On the Logical Analysis of the Foundations of Vector Calculus

Transcription

1 O the Logical Aalysis of the Foudatios of ecto Calculus Temu Z. Kalaov Home of Physical Poblems, Pisatelskaya 6a, Tashket, Uzbekista, Abstact. A citical aalysis of the foudatios of stadad vecto calculus is poposed. The methodological basis of the aalysis is the uity of fomal logic ad of atioal dialectics. It is poved that the vecto calculus is icoect theoy because: (a) it is ot based o a coect methodological basis the uity of fomal logic ad of atioal dialectics; (b) it does ot cotai the coect defiitios of movemet, diectio ad vecto ; (c) it does ot take ito cosideatio the dimesios of physical quatities (i.e., umbe ames, deomiate umbes, cocete umbes), chaacteizig the cocept of physical vecto, ad, theefoe, it has o atual-scietific meaig; (d) opeatios o physical vectos ad the vecto calculus popositios elatig to the physical vectos ae cotay to fomal logic. Keywods: mathematics, vecto calculus, geomety, physics, egieeig, philosophy of sciece MSC: 00A05, 00A30, 00A35, 00A69, 00A79, 03A05, 03A10, 97G70, 97M50, 51P05 Itoductio As is well kow, the mathematical fomalism of vecto calculus is widely ad successfully used i atual scieces [1-7]. Howeve, this does ot mea that the poblem of validity of vecto calculus is ow completely solved, o that the foudatios of vecto calculus ae ot i eed of fomal-logical aalysis. I my view, stadad vecto calculus caot be cosideed as absolute tuth if thee is o fomal-logical substatiatio of this calculus. Recetly, thee has aise a ecessity fo citical aalysis of the foudatios of vecto calculus. But thee ae o woks devoted the aalysis of vecto calculus withi the famewok of the uity of fomal logic ad of atioal dialectics. The pupose of the peset wok is to popose the coect aalysis of the foudatios of vecto calculus. The aalysis is caied out withi the famewok of the coect methodological basis: the uity of fomal logic ad of atioal dialectics. 1. Aalysis of the cocepts of diectio ad vecto" As is well kow, i mathematics, physics, ad egieeig, a vecto (o Euclidea vecto, o geometic vecto, o spatial vecto) is called quatitative chaacteistics which has ot oly a umeical value, but also the diectio [7, 8]. I othe wods, vecto is a lie segmet with a defiite diectio (o gaphically is a aow), coectig a iitial poit with a temial poit. I.e., vecto is a geometic object that has magitude (o legth) ad diectio ad ca be added to othe vectos accodig to vecto algeba. Physical examples of vecto quatities ae mateial poit displacemet, velocity ad acceleatio of a mateial poit, as well as a foce. Theefoe, aalysis of the cocept of vecto is ot possible without the defiitios of cocepts of movemet ad diectio. 1. Movemet is a chage i geeal, ay iteactio of mateial objects. Categoy of movemet is a scietific cocept that eflects the most commo ad essetial popety of pheomea (pocesses), the most commo ad essetial elatios ad coectios i eality. Movemet is a attibute of matte. I accodace with the dialectical piciple of the uity of matte ad movemet, the movemet does ot exist without mateial objects. But the movemet is ot a mateial object. The movemet is maifested as the uity of opposites: chageableess ad stability, cotiuity ad discotiuity. Cocetizatio of the movemet is the mai foms of 1

