{"id":527,"date":"2023-08-18T06:38:07","date_gmt":"2023-08-18T06:38:07","guid":{"rendered":"https:\/\/cch.um6p.ma\/?page_id=527"},"modified":"2023-10-14T20:28:39","modified_gmt":"2023-10-14T20:28:39","slug":"gregory-chaitin-complexity-in-metamathematics-friend-or-foe","status":"publish","type":"page","link":"https:\/\/cch.um6p.ma\/?page_id=527","title":{"rendered":"Gregory Chaitin, Complexity in Metamathematics, Friend or Foe?\u00a0"},"content":{"rendered":"<p>UM6P Science Week, 21 February 2023, 1h26mn<strong><br \/>\n<\/strong>Talk followed by a roundtable with Stephen Wolfram, Herv\u00e9 Zwirn and David Chavalarias<br \/>\nModerator: Fouad Laroui<\/p>\n<p>Introductory texts (English, French and Arabic) by Hind Aboulazm<br \/>\nDrawing by Sophie Lenormand<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/fBmg_iR2394\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">In his talk, Gregory Chaitin delves into the intricate realm of metamathematics, contemplating its role as both a companion and adversary to the world of mathematics. Metamathematics, an endeavor aimed at introspectively scrutinizing mathematics through mathematical means, traces its lineage back to Leibniz, its grandparent, and its parental figure, David Hilbert. At its core, this discipline relies upon formal axiomatic systems <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">frozen versions of mathematics viewed from a vantage point above. Such systems serve as the crucible for reasoning and mathematical proof, rigorously delineated to exclude any subjective elements through the apparatus of symbolic logic. Chaitin sketches the historical underpinnings of metamathematics, with Hilbert&#8217;s assertion that mathematics is inherently binary <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">black or white <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">when executed within a formal axiomatic system. This approach necessitates defining and formalizing the system&#8217;s structure, exemplified by Alfred Tarski&#8217;s work on geometry. The Peano arithmetic system and the Zermelo-Fraenkel set theory stand as emblematic formal axiomatic frameworks.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2191 aligncenter\" src=\"https:\/\/cch.um6p.ma\/wp-content\/uploads\/2023\/09\/Programme-elegant-_-Sophie-Le-Normand.jpg\" alt=\"\" width=\"471\" height=\"508\" srcset=\"https:\/\/cch.um6p.ma\/wp-content\/uploads\/2023\/09\/Programme-elegant-_-Sophie-Le-Normand.jpg 773w, https:\/\/cch.um6p.ma\/wp-content\/uploads\/2023\/09\/Programme-elegant-_-Sophie-Le-Normand-278x300.jpg 278w, https:\/\/cch.um6p.ma\/wp-content\/uploads\/2023\/09\/Programme-elegant-_-Sophie-Le-Normand-768x829.jpg 768w, https:\/\/cch.um6p.ma\/wp-content\/uploads\/2023\/09\/Programme-elegant-_-Sophie-Le-Normand-139x150.jpg 139w\" sizes=\"auto, (max-width: 471px) 100vw, 471px\" \/>Intriguingly, Chaitin introduces the concept of an \u201celegant program\u201d <\/span><span style=\"font-weight: 400;\">\u2014&nbsp; <\/span><span style=\"font-weight: 400;\">minimalistic code that yields precise output <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">providing a gateway to the incompleteness theorem. He expounds on the challenges in proving an elegant program, accentuating the limitations of formal axiomatic systems in achieving such a task. The notion of incompleteness, which perturbs the ambition of a universal mathematical framework, becomes a recurrent motif throughout the presentation. Chaitin&#8217;s intellectual trajectory culminates in his assertion that any formal axiomatic system <\/span><span style=\"font-weight: 400;\">\u2014&nbsp; <\/span><span style=\"font-weight: 400;\">no matter its intricacy <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">can only grasp an infinitesimal fraction of the boundless intricacies intrinsic to the Platonic realm of pure mathematics. He underscores the necessity of expanding axiomatic horizons and embracing new paradigms to engage with the enigma of undecidability and foster further mathematical progress. While complex and often shrouded in philosophical implications, this endeavor offers a richer perspective into the evolving nature of mathematics and its interactions with the human intellect.