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

7、 a special type of“category”=“mathematical country”consisting incities A,B,C,itineraries A B between cities,a law for composing itineraries A B C.As a category,a topos is“complete”in the sense thatanything which can be mathematically extrapolatedfrom elements of the topos exists in the topos.As it i

8、s complete,a topos E is“too big”.It needs to be approximated by“small”categoriesC E.A full topos E can be reconstructed from a small categoryC Eif C is“dense”in E.In that case,there is a unique“topology”(=extrapolation principle)J on C such thatbCJ E.L.LafforgueRealityNovember 24th,20234/11Linguisti

9、c description of toposes:Starting with an unknown topos E,suppose we have identified enough elements of Eto define a sketching by a small categoryC E.Suppose this sketching is“dense”so thatbCJ E for some topology J.For a linguistic description of E,we need a(first-order geometric)theory Twhich is we

10、ll-adapted to talk about E and Cin the sense that there exists a natural“naming functor”C CT=“syntactic category”of T consisting in?cities=formulas in the vocabulary of T,itineraries=T-provable implicationsinducing a topos morphismbCJd(CT)JT=ET.If this morphism is an embeddingbCJ,ET,there is a“quoti

11、ent theory”T0of T(with the same vocabulary and more axioms)such thatbCJ ET0,so that T0describesbCJ=E.L.LafforgueRealityNovember 24th,20235/11Partial sketching of toposes:Starting from an unknown topos E(incarnating some element of reality or semantic content),we would want to draw a“dense”sketchC Eb

12、y a category C which isfinite(or at least can be described with finitely many words).This is not possible in general.This means that we have to accept partial sketchingsC Ewhich are not dense:there is no equivalencebCJ E.Theoretically,such a C E defines a canonical topology J on Cinducing a topos mo

13、rphismbCJ E.But it cannot be constructed on C as E is not known.This means that the interpretation of C(incarnated in a topology J=extrapolation principle)cannot come from E.L.LafforgueRealityNovember 24th,20236/11Joint descriptions of toposes of some type:Suppose we start from a family of toposesEi

14、,i I,which incarnate elements of reality of the same type.For instance,all Eis could be real imageswhich we want to sketch and describe.As all Eis incarnate elements of realityof the same type,it is natural to think that there should exista joint description theory Tfor all Eis.This means that each

15、Eicould bepartially(but quite faithfully)sketchedby a finite categoryCi Eiendowed with a naming functorNi:Ci CT.L.LafforgueRealityNovember 24th,20237/11Interpretation through language:Suppose that there is a general formalized language Tfor describing elements of reality of some type,incarnated in a

16、 family of toposes Ei,i I.This means that there are(quite faithful)sketchesCi Eiby finite categories Ciendowed with naming functorsNi:Ci CT.For each i,the syntactic topology JTof CT(characterized byd(CT)JT=ET)induces a canonical topology Jion Cidefining a cartesian square of toposes:d(Ci)Ji_?/d(CT)J

17、T_?bCi/bCT This means that the interpretation Jion Ciwould come from the general description theory T.L.LafforgueRealityNovember 24th,20238/11General language and partial singular descriptions:Suppose that each topos Eiin the familycan be sketched by a finite categoryCi Eiendowed with a naming funct

18、orNi:Ci CTso that the topology JTof CTinduces an interpretation topology Jion Ci.Each induced morphism of toposesd(Ci)Jid(CT)JT=ETfactorizes canonically as(bCi)Jisurjection/ETi?embedding/ETfor a unique subtopos ETiof ETwhich incarnates the semantic contentof a unique quotient theory Tiof T:the theor

19、y Tihas the same vocabulary as T,but has more axioms.The extra axioms of Timake up a singular partial description of Ei.L.LafforgueRealityNovember 24th,20239/11Defining a description language:Start with a family of elements of realityconsidered of the same type.For example:images.This similarity sho

20、uld be expressed in the formof a joint description theory T.If each“element of reality”in the familyis considered to be incarnated by an unknown topos Ei,we need the vocabulary and the axioms of Tto be rich enough so that:-Each Eican be sketched by a finite categoryCi Eiendowed with a naming functor

21、Ni:Ci CT.-The topology Jion Ciinduced by the topology JTof CTshould be refined enough to define a topos morphismd(Ci)Ji Ei.L.LafforgueRealityNovember 24th,202310/11Starting from a vocabulary without axioms:Starting from a family of elements of realitysupposed to be incarnated by some unknown toposes

22、 Ei,i I,one may first define a vocabulary rich enough so that:Each Eican be sketched by a finite categoryCi Eiendowed with a naming functorNi:Ci C.Then,one may look for a topology J on Csuch that,for any i I,the induced topology Jion Cidefines a topos morphismd(Ci)Ji Ei.This means that any point ofd(Ci)Jishould make sense as a point of Ei.Such a topology J on Ccorresponds to a quotient theory T of(defined by the same vocabulary completed with axioms)which is a description theory for the Eis.L.LafforgueRealityNovember 24th,202311/11