2 movemet: mechaical, physical (themal, electomagetic, gavitatioal, atomic, ad uclea), chemical, biological, ifomatioal, ad social oes. 2. Chage as a pocess ca be of two types: a qualitative chage ad quatitative chage. The qualitative chage (i.e. a chage of qualitative detemiacy) is studied by dialectical logic ad atual scieces. The quatitative chage (i.e. a chage of quatitative detemiacy) withi the limits of cetai qualitative detemiacy is studied by fomal logic ad mathematics. The quatitative chage ca be studied oly withi a efeece system which cotais a clock as compoet pat. 3. A clock (i.e., a device cotaiig a wokig clock mechaism, movig the aow ad the fixed dial) detemies the time ad time chaacteizes the clock. Time is a cocete cocept because it expesses the popety of the clock mechaism (clock pocess). Time t is a uivesal vaiable (with the dimesio of secod ), a ifomatio basis that is used to put i ode of ifomatio about evets ad pocesses i the wold. Time t is defied by the followig mathematical expessio [9]: t = τ whee = 0, 1, 2,...; τ is elemetay (uit) duatio which ca be made as small as desied. Cocete umbes (deomiate umbes) t have oe ad the same qualitative detemiacy (i.e., dimesio of time ). The set of umbes t foms a odeed sequece. A membe of the sequece is called a momet of time. Numeical values of quatity t is chaged due to clock mechaism which cotiuously chages umeical values of the quatity. 4. The mechaical fom of movemet (i paticula, the motio of a mateial poit M ) is studied i a efeece system which epesets the uity of the system of coodiates ad clock. The system of coodiates is a system of measuig devices which detemies the positio (i.e., the set of coodiates) of a mateial poit M i space. (Fo example, the Catesia coodiate system epesets the system of thee coected measuig scales (dawig scales): staight lies O x, O y, O z with pited cocete umbes (deomiate umbes) havig the idetical dimesio of mete ). The space of the object (fo example, geometic space, ad eegy space) is the set of possible (available) states of the mateial object (i paticula, the set of positios of the mateial poit M ). Each state is chaacteized by a cetai cocete umbe (deomiate umbe) havig a dimesio. Movemet of a object i space is a pocess of tasitio fom some states to othe states, i.e. the pocess of tasitio fom some cocete (deomiate) umbes to othe cocete (deomiate) umbes. 5. A pocess has the begiig (i.e., the begiig of the chages) ad the ed (i.e., the ed of the chages). The tasitio fom the iitial state to the fial state epesets the sum of elemetay tasitios ad, theefoe, is chaacteized by a icease of the chages. I othe wods, the total chage is the sum of elemetay chages. Sice elemetay chage is chaacteized by the cocete (deomiate) umbe havig a dimesio, the total chage has dimesio as well ad is expessed by the followig mathematical fomula: s = λ, whee = 0, 1, 2, 3,..., λ is elemetay (sigle) chage which has the dimesio ad is assumed to be costat. The set of deomiate umbes s foms a odeed sequece. The umeical values of the deomiate quatity s ae chaged if the umeical values of the quatity ae chaged. If the umeical values of the quatity ae ot chaged with time, the pocess is ot ealized. 2

3 6. A pocess is chaacteized by the diectio (diectivity) of chage, the ate of chage, ad acceleatio of chage. If the pocess is ot ealized, the diectio (diectivity), the ate, ad acceleatio do ot exist. Explaatio is that the diectio (diectivity), ate, ad acceleatio ae the popeties of the pocess ad ot the popeties of the mateial object. Theefoe, the diectio of chage detemies the ode of the umbe set but a odeed umbe set does ot detemie the diectio. Neithe pue mathematics o applied mathematics (i.e., the mathematical fomalism of the atual scieces) does ot cotai a mathematical (calculatio) pocess because the mathematics does ot epeset a compute o some othe mateial device that ealizes the pocess of chage of the values of the quatity s. Chage of the valuesof the quatity is caied out by a opeato (peso). Theefoe, the coect mathematical fomalism ca ot cotai the cocepts of diectio (diectivity) ad vecto. 7. If oe assumes that the mathematical fomalism cotais the cocepts diectio (diectivity) ad vecto, the the fomula fo the quatity s to be witte i the followig vecto fom: s = whee λ is elemetay (uit) vecto. But sice the umeical values of quatity i this fomula is ot chaged with time, the pocess of chage of the umeical values of the quatity s i mathematical fomalism is ot ealized. Theefoe, this fomula does ot descibe diectio, ad the mathematical fomalism does ot cotai the cocepts of diectio (diectivity), vecto, ad uit vecto. Thus, the diectio (diectivity) ad vecto ae ot mathematical objects (cocepts). The cocepts of diectio ad vecto do ot coespod to ay geometic object (fo example, a lie segmet). Idicatio of the bouday poits of the lie segmet ad desigatio of these poits with the help of tems (wods) begiig ( iitial poit ) ad ed ( temial poit ) do ot defie mathematically a geometic vecto (because the ode of poits do ot defie the diectio of movemet). All poits of the lie segmet have oe ad the same qualitative detemiacy: cocept of iitial poit ad cocept of temial poit ae idetical oes. Theefoe, the tems begiig ad ed of the segmet ae ot mathematical defiitios of the cocept of diectio (diectivity). A aow is a visual (gaphic) image of couse. I othe wods, vebal, liteal, symbolic, umeical, ad gaphical epesetatios (display) of the begiig ad the ed of the segmet ae ot a mathematical defiitio of the cocept of diectio (diectivity). Theefoe, the coect mathematical fomalism ca ot ad must ot cotai the cocepts of diectio (diectivity) ad vecto. The coodiate system epesets a system of thee coected dawig scales: staight lies O x, O y, O z, which ca ot be attibuted to the diectio. Also, staight lies O x, O y, O z ca ot cotai the uit vectos. Fom the poit of view of fomal logic, the tems diectio (diectivity) ad vecto i mathematics ad theoetical physics mea epesetatio, i.e. imagiay image of the pocess, which is depicted with the help of a aow. 2. Aalysis of the stadad popositios of vecto algeba The mathematical cocept of vecto i geeall ca ot be used i the atual scieces: this cocept does ot make sese i the atual scieces. The cocept of vecto as used i the atual scieces is chaacteized by the cocepts of deomiate quatity ad dimesio of quatity. Theefoe, aalysis of the stadad popositios of vecto algeba must be doe fom this viewpoit. λ 3