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">The discussion with Stephen Wolfram, Herv\u00e9 Zwirn and David Chavalarias that followed the presentation witnessed a synthesis of viewpoints from various luminaries, oscillating between optimism and caution regarding the future of mathematics. Contemplations ranged from the compatibility of mathematics with science despite the specter of incompleteness to the intrinsic value of computational irreducibility in the fabric of existence. Chaitin&#8217;s exposition ultimately illuminated the profound interplay between mathematics, cognition, and the boundless tapestry of human inquiry.<\/span><\/p>\n<p style=\"text-align: right;\">Hind Aboulazm<\/p>\n<p style=\"text-align: justify;\"><b>Gregory Chaitin<\/b><span style=\"font-weight: 400;\">, born in 1947, is an Argentine-American mathematician and computer scientist. From the late 1960s onward, he has made significant contributions to algorithmic information theory and metamathematics, including a notable result in computer theory analogous to G\u00f6del&#8217;s incompleteness theorem. Chaitin, along with Andrei Kolmogorov and Ray Solomonoff, is viewed as one of the pioneers of the algorithmic complexity field, often referred to as Solomonoff-Kolmogorov-Chaitin complexity or program-size complexity. The emergence of algorithmic information theory as a core aspect of theoretical computer science, information theory, and mathematical logic owes much to the contributions of figures like Solomonoff, Kolmogorov, Martin-L\u00f6f, and Leonid Levin. This subject is frequently taught in computer science courses. Beyond just computer scientists, Chaitin&#8217;s research has garnered the interest of numerous philosophers and mathematicians, leading them to explore profound questions in mathematical innovation and digital philosophy. <\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gregory Chaitin <\/span>is author of <em>Information, Randomness &amp; Incompleteness<\/em> (1987) <em>Algorithmic Information Theory<\/em> (1987) <em>The Limits of Mathematics<\/em> (1998), <em>The Unknowable<\/em> (1999), <em>Exploring Randomness<\/em> (2001), <em>Meta Math!: The Quest for Omega<\/em> (2005), <em>Mathematics, Complexity and Philosophy<\/em> (2011), <em>G\u00f6del&#8217;s Way<\/em> (2012) and <em>Proving Darwin: Making Biology Mathematical<\/em> (2012).<\/p>\n<p style=\"text-align: center;\">_________________________________<\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gregory Chaitin a explor\u00e9 profond\u00e9ment les m\u00e9tamath\u00e9matiques, les consid\u00e9rant \u00e0 la fois comme alli\u00e9es et rivales des math\u00e9matiques traditionnelles. Les m\u00e9tamath\u00e9matiques, qui scrutent les math\u00e9matiques \u00e0 travers un prisme math\u00e9matique, trouvent leurs racines chez Leibniz, leur pr\u00e9curseur, et David Hilbert, leur principal repr\u00e9sentant. Elles s&#8217;appuient principalement sur des syst\u00e8mes axiomatiques formels, qui sont des repr\u00e9sentations \u00e9lev\u00e9es et structur\u00e9es des math\u00e9matiques. Ces syst\u00e8mes sont les fondations du raisonnement math\u00e9matique, strictement encadr\u00e9s pour \u00e9liminer toute subjectivit\u00e9 gr\u00e2ce \u00e0 la logique symbolique.