, might! be! precisely! here:! Klee! proposes! a! logic! of! transitions! and! this! logic!of!transition!is!based!on!a!language!that!might!sound!very!far!from!the!genesis!of! design!objects.!Let's!analyse!some!of!these!transitions:!the!first!language!is
, define! horizontal! and! vertical! and! relate! that! to! the! ohysical! notion! of! gravity;! having! 2A The! analyses! of! the! two! courses! leads! to! identify! two! critical! features! to! get! a! generic!creative!design!capacity:!! a. A!knowledge!structure!that!is!characterized!by!nonAdeterminism!and!nonA independence! b. A! genesis! process! that! helps! to! progressively! " accumulate " ! languages! on! the! object! in! a! robust! way
, This!is!the!basic! " knowledge!base " !of!the!designer.!! A!The!second!element!is!the!set!of!forcing!conditions!that!will!act!on!M.!To!build! new!sets!from!M,!we!have!to!extract!elements!according!to!some!conditions!that!can!be! defined!in)!a!set!of!candidate!conditions!Q!and!a!partial!order!relation! we! can! extract!conditions!that!can!form!series!of!compatible!and!increasingly!refined!conditions! (q0! qi)! with! for! any! i:! qi! <! qiA1;! this! means! that! each! condition! refines! its! preceding!one.!The!result!of!each!condition!is!a!subset!of!M.!Hence!the!series!(qi)!builds! series! of! nested! sets,! each! one! being! included! in! its! preceding! set! of! the! series.! Such! series!of!conditions!generates!a!filter!!F!on!Q.!And!a!filter!can!be!interpreted!as!a!stepAbyA step! definition! of! some! object! or! some! set! of! objects! where! each! step! refines! the! preceding!definition!by!adding!new!conditions.!Q!is!the!knowledge!structure!used!by!the! designer.! In! the! world! of! industrial! design,! this! could! be! color,! texture,! matters,! lines,! points,!joints,?!(referring!to!the!words!used!by!Klee!or!Itten)!! A!The!third!element!of!Forcing!is!the!dense!subsets!of!(Q,!<):!a!dense!subset!D!of!Q! is!a!set!of!conditions!so!that!any!condition!in!Q!can!be!refined!by!at!least!one!condition! belonging!to!this!dense!subset.!One!property!of!dense!subsets!is!that!they!contain!very! long!(almost! " complete " )!definitions!of!things!(or!sets)!on!M,!since!every!condition!in!Q,! whatever! its! " length " ,! can! always! be! refined! by! a! condition! in! D.! But! a! dense! subset! contains!only!constraints!so!that!it!is!a!way!to!speak!of!all!elements!without! " having " !one! element! and! speaking! of! them! only! from! their! " properties " .! Interestingly! enough! the! notions!of!perspective,!balance,!organism?!could!be!interpreted!as!dense!subset!defined! by!constraints!such!as!line!(for!perspective),!weight!(for!balance),!organ!(for!organism)?! The!set!of!constraints!(lines)!leading!to!a!perspective!is!dense!in!the!set!of!all!constraints! since!whatever!a!sequence!of!constraints!(a!partially!defined!piece)!it!is!always!possible!to! identify! some! additional! constraints! that! lead! to! speak! of! the! " perspective " ! of! this! partially!defined!object.!! A!The!fourth!element!of!Forcing!(its!core!idea!)!is!the!formation!of!a!generic&filter!G! which!step!by!step!completely!defines!a!new!set!not!in!M!!!Now!how!is!it!possible!to!jump! out! the! box! M?! Forcing! uses! a! very! general! technique:! it! creates! an! object! that! has! a! property!that!no!other!object!of!M!can!have!!Technically,!a!generic!filter!is!defined!as!a! filter!that!intersects!all!dense!subsets.!In!general!(see!condition!1!below)!this!generic!filter! defines!a!new!set!that!is!not!, 1966.
, generically " ! different!from!all!elements!of!M!and!still!is!coherent!with!the!rules!of!M.!Our!question!is! then:!what!are!the!conditions!for!a!filter!to!be!a!generic!filter!that!goes!out!of!M?!There! are!actually!two!sufficient!conditions!to!create!a! " generic!filter
, !the!splitting!condition!implies!that!there!q!and!q'!that! refine!p!and!are!incompatible;!so!one!of!the!two!is!not!in!G!hence!is!in!D.!Hence!any! condition!of!Q!is!refined!by!an!element!of!D.!Hence!D!is!dense.!So!G!is!not!generic. ! Figure 11: The splitting condition ! ! ! Condition! 2:! the! countable! condition:! if! M! is! countable, 2002.
, !the!splitting!condition!excludes!two!cases:!there! are! deterministic! chains! of! conditions! (laws)! or! there! is! a! modular! structure! such!that!there!is!finally!a!constraint!r!that!refine!q!as!well!as!q'!(q!and!q'!are! modularized! and! r! is! unsensitive! to! the! choice! or! q! vs! q').! So! it! appears! that! Itten,!providing!a!knowledge!structure!that!characterized!by!nonAdeterminism! and! nonAindependence! finally! enabled! students! to! build! a! knowledge! base! that&meets&the&splitting&condition.!! Conversely! there! appears! that! knowledge! structures! which! favor! laws! and! modularization! (like! classical! engineering! design)! actually! prevent! splitting! condition.!Itten!builds!a!knowledge!structure!that!enable!the!designer!to! " goA outAofAtheAbox " ! whereas! engineering! design! tends! to! build! a! knowledge! structure!that!avoid!to!goAoutAofAtheAbox.!! 2A Regarding! the! second! condition,! if! one! analyses! the! partAwhole! relationships! introduced! by! Klee! as! dense! subsets! (see! above),! then! the! genesis! process! proposed!by!Klee!corresponds!to!a!form!of!countability!of!the!dense!subsets
, (Itten)!and!with!a!method!to!systematically!build!a!generic!filter!(Klee)!! Conclusion$ ! This! mathematical! analysis! helps! us! to! generalize! the! results! obtained! in! the! Bauhaus!context.!We!have!proposed!to!analyse!style!(or!meaning)!creation!as!a!form!of! generic& creativity! (part! 1),! ie! a! creativity! that! will! address! the! generic! languages! of! a! culture.!The!Bauhaus!texts!(and!the!formal!models!of!forcing!in!mathematics)!helped!to! analyse! some! critical! features! that! make! design! capacities! adapted! to! generic! creativity! (parts!2!to!4):!! 1A the! designer(s)! ! relies! on! (or! is! able! to! create)! a! knowledge! structure! that! is! characterized! by! nonAdeterminism! and! nonAindependence.! Technically! speaking,! this! knowledge! structure! meets! the! splitting! condition.! Metaphorically,! this! knowledge!structure!is! " open " ,!it!contains!many! " holes " ,!ie!many!open!concepts! that!will!require!investigations,!learning!and!knowledge!creation.!! 2A the!designer(s)!is!(are)!able!to!identify!(or!build)!a!structure!of!dense!subsets!and! to!follow!a!design!process!that!leads!him!(them)!through!all!dense!subsets!of!the! knowledge!base.!! ! This! comparison! helps! also! to! show! that! systematic! engineering! is! precisely! characterized! by! knowledge! structures! that! prevent! the! splitting! condition,! that! are! characterized!by!independence!(modularity)!and!determinism!(engineering!science).!! But! our! results! also! suggest! an! analytical! framework! for! contemporary! design
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