions and the development of new digital tools to explore them. We develop new tools for Handwritten Text Recognition capable of recognize mathematical equations and new editorial frameworks adapted to genetic editions of complex mathematical drafts.
The main difficulty in realizing this task lies in the access to Leibniz’s mathematical texts. As surprising as it may sound, although Leibniz’s philosophy has been at the centre of a great deal of attention for three centuries, almost half of his mathematical production was still unpublished when our project began. Leibniz published very little during his lifetime (in mathematics, less than 5% of what he wrote) and remained very elusive about some of his deepest insights (such as his great idea of an analysis situs, his work on binary arithmetic, on logical calculi or his elaboration of what we now call “determinants”). Soon after his death, scholars felt compelled to dive into the ocean of his unpublished manuscripts, but mathematics was not well served in this editorial enterprise. Nor was it the easiest part to deal with either. The number of manuscripts was enormous (several thousands of folios) and their deciphering necessitated interdisciplinary skills which were rarely found. As a result, about half of Leibniz’ mathematical manuscripts were still unpublished in 2021 – a dramatic situation which has no equivalent for other great thinkers from that period. A second obstacle is that, even for the published texts, about half of them were established at the end of the XIXth century without adhering to rigorous and scientific standards – some texts of this group being in fact mere artefacts, which it is not possible to work on with any confidence.
Although it would be an exaggeration to say that all the content of the unpublished manuscripts is original, preliminary exploration of the Nachlass have indicated that they provide the seeds for a deep and fruitful reassessment of Leibniz’ philosophy. The main aim of the PHILIUMM project is to complete this task. In order to fulfil this aim, we have divided the corpus into five main topics of research: Dyadica (binary arithmetic), Ars combinatoria, Foundations of differential calculus, Leibniz’ doctrine of mathematical abstraction and ‘Machines and thought’.
The project relies on a research group that has developed over the past ten years in France and is now unparalleled worldwide. It also builds on a close partnership with the Leibniz-Archiv in Hanover and benefits from recent advances in the digitization of Leibniz's mathematical manuscripts (available online since 2016). Preliminary results have already been obtained on specific sets of texts. We have formulated original scientific hypotheses to guide the study of nearly 17,000 folios of unpublished material.
One of our leading hypothesis is a radical reinterpretation of what Leibniz meant by reducing mathematical truths to "identities." This hypothesis resonates strongly with current philosophy of logic and mathematics and, we hope to shed new light on contemporary debates in philosophy of mathematics. For that, we study the place Leibniz gave to identifications (not necessarily based on an absolute notion of logical identity) in his mathematical practice. We explore the way in which this approach to mathematics has echoes in foundational issues in contemporary mathematics, based on the relativization of identity (or “internal identity”) in category theory and, more recently, within Homotopy Type Theory.
We also aim to use this project to make Leibniz's thought more accessible (especially to historians of mathematics, mathematics professors, and students) through the publication of online text editions and the development of new digital tools to explore them. We develop new tools for Handwritten Text Recognition capable of recognize mathematical equations and new editorial frameworks adapted to genetic editions of complex mathematical drafts.
The main achievement of our project is the transcription and commentary of unpublished mathematical manuscripts by Leibniz, as well as the development of digital editing tools capable of processing them.
To date, we have already transcribed and commented on over 400 folios covering the themes announced within the project framework: Dyadica, Ars combinatoria, Foundations of differential calculus and algebra (particularly concerning the question of "reduction to the identicals"). We also contributed to the progress of the upcoming volumes of Leibniz's mathematical writings dedicated to the foundations of geometry and analysis situs, for which we have transcribed about a hundred folios.
We have launched a website (https://eman-archives.org/philiumm/home(s’ouvre dans une nouvelle fenêtre)) where we progressively make our transcriptions of unpublished manuscripts, as well as translations and commentaries, accessible. Concurrently, we present our work at international conferences and publish articles or monographs on the topics addressed.
Our website also features research we conduct in the field of digital humanities. This includes the development of a text recognition model (on the digital platform e-scriptorium) which is now reliable with a success rate of 90%, the development of a segmentation model for texts containing equations, and the development of tools for the genetic edition of manuscripts (in collaboration with the digital platform EMAN). Finally, we have launched a major initiative to propose to the UNICODE consortium the project for a "Leibniz Font" which will be intended for use by the entire community of historians of mathematics, as it will include a very large number of ancient mathematical symbols (over 200).