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Chaitin a trac\u00e9 l&#8217;histoire des m\u00e9tamath\u00e9matiques, citant Hilbert qui voyait les math\u00e9matiques comme \u00e9tant fondamentalement dichotomiques <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">&nbsp;soit vraies, soit fausses <\/span><span style=\"font-weight: 400;\">\u2014 <\/span><span style=\"font-weight: 400;\">&nbsp;lorsqu&#8217;elles sont pratiqu\u00e9es dans un cadre axiomatique formel. Cela n\u00e9cessite une structuration et une formalisation pr\u00e9cises, comme le montrent les contributions d\u2019Alfred&nbsp; Tarski en g\u00e9om\u00e9trie. Le syst\u00e8me arithm\u00e9tique de Peano et la th\u00e9orie des ensembles de Zermelo-Fraenkel sont des exemples notables de tels cadres.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Chaitin a pr\u00e9sent\u00e9 l&#8217;id\u00e9e du &#8220;programme \u00e9l\u00e9gant&#8221;, un code \u00e9pur\u00e9 g\u00e9n\u00e9rant des r\u00e9sultats d\u00e9termin\u00e9s, le reliant au th\u00e9or\u00e8me d&#8217;incompl\u00e9tude. Il a mis en \u00e9vidence les d\u00e9fis de prouver un tel programme, montrant les limites des syst\u00e8mes axiomatiques formels pour cette mission. L&#8217;incompl\u00e9tude, qui remet en question l&#8217;id\u00e9e d&#8217;un syst\u00e8me math\u00e9matique universel, reste un th\u00e8me r\u00e9current dans son argumentation.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Chaitin a conclu en affirmant que tout syst\u00e8me axiomatique, aussi \u00e9labor\u00e9 soit-il, ne peut appr\u00e9hender qu&#8217;une minuscule partie des immenses complexit\u00e9s des math\u00e9matiques pures platoniciennes. Il a insist\u00e9 sur la n\u00e9cessit\u00e9 de repenser les axiomes et d&#8217;embrasser de nouveaux paradigmes pour aborder le myst\u00e8re de l&#8217;ind\u00e9cidabilit\u00e9 et stimuler l&#8217;avancement math\u00e9matique. Bien que ce sujet soit dense et empreint de r\u00e9flexions philosophiques, il offre une vision \u00e9largie de l&#8217;\u00e9volution des math\u00e9matiques et de leur relation \u00e0 la pens\u00e9e humaine.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">La discussion post-pr\u00e9sentation poursuivie avec Stephen Wolfram, Herv\u00e9 Zwirn et David Chavalarias a permis de fusionner les opinions de diff\u00e9rents experts, oscillant entre espoir et r\u00e9serve sur le futur des math\u00e9matiques. Les r\u00e9flexions ont vari\u00e9 de la synergie entre math\u00e9matiques et science malgr\u00e9 l&#8217;incompl\u00e9tude, \u00e0 l&#8217;importance de l&#8217;incompressibilit\u00e9 informatique dans l&#8217;essence de la r\u00e9alit\u00e9. L&#8217;intervention de Chaitin a brillamment soulign\u00e9 le lien entre les math\u00e9matiques, la pens\u00e9e et l&#8217;infini voyage de la d\u00e9couverte humaine.<\/span><\/p>\n<p style=\"text-align: right;\">H.A<b><br \/>\n<\/b><\/p>\n<p style=\"text-align: justify;\"><b>Gregory Chaitin<\/b><span style=\"font-weight: 400;\">, n\u00e9 en 1947, est un math\u00e9maticien et informaticien argentin-am\u00e9ricain. Depuis la fin des ann\u00e9es 1960, il a apport\u00e9 d&#8217;importantes contributions \u00e0 la th\u00e9orie de l&#8217;information algorithmique et \u00e0 la m\u00e9tamath\u00e9matique, notamment un r\u00e9sultat notable en th\u00e9orie informatique analogue au th\u00e9or\u00e8me d&#8217;incompl\u00e9tude de G\u00f6del. Chaitin, avec Andrei Kolmogorov et Ray Solomonoff, est consid\u00e9r\u00e9 comme l&#8217;un des pionniers du domaine de la complexit\u00e9 algorithmique, souvent appel\u00e9e complexit\u00e9 de Solomonoff-Kolmogorov-Chaitin, ou complexit\u00e9 de taille de programme. L&#8217;\u00e9mergence de la th\u00e9orie de l&#8217;information algorithmique comme aspect central de l&#8217;informatique th\u00e9orique, de la th\u00e9orie de l&#8217;information et de la logique math\u00e9matique doit beaucoup aux contributions de figures comme Solomonoff, Kolmogorov, Martin-L\u00f6f et Leonid