Laurent Lafforgue_Expose_Lafforgue_AI-Theory_24_novembre_2023_watermark.pdf

1、Reality and its representations:a mathematical modelLaurent Lafforgue(Huawei Paris Research Center,Boulogne-Billancourt,France)AI Theory WorkshopFriday November 24th,2023L.LafforgueRealityNovember 24th,20231/11The double expression of semantic contentsand its modelling by topos theory:mind images(=s

2、emantic contents)linguistic description#sketchingdrawings,schemesextrapolation77texts(=syntactic data)imaginationhh based oninterpretation!?understanding?Proposed mathematical model:Grothendieck toposesEsyntactic description%sketchingsites(C,J)completionC bCJ=E66(first-order“geometric”)theoriesseman

3、ticincarnationT 7 ET=Ejj=“small”category CT+J=topology=vocabulary(words)=extrapolation principle on C+axioms(grammar rules)L.LafforgueRealityNovember 24th,20232/11Why can we propose Grothendieck toposesfor modelling elements of reality?For us human beings:-Any aspects or elements of realitycan be de

4、scribed or at least talked aboutby appropriate forms of human language.-On the other hand,these linguistic descriptions are not unique.Reality is independent of its multiple descriptions.In topos theory:-Any topos E can be presented asa geometric incarnation of the semantic contentsof some formalize

5、d language T(technically,a“first-order geometric theory”)in the sense that there is an identificationpoints of the topos E models of the theory T.-Such a linguistic description of a topos E is not unique.Any topos E incarnates the semantics of infinitely many theories T.-This correspondence is compl

6、ete in the sense thatthe semantics of any such theory T is incarnated by a topos ET.L.LafforgueRealityNovember 24th,20233/11Geometric sketching of toposes:Start with an element of reality or semantic contentwhich is supposed to be mathematically incarnated byan unknown topos E.Technically,a topos is