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**Geometry and monadology. Leibniz’s analysis situs and philosophy of space.**
*(English)*
Zbl 1180.51002

Science Networks. Historical Studies 33. Basel: Birkhäuser (ISBN 978-3-7643-7985-8/hbk). xx, 658 p. (2007).

The book under review is based on a doctoral dissertation defended at the Scuola Normale Superiore in Pisa, Italy. It deals with G. W. Leibniz’s lifelong project of founding a new geometrical science, the analysis situs. The author supports the opinion that the study of Leibniz’s mathematics is helpful for understanding Leibniz’s philosophy. But whereas infinitesimal analysis appears to be not more than a heuristic instrument, his geometrical studies “become indispensable metaphysical instruments for objectively determining space. By his analysis situs, Leibniz argues he can demonstrate the continuity of space, its tridimensionality, the possibility of rigid motion in it, its Euclidean nature, or its absolute necessity” (p.xii).

The work is divided into four chapters. Chapter 1 deals with the possible influences on Leibniz and the main steps in the development of his new geometrical discipline. Chapter 2 gives a reconstruction of Leibniz’s main results in modern terminology. Chapter 3 presents situational analysis as a requisite for Leibniz’s theory of expressions forming the heart of his late monadological metaphysics. Chapter 4, finally, attempts to give a more concrete foundation of Leibniz’s theory of extension. The volume is closed by a selection of Leibniz’s writings relevant for situational analysis.

The first chapter gives the historical background of the analysis situs. It is argued for a rebirth of geometry during the 17th century scientific revolution. This rebirth can be understood, and Leibniz did it this way, as a reform and improvement of ancient mathematics. The author identifies different stages of development in Leibniz’s thoughts on geometry, from early studies in the 1670s, a more definitional period in the 1680s up to mature thoughts until his death. The chapter is closed with a tentative chronology of Leibniz’s last writings on analysis situs, and a list of Leibniz’s essays and letters relevant for the topic.

The second chapter deals in detail what Leibniz calls “analysis situs”, an analysis of situation, obviously regarding “situs” as a primitive notion. It cannot be defined directly. There is no absolute situation of a unique object, but the relative situation of a set of objects (p.133). Basic concepts in the situational analysis are congruence, coincidence and superposition. Of special interest is the relation between situation, space and point. In the course of the conceptual development, Leibniz achieved the understanding of a point as an absolute situation which “is in fact one of the greatest achievements of the analysis of situation, and it is the one that actually permits us to apply situational relations to geometric loci and space in general” (p.167). Further topics are the straight line and the metric structure of space. In an addition “On extension of situation” (pp.265–295), the application of analysis situs to domains outside geometry is discussed, in particular the application to intensive quantities, time, congruent and incongruent counterparts, and number.

Chapter 3 “Phenomenology” relates the analysis situs to Leibniz’s metaphysical system. The author intends to show “whether from Leibniz’s definition of a monad as the substance whose essential activity is percipere, one can (or cannot) derive the transcendental need for representing phenomena in space, and therefore space itself” (p.300). Among the concepts discussed are the potential and the actual infinity, the concept of God, indiscernibles, and continuity. In an addition “The algebra of expression” the author gives a formulation of Leibniz’s theory of expression by expounding the correspondence between phenomena and noumena with first sketchy steps to model this theory mathematically. He uses two domains to express the relation of expression. The first is the set of all monads or the set of all monadic properties, the second the set of all representations of a monad.

In Chapter 4 the author presents a study of the “determination of real space, and the deduction of the variegated, actually experienced phenomenal world” (p.437). The notion of existence is added to the concepts of Leibniz’s theory of knowledge and compared with Kant’s conception. Among the further topics discussed are real space and corporal substances, leading to considerations on ideal space, having the properties of real space that can be deduced a priori (p.551).

The volume closes with an edition of 21 of Leibniz’s manuscripts on the analysis situs, all but one published here for the first time.

In sum the author has provided a great survey of Leibniz’s writings on geometry presented with deep analyses in several fields of mathematics and metaphysics, providing a vivid impression of the interconnectedness of Leibniz’s general system of science.

The work is divided into four chapters. Chapter 1 deals with the possible influences on Leibniz and the main steps in the development of his new geometrical discipline. Chapter 2 gives a reconstruction of Leibniz’s main results in modern terminology. Chapter 3 presents situational analysis as a requisite for Leibniz’s theory of expressions forming the heart of his late monadological metaphysics. Chapter 4, finally, attempts to give a more concrete foundation of Leibniz’s theory of extension. The volume is closed by a selection of Leibniz’s writings relevant for situational analysis.

The first chapter gives the historical background of the analysis situs. It is argued for a rebirth of geometry during the 17th century scientific revolution. This rebirth can be understood, and Leibniz did it this way, as a reform and improvement of ancient mathematics. The author identifies different stages of development in Leibniz’s thoughts on geometry, from early studies in the 1670s, a more definitional period in the 1680s up to mature thoughts until his death. The chapter is closed with a tentative chronology of Leibniz’s last writings on analysis situs, and a list of Leibniz’s essays and letters relevant for the topic.

The second chapter deals in detail what Leibniz calls “analysis situs”, an analysis of situation, obviously regarding “situs” as a primitive notion. It cannot be defined directly. There is no absolute situation of a unique object, but the relative situation of a set of objects (p.133). Basic concepts in the situational analysis are congruence, coincidence and superposition. Of special interest is the relation between situation, space and point. In the course of the conceptual development, Leibniz achieved the understanding of a point as an absolute situation which “is in fact one of the greatest achievements of the analysis of situation, and it is the one that actually permits us to apply situational relations to geometric loci and space in general” (p.167). Further topics are the straight line and the metric structure of space. In an addition “On extension of situation” (pp.265–295), the application of analysis situs to domains outside geometry is discussed, in particular the application to intensive quantities, time, congruent and incongruent counterparts, and number.

Chapter 3 “Phenomenology” relates the analysis situs to Leibniz’s metaphysical system. The author intends to show “whether from Leibniz’s definition of a monad as the substance whose essential activity is percipere, one can (or cannot) derive the transcendental need for representing phenomena in space, and therefore space itself” (p.300). Among the concepts discussed are the potential and the actual infinity, the concept of God, indiscernibles, and continuity. In an addition “The algebra of expression” the author gives a formulation of Leibniz’s theory of expression by expounding the correspondence between phenomena and noumena with first sketchy steps to model this theory mathematically. He uses two domains to express the relation of expression. The first is the set of all monads or the set of all monadic properties, the second the set of all representations of a monad.

In Chapter 4 the author presents a study of the “determination of real space, and the deduction of the variegated, actually experienced phenomenal world” (p.437). The notion of existence is added to the concepts of Leibniz’s theory of knowledge and compared with Kant’s conception. Among the further topics discussed are real space and corporal substances, leading to considerations on ideal space, having the properties of real space that can be deduced a priori (p.551).

The volume closes with an edition of 21 of Leibniz’s manuscripts on the analysis situs, all but one published here for the first time.

In sum the author has provided a great survey of Leibniz’s writings on geometry presented with deep analyses in several fields of mathematics and metaphysics, providing a vivid impression of the interconnectedness of Leibniz’s general system of science.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

51-03 | History of geometry |

00A30 | Philosophy of mathematics |

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A45 | History of mathematics in the 17th century |

01A50 | History of mathematics in the 18th century |