Modern Geometry and the Concept of Space

Vincenzo de Risi

Source: Oliver Byrne, The first six books of The Elements of Euclid: in which coloured diagrams and symbols are used instead of letters for the greater ease of learners, London: Pickering, 1847

Although a divide between ancient and modern geometry can be framed in different ways, the most useful one may well be the emergence of the consideration of space itself as an object of geometrical investigation.
Greek mathematics understood geometry as a study of straight lines, angles, circles and planes, or in more general terms as a science of figures conceived against an amorphous background space whose definition lies outside the limits of the theory. This understanding was superseded by a conception of space (and spaces, now plural for the first time) itself endowed with geometrical properties. The main concern of this new geometrical science is to characterize the structures and features of geometrical space in axioms and demonstration.
Although it is quite clear that this revolution in geometry helped shape the scientific world such that contemporary mathematics remains incomprehensible without it, the questions of when, why and how this revolution took place are still to some extent obscure.

The earliest record of a geometrical understanding of space itself is probably to be found in some Renaissance treatises on perspective, and then in Pascal’s pioneering studies on the geometry of projections. The first comprehensive and conscious attempt in a new science of space, however, appears in the hundreds of Leibniz’s essays on the analysis situs, in which the philosopher tries to define space and its properties axiomatically, and to link Euclidean axioms to the very definition of space.
A century and a half later, this geometrical understanding of space had become commonplace: non-Euclidean geometries and projective techniques form part of mainstream mathematical research, and a modern conception of space and spatial intuition has helped reshape pure and applied science, philosophy, and even the Visual Arts.
Against this background, the research group aims to reconstruct this 150-year-long mathematical, philosophical, scientific and cultural revolution, from the time the ancient view of geometry as a science of figures and magnitudes began to fracture to the flowering of a full-fledged expression a genuine science of space.

The main body of the research will naturally focus on history of mathematics itself, since this history registers, year by year, the core developments in the field. The very nature of the investigation, however, encourages a more complex and structural approach.
This may include, for example, some of the following: (1) An analysis of 18th-century developments towards a geometry of projections, that might reveal how the idea of a plurality of spatial structures can be traced from the followers of Desargues to Monge and Poncelet. This approach could be fruitfully combined with parallel research into other interconnected fields whose development sheds light on the growth of mathematics: (2) optics and (3) the physiology of perception are obvious candidates, but almost as interesting is (4) the philosophical debate about the theory of vision (in Berkeley or Reid, for instance) or the different space-sensoria in general (e.g. the quarrel about Molyneaux’ s case). A different line of research might draw on (5) the prehistory of non-Euclidean geometries in the 18th century (e.g. Saccheri or Lambert) and (6) the beginnings of philosophic reflection on the topic (Reid, Lambert, and others). It would also be useful to explore (7) the contribution made by the development of mechanics to the concept of space, in the Newton-Leibniz controversy and its consequences, and (8) the effect of this new geometry on physics itself. One might even address (9) Kant’s famous synthesis of these debates in his writings about space and geometry.

The aim of the group will be to bring together scholars in different disciplines to construct a comprehensive picture of the developments that gave rise to contemporary geometry and the modern concept of space.


The Research Group is accepting proposals for non-funded Visiting Fellowships from one to six months. They are normally open to doctoral candidates or post-docs (who discussed their dissertation since no more than five years at the time of the application), who have external funding from the doctoral program they are enrolled in or other international research grants. The Max Planck Institute offers to successful candidates office space and use of the Institute facilities. Visiting Fellows are expected in turn to take part to the cultural and scientific life of the Institute, to advance their own research project, and to actively contribute to the Group researches.

The submitted research project should strictly be related with the Research Group interests in epistemology of mathematics and philosophy of space in the 17th and 18th centuries, or with the history of geometry in general (from Antiquity to the Modern Age).

Candidates are requested to submit a curriculum vitae (including list of publications), the research proposal, a sample writing representative of the candidate’s work (such as a chapter of the doctoral dissertation or a scientific article), an indication about the preferred length and starting date of the fellowship, to Vincenzo De Risi (by surface mail addressed to the Institute, or by e-mail submission). Applications may be submitted in German, English, French, or Italian; candidates are however expected to be able to present their own work and discuss that of others fluently in English.

Please contact Vincenzo De Risi by e-mail for any enquiry.