{"id":4955,"date":"2015-08-10T17:37:42","date_gmt":"2015-08-10T15:37:42","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=4955"},"modified":"2015-08-10T17:41:03","modified_gmt":"2015-08-10T15:41:03","slug":"resena-parallel-postulates-and-decidability-of-intersection-of-lines-a-mechanized-study-within-tarskis-system-of-geometry","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/resena-parallel-postulates-and-decidability-of-intersection-of-lines-a-mechanized-study-within-tarskis-system-of-geometry\/","title":{"rendered":"Rese\u00f1a: Parallel postulates and decidability of intersection of lines: a mechanized study within Tarski\u2019s system of geometry"},"content":{"rendered":"<p>Se ha publicado un art\u00edculo de razonamiento formalizado en <a href=\"http:\/\/coq.inria.fr\/\">Coq<\/a> sobre geometr\u00eda titulado <a href=\"http:\/\/bit.ly\/1JnGKVb\">Parallel postulates and decidability of intersection of lines: a mechanized study within Tarski\u2019s system of geometry<\/a>.<\/p>\n<p>Sus autores son Pierre Boutry, <a href=\"http:\/\/dpt-info.u-strasbg.fr\/~narboux\/\">Julien Narboux<\/a> y <a href=\"https:\/\/sites.google.com\/site\/pascalschreck\/\">Pascal Schreck<\/a> (del <a href=\"http:\/\/bit.ly\/1hsWEmk\">\u00c9quipe Informatique G\u00e9om\u00e9trique et Graphique<\/a> en la Universidad de Estrasburgo, Francia)<\/p>\n<p>Su resumen es<\/p>\n<blockquote><p>\n  In this paper we focus on the formalization of the proof of equivalence between different versions of Euclid&#8217;s 5 th postulate. This postulate is of historical importance because for centuries many mathematicians believed that this statement was rather a theorem which could be derived from the first four of Euclid&#8217;s postulates and history is rich of incorrect proofs of Euclid&#8217;s 5 th postulate. These proofs are incorrect because they assume more or less implicitly a statement which is equivalent to Euclid&#8217;s 5 th postulate and whose validity is taken for granted. Even though these proofs are incorrect the attempt was not pointless because the flawed proof can be turned into a proof that the unjustified statement implies the parallel postulate. In this paper we provide formal proofs verified using the Coq proof assistant that 10 different statements are equivalent to Euclid&#8217;s 5 th postulate. We work in the context of Tarski&#8217;s neutral geometry without continuity nor Archimedes&#8217; axiom. The formalization provide a clarification of the hypotheses used for the proofs. Following Beeson, we study the impact of the choice of a particular version of the parallel postulate on the decidability issues.\n<\/p><\/blockquote>\n<p>El c\u00f3digo de las correspondientes teor\u00edas en Coq se encuentra <a href=\"http:\/\/geocoq.github.io\/GeoCoq\">aqu\u00ed<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Se ha publicado un art\u00edculo de razonamiento formalizado en Coq sobre geometr\u00eda titulado Parallel postulates and decidability of intersection of lines: a mechanized study within Tarski\u2019s system of geometry. Sus autores son Pierre Boutry, Julien Narboux y Pascal Schreck (del \u00c9quipe Informatique G\u00e9om\u00e9trique et Graphique en la Universidad de Estrasburgo, Francia) Su resumen es In&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[100],"tags":[45,285],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/4955"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=4955"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/4955\/revisions"}],"predecessor-version":[{"id":4959,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/4955\/revisions\/4959"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=4955"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=4955"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=4955"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}