The project has played an important role in structuring the community of researchers working on the relationship between mathematics and philosophy in Leibniz, notably through the organization of major international conferences, but more generally by supporting numerous activities dedicated to the study of mathematics in the early modern period. We have been involved in organizing three summer schools for young researchers in the history of mathematics, the philosophy of mathematics, or Leibniz studies.
To date, we have already transcribed and commented on over 400 folios covering the themes announced within the project framework: Dyadica, Ars combinatoria, Foundations of differential calculus and algebra (particularly concerning the question of "reduction to the identicals"). We also contributed to the progress of the upcoming volumes of Leibniz's mathematical writings dedicated to the foundations of geometry and analysis situs, for which we have transcribed about a hundred folios.
We have launched a website (https://eman-archives.org/philiumm/home(s’ouvre dans une nouvelle fenêtre)) where we progressively make our transcriptions of unpublished manuscripts, as well as translations and commentaries, accessible. Concurrently, we present our work at international conferences and publish articles or monographs on the topics addressed.
Our website also features research we conduct in the field of digital humanities. This includes the development of a text recognition model (on the digital platform e-scriptorium) which is now reliable with a success rate of 90%, the development of a segmentation model for texts containing equations, and the development of tools for the genetic edition of manuscripts (in collaboration with the digital platform EMAN). Finally, we have launched a major initiative to propose to the UNICODE consortium the project for a "Leibniz Font" which will be intended for use by the entire community of historians of mathematics, as it will include a very large number of ancient mathematical symbols (over 200).
The project has played an important role in structuring the community of researchers working on the relationship between mathematics and philosophy in Leibniz, notably through the organization of major international conferences, but more generally by supporting numerous activities dedicated to the study of mathematics in the early modern period. We have been involved in organizing three summer schools for young researchers in the history of mathematics, the philosophy of mathematics, or Leibniz studies.
We have achieved entirely new results in interpreting the relationship between philosophy and mathematics in Leibniz's through the exploration of unpublished texts. This includes:
- A better understanding of Leibniz's creation of a binary arithmetic, or Dyadica, through which he anticipated one of the major developments in current information processing via computers.
- A renewed interpretation of what Leibniz means by mathematical statements being "reducible to identicals" and an original parallel with developments in current logic (Homotopy Type Theory).
- A clarification of his understanding of the status of infinitesimals, made possible both by the transcription of texts on the "angle of contact" (previously unknown) and some later unpublished texts that we have transcribed and translated into English.
- A better understanding of the reflections that led Leibniz to create the notion of "transcendent number."
- The discovery of a highly original conception of what he meant by "abstraction" and the fertility of this approach for engaging in contemporary debates on the philosophical question of the application of mathematics.
The tools developed in the field of digital humanities are also far beyond what previously existed in the state of the art. We have developed the first HTR (Handwritten Text Recognition) model for recognizing Leibniz's manuscripts, which is reliable up to 90 percent, as well as the first tools for equation recognition. The tools for genetic editing are also entirely new and open up very fruitful avenues for anyone working on digital edition of manuscript. Finally, the creation of a Leibniz font, currently under review by the UNICODE consortium, constitutes a highly important prospect for the entire community of mathematicians' historians worldwide.
Several monographs are now being written concerning the various topics studied in the project.
- A better understanding of Leibniz's creation of a binary arithmetic, or Dyadica, through which he anticipated one of the major developments in current information processing via computers.
- A renewed interpretation of what Leibniz means by mathematical statements being "reducible to identicals" and an original parallel with developments in current logic (Homotopy Type Theory).
- A clarification of his understanding of the status of infinitesimals, made possible both by the transcription of texts on the "angle of contact" (previously unknown) and some later unpublished texts that we have transcribed and translated into English.
- A better understanding of the reflections that led Leibniz to create the notion of "transcendent number."
- The discovery of a highly original conception of what he meant by "abstraction" and the fertility of this approach for engaging in contemporary debates on the philosophical question of the application of mathematics.
The tools developed in the field of digital humanities are also far beyond what previously existed in the state of the art. We have developed the first HTR (Handwritten Text Recognition) model for recognizing Leibniz's manuscripts, which is reliable up to 90 percent, as well as the first tools for equation recognition. The tools for genetic editing are also entirely new and open up very fruitful avenues for anyone working on digital edition of manuscript. Finally, the creation of a Leibniz font, currently under review by the UNICODE consortium, constitutes a highly important prospect for the entire community of mathematicians' historians worldwide.
Several monographs are now being written concerning the various topics studied in the project.