Article ID: CBB000773900

La réception des Vorlesungen über neuere Geometrie de Pasch par Peano (2006)

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The Reception by Peano of Pasch's Vorlesungen über neuere Geometrie Peano wrote the Calcolo geometrico in 1888. One year later, in 1889, he published I principii di geometria , in which he developed, in Pasch's wake, an axiomatisation of geometry. What is the relationship between this work and the previous elaboration of a geometrical calculus ? In this article, we outline the deep difference between the two methods : although the construction of a geometrical algebra aimed at showing the specificity of spatial magnitudes, the axiomatisation does not refer to the ontological distinction between geometrical and numerical entities. However, we show that the way Peano read Pasch's Vorlesungen depended on his previous involvement in the Grasmannian tradition : for him, the segment AB does not refer (as it did for Pasch) to an observable object, rather, AB designates the result of a new geometrical product. In the end, the situation is quite complicated : on the one hand, the algebra and the axiomatic are grounded, in Peano's thought, on completely opposite conceptions of geometrical objects ; on the other, the axiomatic method, as it is developed in I principii , results directly from an interpretation of Pasch entirely based on the Calcolo .

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Authors & Contributors
Luciano, Erika
Contro, Walter S.
Sørensen, Henrik Kragh
Guitart, René
Barbin, Évelyne
Nabonnand, Philippe
Journals
Archive for History of Exact Sciences
Revue d'Histoire des Mathématiques
Centaurus: International Magazine of the History of Mathematics, Science, and Technology
Bollettino di Storia delle Scienze Matematiche
Historia Mathematica
Intellectual History Review
Publishers
Leo S. Olschki
American Mathematical Society
Mathematical Association of America
Oxford University Press
Raffaello Cortina Editore
Edizioni Cadmo
Concepts
Mathematics
Geometry
Algebra
Logic
Calculus
Correspondence and corresponding
People
Peano, Giuseppe
Pasch, Moritz
Euclid
Olivier, Louis
Staudt, Karl Georg Christian von
Schwarz, Hermann Amandus
Time Periods
19th century
20th century, early
Medieval
18th century
20th century
Places
Europe
France
Great Britain
Germany
Mediterranean region
North America
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