{"id":202,"date":"2015-05-04T09:02:19","date_gmt":"2015-05-04T09:02:19","guid":{"rendered":"http:\/\/sicv.activearchives.org\/logbook\/?p=202"},"modified":"2015-05-06T08:12:31","modified_gmt":"2015-05-06T08:12:31","slug":"unpredictable-enough","status":"publish","type":"post","link":"https:\/\/sicv.activearchives.org\/logbook\/unpredictable-enough\/","title":{"rendered":"Unpredictable enough"},"content":{"rendered":"<p><a href=\"http:\/\/sicv.activearchives.org\/logbook\/wp-content\/uploads\/2015\/05\/1925_kurt_g\u00f6del.png\"><img loading=\"lazy\" class=\"alignleft size-thumbnail wp-image-209\" src=\"http:\/\/sicv.activearchives.org\/logbook\/wp-content\/uploads\/2015\/05\/1925_kurt_g\u00f6del-150x150.png\" alt=\"1925_kurt_g\u00f6del\" width=\"150\" height=\"150\" \/><\/a>G\u00f6del proved that within any formal system sufficiently powerful to include ordinary arithmetic, there will always be\u00a0 undecidable statements that cannot be proved true, yet cannot be proved false. Turing proved that within any formal (or mechanical) system, not only are there functions that can be given a finite description yet cannot be computed by any finite machine in a finite amount of time, but there is no definite method to distinguish computable from noncomputable functions in advance. That&#8217;s the bad news. The good news is that, as Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the noncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance.<\/p>\n<p>George Dyson, Turing\u2019s Cathedral, The origins of the Digital Universe, Vintage, 2011, p50.<\/p>\n<p><!--more--><\/p>\n<p><img loading=\"lazy\" class=\"alignleft size-thumbnail wp-image-211\" src=\"http:\/\/sicv.activearchives.org\/logbook\/wp-content\/uploads\/2015\/05\/Von_Neumann_2-150x150.jpeg\" alt=\"Von_Neumann_2\" width=\"150\" height=\"150\" \/>Comment sortir de l&#8217;incompl\u00e9tude radicale qu&#8217;instaure le th\u00e9or\u00e8me de G\u00f6del? Doit-on se r\u00e9signer \u00e0 se taire? [&#8230;] Comme on prouve, contre Z\u00e9non, le mouvement en marchant, les math\u00e9matiques se justifient et se fondent par leur succ\u00e8s dans leurs applications. C&#8217;est \u00e0 traduire les math\u00e9matiques dans le d\u00e9voilement scientifique et technique du monde que von Neumann va d\u00e8s lors s&#8217;employer. Comprenons bien la d\u00e9marche \u00e0 laquelle nous assistons ici: il ne s&#8217;agit pas d&#8217;une renonciation simple mais de l&#8217;invention d&#8217;un\u00a0 nouveau chemin qui passant par la physique (son article fondamental sur la m\u00e9canique quantique), la th\u00e9orie ergodique, la g\u00e9om\u00e9trie, la th\u00e9orie des jeux et l&#8217;\u00e9conomie vont le conduire \u00e0 la construction des premiers ordinateurs. Les math\u00e9matiques se fondent en se r\u00e9alisant dans la d\u00e9marche scientifique et technique, telle est la le\u00e7on de von Neumann. Le math\u00e9maticien s&#8217;accomplit en devenant ing\u00e9nieur. Dans les travaux les plus pratiques de von Neumann, ceux qui aboutissent \u00e0 l&#8217;ordinateur s\u00e9quentiel comme ceux qui pr\u00e9figurent le r\u00e9seau de neurones formels, une question est perp\u00e9tuellement pos\u00e9e: notre univers est-il algorithmique? Notre pens\u00e9e est-elle compl\u00e8tement formalisable et simulable par une machine de Turing? [&#8230;] Nous sommes bien, au plus pr\u00e8s des r\u00e9alisations techniques, au coeur des math\u00e9matiques appliqu\u00e9es, engag\u00e9s dans un probl\u00e8me de fondement. C&#8217;est \u00e0 travers la conception et la construction de machines, un probl\u00e8me philosophique fondamental qui se trouve expos\u00e9. [&#8230;] La recherche sur les fondements n&#8217;est pas arr\u00eat\u00e9\u00e9 par le th\u00e9or\u00e8me de G\u00f6del, pas plus que par n&#8217;importe quel autre th\u00e9or\u00e8me d&#8217;incompl\u00e9tude, elle se d\u00e9ploie ailleurs, autrement, sur le terrain o\u00f9 les machines s&#8217;attellent \u00e0 des t\u00e4ches intellectuelles jusqu&#8217;alors privil\u00e8ges de l&#8217;esprit humain.<\/p>\n<p><a href=\"http:\/\/gerard.chazal.pagesperso-orange.fr\/von_neumann.htm\" target=\"_blank\">La pens\u00e9e et les machines: le m\u00e9canisme algorithmique de John von Neumann<\/a>, G\u00e9rard Chazal, in Th\u00e9orie g\u00e9n\u00e9rale et logique des automates, John von Neumann, collection Milieux, Champ Vallon, 1996, pp14-15.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>G\u00f6del proved that within any formal system sufficiently powerful to include ordinary arithmetic, there will always be\u00a0 undecidable statements that cannot be proved true, yet cannot be proved false. Turing proved that within any formal (or mechanical) system, not only are there functions that can be given a finite description yet cannot be computed by [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[5,4],"tags":[30,31,29,34],"_links":{"self":[{"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/posts\/202"}],"collection":[{"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/comments?post=202"}],"version-history":[{"count":10,"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/posts\/202\/revisions"}],"predecessor-version":[{"id":216,"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/posts\/202\/revisions\/216"}],"wp:attachment":[{"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/media?parent=202"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/categories?post=202"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sicv.activearchives.org\/logbook\/wp-json\/wp\/v2\/tags?post=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}