Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica.(3)

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

ONAXIOMATICFOUNDATIONSCOMMONTOCLASSICALPHYSICS...145

PARTB:MAINREASONSFORCHANGINGSOMEPARTSOF

SECTION2IN(BR&MONT'S D)[17].ASTRANGEBUTLUCKYSITUATION

AS2Ontheonehand,thearticleBr&Mont-seeAS1;6-treatsparticlesystems(withoutconstraints),isoftheMach-PainleveÂtype(1),andisbasedonA.Bressan'smodallogic(althoughforpracticalmotivesthislogicissometimesconsciouslytreatednon-rigorouslyinBr&Mont-seeftn.7onp.168there-).

AS2,1Furthermore,IbelievethattheaforementionedconscioustolerationofmistakesofmodallogicwasevencompulsoryinBr&Mont,inordernottocompel,e.g.,rigorousmathematicalphysicistsinterestedinBr&Mont(asaworkonmechanics)toknowmy(powerfulbutcomplex)theoryofmodallogic(presentedinGIMC-seeAS1;5-).AS3Ontheotherhand,inthepresentworkthelogicalmodalcalculusMCypresentedinGIMCisinsteadcarefullytakenintoaccount(andalsowithsomeusefulresultsformechanicsappreciablebythewellknownmathematicalphysicistC.Truesdell-seeAS7belowinpartC).

AS4However,inAxFound-seeAS1-acontributingworkwillperformsomechangesimprovingBr&Montnotonlyinconnectionwith(modal)logic;andthisholdsespeciallyforDef.2.3andthesubsequentpartofSect.2.FurthermorethesechangesareinaccordwiththeproofswritteninSects.3to8forthetheoremsconsideredinBr&Mont,whilethesetheoremsaswellastheirproofsstronglycontrastwiththeoriginalversionofDef.2.3andthesubsequentpartofSect.2.Thisisthe(strange)situationmentionedin(PartB)'stitle.

AS4,1NeitherTruesdellnorPitteriattendedanytechnicalcourseonmodallogic.Howevertheyusedthislogicintuitivelyandcorrectly.

AS4,2Inparticular,intheyear2004InotedaremarkableintuitionofTruesdellthathewroteascontributor,withina(polemic)ftn.inapaperappearedin1953-seeAS6belowinpartC.

AS4,3InspiteofAS3,byAS4toAS4;2(henceAS6),IthinkthatalsoreadersnotinterestedinmodallogicmaybeinterestedinmanycontributionsofminetoAxFound.

PARTC:REASONSFORUSING``AXFOUND''ANDFORWRITING

THEPRESENTWORKINMEMORYOFPROF.C.TRUESDELL

(1)In(PartB)'stitlesomechangesneededbyBr&Montarementioned,howevertogether

(1)(a)Abookorpaper,e.g.Met D[3]issaidtobeaÁlaMach-PainleveÂoroftheMach-PainleveÂtype,ifinitnotionssuchasmassandforce(andpossiblyalso,e.g.,inertialspaceand(inertial)instant)arenotregardedasprimitives,unlikewhathappensinmostworks.

(b)E.g.MetisaÁlaMach-PainleveÂatahighlevel,especiallywithrespecttoPainleveÂ1922,i.e.[59].