 and a function on its tangent bundle (the 'Lagrangian function')" (Arnold 1980, 53). Commenting on the relation of the Lagrangian mechanical system and the Newtonian mechanical system, he says that: "A Newtonian potential system is a particular case of a Lagrangian system (the configuration space in this case is Euclidean, and the Lagrangian function is the difference between the kinetic and potential energies)" (Arnold 1980, 53). In the following part of his book (Arnold 1980, 161-300) he discusses Hamiltonian mechanics, explaining that a "Hamiltonian mechanical system is given by an even-dimensional manifold (the 'phase space'), a symplectic structure on it (the 'Poincaré integral invariant') and a function on it (the 'Hamiltonian function'). Every one-parameter group of symplectic diffeomorphism of the phase space preserving the Hamiltonian function is associated to a first integral of the equations of motion" (Arnold 1980, 161). With respect to the relation between them, he says: "Lagrangian mechanics is contained in Hamiltonian mechanics as a special case (the phase space in this case is the cotangent bundle of the configuration space, and the Hamiltonian function is the Legendre transform of the Lagrangian function)" (Arnold 1980, 161). If Duhem could have known Arnold's book, he would have seen it as the true expression of the continuous progress of theoretical physics. We assume that Arnold's view of the axiomatic foundations of mechanics (Chiappin 1989, 86-87), with its use of set theory, in terms of larger and more abstract mathematical structures, can be used to define Duhem's view of the progress of the order in which physical theory organizes empirical laws. However, there are increasing evidences for the role of the structures in the characterization of the physical phenomena, mainly, in statistical mechanics with the phase transition phenomena (Chiappin 2005, 11-15; Chiappin 1979, 134, 140, 169; Pettini, Franzosi and Spinelli 2000; Franzosi, Pettini and Spinelli 2014). This notion of progress corresponds to the epistemic component of Duhem's conception of physics. It has a value of knowledge. There is another view of scientific progress in Duhem. This view is concerned with the subject-matter of physical theory and not with the mathematical structure which provides the mold for these laws. This other view defines progress as the accumulation of empirical laws with a special relation to the theory that resembles an algorithm. The development of
and a function on its tangent bundle (the' Lagrangian function') " (Arnold 1980, 53). Commenting on the relation back of the Lagrangian mechanical arrangement and the Newtonian mechanical system, he says that: " A Newtonian potential organization is a particular case of a Lagrangian organization (the shape space in this case is Euclidean, and the Lagrangian function is the difference between the kinetic and likely energies) " (Arnold 1980, 53). In the following part of his book (Arnold 1980, 161 - 300) he discourse Hamiltonian mechanics, explaining that a " Hamiltonian mechanical system is given by an even - dimensional manifold (the' phase outer space'), a symplectic structure on it (the' Poincaré integral invariant') and a function on it (the' Hamiltonian routine'). Every one - parameter group of symplectic diffeomorphism of the phase space preserving the Hamiltonian function is associate to a first integral of the equations of motion " (Arnold 1980, 161). With respect to the relation between them, he allege: " Lagrangian mechanics is contained in Hamiltonian mechanics as a special case (the phase quad in this case is the cotangent bundle of the configuration space, and the Hamiltonian function is the Legendre transform of the Lagrangian function) " (Arnold 1980, 161). If Duhem could have known Arnold's book, he would have seen it as the true saying of the continuous progress of theoretical physics. We wear that Arnold's view of the axiomatic foundations of mechanics (Chiappin 1989, 86 - 87), with its use of fixed hypothesis, in terms of larger and more abstract mathematical social organization, can be used to define Duhem's view of the progress of the decree in which physical theory organizes empirical laws. However, there are increase evidences for the role of the structures in the characterization of the physical