Levin. Au-del\u00e0 du domaine informatique, les recherches de Gregory Chaitin ont suscit\u00e9 l\u2019int\u00e9r\u00eat de nombreux philosophes et math\u00e9maticiens, les incitant \u00e0 explorer des questions profondes d\u2019innovation math\u00e9matique et de philosophie digitale. <\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gregory Chaitin est l&#8217;auteur de <em>Information, Randomness &amp; Incompleteness<\/em> (1987) <em>Algorithmic Information Theory<\/em> (1987) <em>The Limits of Mathematics<\/em> (1998), <em>The Unknowable<\/em> (1999), <em>Exploring Randomness<\/em> (2001), <em>Meta Math!: The Quest for Omega<\/em> (2005), <em>Mathematics, Complexity and Philosophy<\/em> (2011), <em>G\u00f6del&#8217;s Way<\/em> (2012) et de <em>Proving Darwin: Making Biology Mathematical<\/em> (2012).<\/span><\/p>\n<p style=\"text-align: center;\">_________________________________<\/p>\n<div dir=\"rtl\" style=\"text-align: right;\"><span lang=\"AR-SA\"><span 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\u062f\u0648\u0631\u0627\u062a \u0639\u0644\u0648\u0645 \u0627\u0644\u062d\u0627\u0633\u0648\u0628. \u0625\u0644\u0649 \u062c\u0627\u0646\u0628 \u0639\u0644\u0645\u0627\u0621 \u0627\u0644\u062d\u0627\u0633\u0648\u0628 \u0641\u0642\u0637\u060c \u0623\u062b\u0627\u0631\u062a \u0623\u0628\u062d\u0627\u062b \u0634\u0627\u064a\u062a\u064a\u0646 \u0627\u0647\u062a\u0645\u0627\u0645 \u0627\u0644\u0639\u062f\u064a\u062f \u0645\u0646 \u0627\u0644\u0641\u0644\u0627\u0633\u0641\u0629 \u0648\u0639\u0644\u0645\u0627\u0621 \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a\u060c \u0645\u0645\u0627 \u062f\u0641\u0639\u0647\u0645 \u0625\u0644\u0649 \u0627\u0633\u062a\u0643\u0634\u0627\u0641 \u0623\u0633\u0626\u0644\u0629 \u0639\u0645\u064a\u0642\u0629 \u0641\u064a \u0627\u0644\u0627\u0628\u062a\u0643\u0627\u0631 \u0627\u0644\u0631\u064a\u0627\u0636\u064a \u0648\u0627\u0644\u0641\u0644\u0633\u0641\u0629 \u0627\u0644\u0631\u0642\u0645\u064a\u0629.<\/span><\/span><\/div>\n<div style=\"text-align: right;\">&nbsp;<\/div>\n<div style=\"text-align: right;\">Source: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Gregory_Chaitin\">https:\/\/en.wikipedia.org\/wiki\/Gregory_Chaitin<\/a><strong><br \/>\n<\/strong><\/div>\n","protected":false},"excerpt":{"rendered":"<p>UM6P Science Week, 21 February 2023, 1h26mn Talk followed by a roundtable with Stephen Wolfram, Herv\u00e9 Zwirn and David Chavalarias Moderator: Fouad Laroui Introductory texts (English, French and Arabic) by Hind Aboulazm Drawing by Sophie Lenormand In his talk, Gregory Chaitin delves into the intricate realm of metamathematics, contemplating its\u2026 <a class=\"continue-reading-link\" href=\"https:\/\/cch.um6p.ma\/?page_id=527\">Lire plus \/ Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":871,"parent":1897,"menu_order":16,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"ngg_post_thumbnail":0,"footnotes":""},"categories":[81,34],"tags":[29,68],"class_list":["post-527","page","type-page","status-publish","has-post-thumbnail","hentry","category-complexus","category-science-week","tag-gregory-chaitin","tag-metamathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/527","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=527"}],"version-history":[{"count":20,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/527\/revisions"}],"predecessor-version":[{"id":2658,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/527\/revisions\/2658"}],"up":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/1897"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/media\/871"}],"wp:attachment":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=527"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=527"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=527"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}