From Magnitudes to Geometry and Back: De Zolt's Postulate

Autores
Giovannini, Eduardo Nicolás; Lassalle-Casanave, Abel
Año de publicación
2022
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon.” This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes “the whole is greater than the part.” On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.
Fil: Giovannini, Eduardo Nicolás. Universidad de Viena; Austria. Universidad Nacional del Litoral. Instituto de Humanidades y Ciencias Sociales del Litoral. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Humanidades y Ciencias Sociales del Litoral; Argentina
Fil: Lassalle-Casanave, Abel. Universidade Federal da Bahia; Brasil
Materia
DE ZOLT'S POSTULATE
EUCLIDEAN GEOMETRY
GENERAL MAGNITUDES
HILBERT
LOGICAL ANALYSIS
PLANE AREA
PURITY OF METHOD
WHOLE AND PARTS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/203766

id CONICETDig_65f82a7f1d38be0cba3178bcbb90c0b5
oai_identifier_str oai:ri.conicet.gov.ar:11336/203766
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling From Magnitudes to Geometry and Back: De Zolt's PostulateGiovannini, Eduardo NicolásLassalle-Casanave, AbelDE ZOLT'S POSTULATEEUCLIDEAN GEOMETRYGENERAL MAGNITUDESHILBERTLOGICAL ANALYSISPLANE AREAPURITY OF METHODWHOLE AND PARTShttps://purl.org/becyt/ford/6.3https://purl.org/becyt/ford/6A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon.” This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes “the whole is greater than the part.” On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.Fil: Giovannini, Eduardo Nicolás. Universidad de Viena; Austria. Universidad Nacional del Litoral. Instituto de Humanidades y Ciencias Sociales del Litoral. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Humanidades y Ciencias Sociales del Litoral; ArgentinaFil: Lassalle-Casanave, Abel. Universidade Federal da Bahia; BrasilWiley2022-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/203766Giovannini, Eduardo Nicolás; Lassalle-Casanave, Abel; From Magnitudes to Geometry and Back: De Zolt's Postulate; Wiley; Theoria; 88; 3; 6-2022; 629-6521755-2567CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1111/theo.12385info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:01:07Zoai:ri.conicet.gov.ar:11336/203766instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:01:08.209CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv From Magnitudes to Geometry and Back: De Zolt's Postulate
title From Magnitudes to Geometry and Back: De Zolt's Postulate
spellingShingle From Magnitudes to Geometry and Back: De Zolt's Postulate
Giovannini, Eduardo Nicolás
DE ZOLT'S POSTULATE
EUCLIDEAN GEOMETRY
GENERAL MAGNITUDES
HILBERT
LOGICAL ANALYSIS
PLANE AREA
PURITY OF METHOD
WHOLE AND PARTS
title_short From Magnitudes to Geometry and Back: De Zolt's Postulate
title_full From Magnitudes to Geometry and Back: De Zolt's Postulate
title_fullStr From Magnitudes to Geometry and Back: De Zolt's Postulate
title_full_unstemmed From Magnitudes to Geometry and Back: De Zolt's Postulate
title_sort From Magnitudes to Geometry and Back: De Zolt's Postulate
dc.creator.none.fl_str_mv Giovannini, Eduardo Nicolás
Lassalle-Casanave, Abel
author Giovannini, Eduardo Nicolás
author_facet Giovannini, Eduardo Nicolás
Lassalle-Casanave, Abel
author_role author
author2 Lassalle-Casanave, Abel
author2_role author
dc.subject.none.fl_str_mv DE ZOLT'S POSTULATE
EUCLIDEAN GEOMETRY
GENERAL MAGNITUDES
HILBERT
LOGICAL ANALYSIS
PLANE AREA
PURITY OF METHOD
WHOLE AND PARTS
topic DE ZOLT'S POSTULATE
EUCLIDEAN GEOMETRY
GENERAL MAGNITUDES
HILBERT
LOGICAL ANALYSIS
PLANE AREA
PURITY OF METHOD
WHOLE AND PARTS
purl_subject.fl_str_mv https://purl.org/becyt/ford/6.3
https://purl.org/becyt/ford/6
dc.description.none.fl_txt_mv A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon.” This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes “the whole is greater than the part.” On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.
Fil: Giovannini, Eduardo Nicolás. Universidad de Viena; Austria. Universidad Nacional del Litoral. Instituto de Humanidades y Ciencias Sociales del Litoral. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Humanidades y Ciencias Sociales del Litoral; Argentina
Fil: Lassalle-Casanave, Abel. Universidade Federal da Bahia; Brasil
description A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon.” This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes “the whole is greater than the part.” On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.
publishDate 2022
dc.date.none.fl_str_mv 2022-06
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/203766
Giovannini, Eduardo Nicolás; Lassalle-Casanave, Abel; From Magnitudes to Geometry and Back: De Zolt's Postulate; Wiley; Theoria; 88; 3; 6-2022; 629-652
1755-2567
CONICET Digital
CONICET
url http://hdl.handle.net/11336/203766
identifier_str_mv Giovannini, Eduardo Nicolás; Lassalle-Casanave, Abel; From Magnitudes to Geometry and Back: De Zolt's Postulate; Wiley; Theoria; 88; 3; 6-2022; 629-652
1755-2567
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1111/theo.12385
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Wiley
publisher.none.fl_str_mv Wiley
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1844613801864855552
score 13.070432