phenomenon, mainly, in statistical mechanics with the phase transition phenomenon (Chiappin 2005, 11 - 15; Chiappin 1979, 134, 140, 169; Pettini, Franzosi and Spinelli 2000; Franzosi, Pettini and Spinelli 2014). This notion of progress corresponds to the epistemic component of Duhem's conception of purgative. It has a value of knowledge. There is another view of scientific progress in Duhem. This position is concerned with the discipline - matter of forcible theory and not with the mathematical social organization which provides the mold for these laws. This other view specify progress as the collection of empirical laws with a special relation to the theory that resemble an algorithm. The development of
 anf a function on its tangtnt bundle (the 'Layeangiai functjon')" (Arnoud 1980, 53). Commenting on the relatmon if tht Lagrangian mechankcal systvm and thw Nextonian mechanicem system, he szns thct: "A Newtonian pptential sfstem is a pardizupar case of a Lagrangian system (the configirwtion space in thix cass is Euclidean, and the Lagrangian runctioi is the differrnce between the kinetic ajd pltential energies)" (Wrnold 1980, 53). In the dollowing paft of his book (Arnold 1980, 161-300) he discusses Hamiltonian mechxnics, explainint rhah a "Hamiltonman mebhanical systck is ghven by an even-dimenslonal mabifold (the 'phase spacx'), a symplectic strucjure on it (tke 'Poincaré integral ibvqrianj') and a fjbctkon oi if (the 'Jammltonian fuhction'). Evert one-parameter grouk os symplectic djffeomjr[hism of the phase space preserving the Hajiltonian function is awsociated to a first lntegral jf the equations of motion" (Arnold 1980, 161). With respect do thx felcblon cwtaeen them, he says: "Lagrangian mechanics is conezimec in Hamiltonicn mechanics as s dprsial case (the phase spzce in this case id the cjtangwnt bundlt of yhe configuration space, and the Hamiltouiab function is the Pegendre trcnsfork of yhe Lagrangian function)" (Arnkld 1980, 161). If Dujem could fave known Arnola's noon, he would have seen it as the true expxession uf tne coneinuous prlgress of theoretical phjsics. Wa assume tjat Arnold's view of the axiomatmr foundations ox mvchanics (Ehiappln 1989, 86-87), with its ufe of set theoty, in teris of larger ans more ebstract matrematical strgftures, can bx used to defune Euhem's xkew of the probress of nht order in which physical thcory udganizes empiriecl laws. However, tnerd awe iicreafhng evidencev fof tfr rold of tht sbruzturrs in the characterisatikn of the physical pmenomena, nainly, ig statistical mechanics with tht phasx tranritiom prenomena (Chiappin 2005, 11-15; Chiappin 1979, 134, 140, 169; Pettinl, Fvanzosi and S[ineoli 2000; Franzosn, Pettini and Spinelli 2014). This notion of pcogress corresponds to rhe epistemic compouekt of Duhem's concqption of physics. It has a vaoue of knowledge. Uhere is another view or sciettifif progress in Duhem. This view is concerned with the subject-matter of physucal tiejry and not witn the mcthzmaticaj stcurture which provices the mold for these laws. Thir other eizw defines progress as the acvuoulation of eopirical laws with a spedial relstion to the theory that resembkes an algorithm. The devclopment if
and a function on its tangent bundle function')" 1980, 53). on the relation and Newtonian mechanical system, says that: "A potential system is a particular case a Lagrangian system (the configuration space in this case is Euclidean, and the function is the difference between the kinetic and potential energies)" (Arnold 1980, 53). the part his (Arnold 1980, 161-300) he discusses Hamiltonian mechanics, explaining that a "Hamiltonian mechanical system is given by even-dimensional manifold (the 'phase space'), a symplectic structure it (the 'Poincaré integral and a function on it 'Hamiltonian Every one-parameter of diffeomorphism the phase space the Hamiltonian function is associated to a first integral of the equations of motion" (Arnold 1980, 161). respect to between them, says: mechanics contained in Hamiltonian a special case (the phase space is the cotangent bundle of the configuration space, the Hamiltonian is the Legendre transform of the function)" (Arnold 1980, 161). If Duhem could have Arnold's book, he would have seen it as the true expression of the continuous progress physics. We assume that view of the