{"title":"Graphs and lattices in the early versions of Ramon Llull's Art","authors":"A. Bonner, P.J. Planas Mulet, L. Badia","doi":"10.1016/j.hm.2025.02.001","DOIUrl":"10.1016/j.hm.2025.02.001","url":null,"abstract":"<div><div>Even though Ramon Llull (c. 1232–1316) made poor use of explicit mathematical formulations, since the times of G.W. Leibniz his Art of finding and demonstrating the truth has been considered an intellectual tool forerunning varied mathematical deployments, from combinatorics, to voting theory and even to computing sciences. This article shows how Llull used implicitly two mathematical formulations, graph theory and lattice theory, not known to have made their appearance before the 20th century: <em>Ars demonstrativa</em> (c.1283) displays some devices of graph theory and <em>Liber principiorum medicinae</em> (c.1274–1283) develops lattice theory in the context of the University of Montpellier.</div></div>","PeriodicalId":51061,"journal":{"name":"Historia Mathematica","volume":"70 ","pages":"Pages 31-52"},"PeriodicalIF":0.5,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143609591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mathematical imagination in 14th-century natural philosophy. The case of the endless, infinite helix line","authors":"Elżbieta Jung, Robert Podkoński","doi":"10.1016/j.hm.2025.02.005","DOIUrl":"10.1016/j.hm.2025.02.005","url":null,"abstract":"<div><h3>Résumé</h3><div>Dans cet article, nous listons et discutons en détail les exemples géométriques apparaissant dans des textes de philosophie naturelle du XIVe siècle dans le contexte des débats relatifs à l'infini, au continu et à l'existence d'entités indivisibles. Dans la première partie de l'article, nous concentrons sur les propriétés de la spirale 'proportionnelle' (<em>linea girativa</em>) inventée par Richard Kilvington, qui décrit clairement cette ligne comme une entité géométrique infiniment longue. Beaucoup de ses contemporains et d'auteurs scolastiques plus tardifs tentèrent ardemment de rejeter cette conclusion, en dépit de sa cohérence mathématique. Dans la seconde partie de l'article, nous présentons les arguments géométriques employés contre les théories atomistes apparues dans la philosophie naturelle oxfordienne du début du XIVe siècle. Ces arguments géométriques furent utilisés par leurs auteurs en raison de leur plus grande efficacité, comparée aux arguments logiques ou philosophiques, pour réfuter l'atomisme. Sur la base de ces discussions, nous présentons enfin quelques conclusions à propos de l'attitude des penseurs de la fin du Moyen Âge concernant l'utilité des preuves mathématiques dans le contexte de la philosophie naturelle aristotélicienne.</div></div><div><h3>Summary</h3><div>In the present paper we compile and discuss in detail the geometrical examples presented in fourteenth-century natural philosophical texts in the context of discussions on infinity, continuity, and the existence of indivisible entities. In the first part of this article we focus on the controversies over the properties of the “proportional” helix line (<em>linea girativa</em>), invented by Richard Kilvington, who unambiguously described this line as an actually infinitely long geometrical entity. Many of his contemporary as well as later scholastic authors tried fervently to deny that conclusion, in spite of its mathematical consistency. In the second part of the article we present the geometrical arguments used against the atomistic theories appearing in Oxford natural philosophy at the beginning of the fourteenth century. These geometrical arguments were recognized by their authors as “more efficacious” in proving atomism false than logical or philosophical ones. On the basis of these discussions we finally present a few conclusions concerning the attitude of later medieval scholars about the usefulness of mathematical proofs in the context of Aristotelian natural philosophy.</div></div>","PeriodicalId":51061,"journal":{"name":"Historia Mathematica","volume":"70 ","pages":"Pages 53-72"},"PeriodicalIF":0.5,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143609592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
José Gil-Férez , Piotr Błaszczyk , M. Andrew Moshier , Alberto Naibo , Marco Panza , Jean-Michel Salanskis
{"title":"A Euclidean proof for the Fourth Postulate","authors":"José Gil-Férez , Piotr Błaszczyk , M. Andrew Moshier , Alberto Naibo , Marco Panza , Jean-Michel Salanskis","doi":"10.1016/j.hm.2024.12.001","DOIUrl":"10.1016/j.hm.2024.12.001","url":null,"abstract":"<div><div>We discuss some classical conundrums about Euclid's Fourth Postulate. Our inquire sheds lights on the role the postulate is playing within the deductive structure of Book I of the <em>Elements</em> and provides a proof of it fully admissible within Euclid's original setting.</div></div>","PeriodicalId":51061,"journal":{"name":"Historia Mathematica","volume":"70 ","pages":"Pages 10-30"},"PeriodicalIF":0.5,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143609590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fabricius's theory for Mars: The model that shocked Kepler","authors":"Christián C. Carman","doi":"10.1016/j.hm.2024.05.002","DOIUrl":"10.1016/j.hm.2024.05.002","url":null,"abstract":"<div><div>David Fabricius is recognized in the history of astronomy for his role in the discoveries of sunspots and the first variable star. As the around 50 letters between them show, he also played a significant role as Kepler's interlocutor when the latter was writing <em>Astronomia Nova</em>. One year before the publication of <em>Astronomia Nova</em>, Fabricius shared with Kepler a new model for Mars that he had developed. The model shocked Kepler because, using the traditional tools of circles and uniform motion, Fabricius had been able to find a model as accurate as Kepler's model involving his first two laws. The official story, based on the reconstruction of Fabricius's model that Apelt made in 1852, asserts that Kepler need not worry because, in fact, he misinterpreted the model. The true model that Fabricius proposed was not predictively accurate. In this paper I offer a new interpretation of Fabricius's model that shows that Kepler did indeed have reason to worry since Fabricius's model was extraordinarily accurate.</div></div>","PeriodicalId":51061,"journal":{"name":"Historia Mathematica","volume":"69 ","pages":"Pages 1-21"},"PeriodicalIF":0.5,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Negatives as fictions in 16th and 17th century mathematics","authors":"David Rabouin","doi":"10.1016/j.hm.2024.09.004","DOIUrl":"10.1016/j.hm.2024.09.004","url":null,"abstract":"<div><div>The use of “negative” entities is a fact attested in various mathematical practices across various contexts since antiquity. Although we now have a better knowledge of this history in its complexity and its richness, many phenomena occurring in it are still blurred and oversimplified by the retrospective look we tend to posit on it under the heading of the history of negative <em>numbers</em>. In the first section, I detail several historical studies relying on a specific kind of conceptualization in terms of “domain extension” and point to several of their shortcomings. In the second section, I propose to turn to the actor's categories and put particular emphasis on the fact that negative entities were conceived as “fictitious” entities. In the third section, I show how this context allows a different reading of the various discussions and reflections on negatives occurring in the second half of the 17<sup>th</sup> century.</div></div>","PeriodicalId":51061,"journal":{"name":"Historia Mathematica","volume":"69 ","pages":"Pages 41-61"},"PeriodicalIF":0.5,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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