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at the centre of a great deal of attention for three centuries, almost half of his mathematical production was still unpublished when our project began.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 Soon after his death, scholars felt compelled to dive into the ocean of his unpublished manuscripts, but mathematics was not well served in this editorial enterprise.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 As a result, about half of Leibniz’ mathematical manuscripts were still unpublished in 2021 – a dramatic situation which has no equivalent for other great thinkers from that period.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Although it would be an exaggeration to say that all the content of the unpublished manuscripts is original, preliminary exploration of the Nachlass have indicated that they provide the seeds for a deep and fruitful reassessment of Leibniz’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.