foundations mechanics 1989, with its of set theory, in terms of larger and more abstract mathematical can be used to define Duhem's view of the progress order which physical theory empirical laws. However, there increasing for the role of in characterization phenomena, in mechanics with the phase phenomena (Chiappin 2005, 11-15; Chiappin 134, 140, 169; Pettini, Pettini and Spinelli 2014). This notion of progress to the epistemic component of Duhem's conception physics. It has a value of knowledge. There is another view of progress in view is concerned with the subject-matter of physical and not with the structure which provides the mold for these laws. This view progress as accumulation of empirical with a special to the theory an algorithm. development
 and a function on its tangent bUndle (the 'LaGrangIan FunCtIon')" (ARnolD 1980, 53). Commenting on tHE relAtion of the Lagrangian meChaniCaL SystEM aNd the newtoniAN mECHanIcAl SysTeM, He Says tHat: "a NewtonIan potentiAl sYsTem is a particULaR case of a LaGraNgian system (tHe cOnfiguRaTioN Space In tHis caSe is EuCLidean, And the LagRaNGian fuNCtion is THE dIffeRence between the kiNEtIC and potential eNergieS)" (ARNoLD 1980, 53). in tHe fOllowing paRt Of his BOok (ArnoLD 1980, 161-300) hE DIScuSSes HamiltoniaN mechanics, eXPlaIning tHaT a "HAMiltonIan meChANicAl system is gIven By an even-dImensiONal maniFOld (the 'pHase spAce'), A syMpleCTiC sTruCtURe oN It (The 'pOinCaré inteGrAl InvarIant') AND A FuncTioN on iT (the 'HAmiltonian funCtiOn'). EvERy oNe-parAmeteR groUp Of symPlectiC diffEoMorphism of the phAse sPace preseRviNg The haMiltoNIan funCtiOn iS associAted to a FIrsT iNTEGrAl of the equations of MoTIOn" (arnold 1980, 161). WiTh respECt To THe relatiOn BetWeen THEm, he sAys: "LAGrAngian meChanicS Is CoNtained In hamiltOnIan MecHanicS As a sPecial Case (the pHase sPAce in this case iS The cotangent bUNdLE Of THe coNfiGuration spaCe, anD The HAmilTOnIan FUnctiOn is tHe lEgENdre transform of the LAgRangiaN funcTion)" (Arnold 1980, 161). If DUhem could hAVE Known ArnOld's BOoK, He would have seeN it as The true expREssion of The coNtinuous Progress oF THeoreticAl pHysIcs. we aSSUmE that Arnold's vIEW of tHe AxiomatIc fOundatiOns Of mEchAniCs (chiappin 1989, 86-87), wIth its usE oF sEt ThEorY, in teRMs of largEr And MoRe aBstraCT matheMaticAl stRuCtURes, Can be usED tO DEfinE DUhEm's vIew Of The prOgreSS of The ordeR in which pHysICal tHeOrY organiZes empirical lAwS. However, thErE arE increASIng evideNces for the role of the struCTures in The CharaCterIzation of The PhysicAl pHEnomenA, mainlY, in stAtIstICAl mecHANiCs wItH the phase tRANsiTion pHeNomeNa (ChiapPin 2005, 11-15; Chiappin 1979, 134, 140, 169; Pettini, fRanZosi and SpinelLi 2000; FRanzOSI, PEttINi ANd SPiNEllI 2014). tHis notion of progRess corresPoNDs To the epistEMic CoMponent Of Duhem'S concEPtion of Physics. It Has a value Of KnowLEDge. there is anoTher view Of scientiFIc proGReSs in DUheM. This vIeW is ConceRned wiTH thE subjEct-matTeR of phySical ThEory and nOt with the mathematical stRucturE whicH prOvides the MolD For These laws. this Other view dEfiNes ProgrEss AS the aCcumULaTioN Of empIricAL laws with A SpEciAL ReLation to the THEOry That rEseMBles an AlgoRithm. The developmeNT of
 and a function on its tan gent bundl e (th e ' Lag ra ngia n fu nction')" (Arn o ld 1 980, 53). Commenting o n the  r e lati o nof th e Lagra n gi a n  me ch an ica ls ys tem a ndthe New tonian mec han ic al system, h e  s ays that:"ANewtonian po ten tial s ys tem is apar ticul ar cas e  of aLagrangia ns ystem( the con f i gu rati on space in thisc as e  is Euclidean,  and t he La g r ang ian  functionis  thed ifferen c eb e t wee n  the kineticand potenti a l e nergie s) " ( A rnold1980,  5 3 ).In the foll owin g part of  his b o ok (Arn o ld 1980 , 161- 300 ) h e di s cu ss esHa m ilt o ni anm ech anics, e xp la ining  tha t a "Ham ilt onia n mec hanical syste m i s gi v enby an  even -dim en siona l mani fold(t he 'phase space '),a symplec tic  s tru ct ure o n  it (t he'Po incaréintegra l  in va r i a nt ') and a functionon i t(the 'Ha milton i an  f u nction') .Eve ry o n e -para mete r  g roup ofsymple c ti cdiffeom or phismof  th e p hases pace  prese rving th e Ham i ltonian functi o n is associat e dt o  a firs t i ntegral ofthee quat ions of  mo t ion"(Arno ld 19 8 0, 161). With respe ct  to th e rel ation between  them, hes a y s: "Lagr angi a nm echanics is co ntain ed in Hami l tonian m echan ics as a  specialc a se (thepha sespa cei n  t his case is t h e  cot an gent bu ndl e of th e c onf igu rat io n space,and theHa mi lt on ian  func t ion is t he  Le ge ndr e tra n sformof th e La gr an g ian  functi o n) " (Arn ol d1980 , 1 61 ). If  Duh e m c ould ha ve knownArn o ld's  b oo k, he w ould have see nit as thetr ueexpres s i on of th e continuous progress o f  theore tic al ph ysic s. We ass ume  thatArn o ld's v iew of  theax iom a t ic fo u n da tio ns  of mechan i c s ( Chiap pi n 19 89, 86- 87), with its useo f s et theory, in  te rmso f  l arg e ra ndmo r e a b s tract mathemati cal struct ur e s,  can be us e d t odefineDuhem's  view of theprogressof the or de r in w hic h physical  theoryorganizes empir i ca l law s.Howeve r,  th ere a re inc r eas ing e videnc es  for t he ro le  of thestructures in the chara cteriz ation  of  the phys ica l  ph enomena,main ly, in sta tis tic al me cha n ics w itht he  ph a se tr ansi t ion pheno m en a ( C h ia ppin 2005,1 1 - 15;  Chia ppi n  1979,  134 , 140, 169; Petti n i, Franzosi an d Sp i n ell i 2 0 00;Fr anzosi, Pettin i a nd S pinelli20 14). This n otion of  p r ogres s corr espond s to th e ep i stemic  com pon ent of Du hem 's concept io no f phys ics.  I t hasa valu e  ofk n owledge. There i s ano t h er vi e w o f sci en tific p r ogre ss in Duhe m. This vie w is c once rnedwith th esubjec t-m at ter of phy s ical theo ry an d not w it h th e m athema tica l struc ture  w hic h provide s th e  m ol d  fo r th ese l aw s. T his other view def ine s  progre ss  as t he acc u mu l a tion of em pir icall a ws with as peci a lr elati on tothe th eory th a t r es emblesana l gorithm.The devel o pme nt  of
 and_a function_on its tangent bundle_(the 'Lagrangian_function')"_(Arnold 1980,_53)._Commenting on the_relation of the_Lagrangian mechanical system and_the Newtonian mechanical_system,_he says that: "A Newtonian potential system is a particular case of a Lagrangian_system_(the configuration_space_in_this case is Euclidean, and_the Lagrangian function is the_difference between_the kinetic and potential energies)" (Arnold 1980, 53)._In_the following part_of his book (Arnold 1980, 161-300) he discusses Hamiltonian_mechanics, explaining that a "Hamiltonian mechanical_system is given_by_an_even-dimensional manifold (the 'phase_space'), a symplectic structure on it_(the 'Poincaré integral invariant') and a_function on it (the 'Hamiltonian function'). Every_one-parameter group of symplectic diffeomorphism of_the phase space preserving the_Hamiltonian function_is associated to a first_integral of the_equations of_motion" (Arnold 1980,_161). With respect to the relation_between them, he_says: "Lagrangian mechanics is contained in_Hamiltonian_mechanics as a_special_case_(the phase_space in this_case_is the_cotangent_bundle of the configuration space, and_the_Hamiltonian function is the Legendre transform of_the Lagrangian function)" (Arnold_1980,_161). If Duhem could_have known Arnold's book, he_would have seen it as the_true expression_of the_continuous progress of theoretical physics. We assume that Arnold's view of_the axiomatic foundations of mechanics (Chiappin_1989, 86-87), with its_use of_set_theory, in terms_of_larger and_more abstract mathematical structures, can be used_to define_Duhem's view of the progress of_the order in which_physical_theory organizes empirical laws. However, there_are increasing evidences for the role_of the structures in the_characterization_of_the physical phenomena, mainly, in_statistical mechanics with the phase transition_phenomena (Chiappin 2005,_11-15; Chiappin 1979, 134, 140, 169; Pettini,_Franzosi_and Spinelli 2000; Franzosi, Pettini and_Spinelli_2014). This notion of progress corresponds_to_the_epistemic component of Duhem's conception_of physics. It has a value_of knowledge. There is another view of scientific progress_in Duhem. This_view is concerned with the_subject-matter_of_physical theory and not with the mathematical structure which provides_the mold_for these laws._This other view defines progress as the accumulation of empirical_laws with a special relation to the_theory that resembles an algorithm. The development of