4 1. As is kow, the positio of a vecto i the Catesia coodiate system O x yz is detemied by its pojectios. The pojectio of the vecto o the axis is defied as follows. Thee ae vecto whose module has the dimesio (fo example, the dimesio of speed, acceleatio dimesio, the dimesio of powe) ad the axis O x with the deomiate umbes which have dimesio of legth. The pojectio of the vecto o the axis quatity (module, legth) O x is called the x of diected segmet x located (placed, putted) o the axis O x : x = cos α whee α is the agle betwee the vecto ad the axis O x. I geeal case, the vecto is decomposed ito compoets ude the Catesia othoomal basis i, j, k as follows: = x i + y j + z k whee i, j, k ae the uit vectos of the Catesia coodiate system; x, y, z ae the pojectios of the vecto o the coespodig axes. The picipal impotace of the basis i, j, k is that the liea opeatios o vectos ude the give basis become the usual liea opeatios o umbes the coodiates of these vectos. I my opiio, these stadad expessios ae ot fee fom objectio. The objectio is that the stadad expessios ae cotay to the fomal-logical laws. segmet Really, the stadad expessios asset that segmet x coicides with the segmet of axis x lies o the axis O x (i.e., O x ). Fom the poit of view of fomal-logical law of idetity, this implies that these segmets have the same qualitative detemiacy (i.e., the same sese, the same dimesios): But the segmet (qualitative detemiacy of the segmet x ) = (qualitative detemiacy of the segmet of the axis x ca ot lie o the axis O x (i.e., the segmet O x ). x ca ot be coicided with a segmet of the axis O x ) because these segmets have diffeet dimesios ad, theefoe, diffeet qualitative detemiacy (i.e., diffeet seses). This statemet is expessed by fomallogical law of absece of cotadictio: (qualitative detemiacy of the segmet x ) (qualitative detemiacy of the segmet of the axis O x ). Cosequetly, the mathematical opeatio of fidig the pojectio of the vecto o the coodiate axes epesets the fomal-logical eo: violatio of the law of absece of cotadictio. 2. As is kow, the ule of additio of vectos havig the same qualitative detemiacy is called the tiagle ule o paallelogam ule. Stadad opeatio of additio of two vectos 4

5 is defied as follows: the sum of two vectos 1 ad 2 is called the vecto uig fom the begiig of the vecto 1 to the ed of the vecto 2 ude the coditio that the vecto 2 is applied to the ed of the vecto 1. Ude the additio of two vectos, thei pojectios o a abitay axis ae added, ad ude the multiplicatio of a vecto by ay umbe, its pojectio o a abitay axis is multiplied by this umbe. I my view, these stadad assetios ae ot fee fom objectio. The objectio is that the stadad assetios ae cotay to the fomal-logical laws. Really, segmets of vectos ad segmets of abitay axis have diffeet qualitative detemiacy (i.e., diffeet seses). This implies that the segmets of the vectos 1 ad 2 ca ot lie o a segmet of a abitay axis (i.e., the segmets of the vectos ca ot coicide with a segmet of a abitay axis). Fom the poit of view of fomallogical law of idetity, these segmets ca be coicided if oly they have idetical qualitative detemiacy (i.e., the same dimesio, the same meaig). 3. As is kow, the scala poduct of two vectos ad F is defied as follows: (a) oe bigs the iitial poits of vectos i coicidece with each othe (i.e., the iitial poits ae coected); (b) oe postulates the elatio F = F cos ϕ whee the poit betwee symbols of vectos deotes the opeatio of scala multiplicatio of vectos, ϕ is agle betwee the vectos. The expessio cos ϕ epesets a deomiate umbe: the pojectio of the vecto o the vecto F. Also, the expessio F cos ϕ epesets a deomiate umbe: the pojectio of the vecto F o the vecto. I my opiio, the stadad defiitio of the scala poduct of vectos is ot fee fom objectio. The objectio is that the stadad defiitio is cotay to the fomal-logical laws. Really, the coicidece (coectio) of iitial poits of vectos ad the fomatio of the pojectios imply that the dimesio of legth (i.e., the qualitative detemiacy) of the vecto is idetical to the dimesio of legth (i.e., the qualitative detemiacy) of the vecto F : (qualitative detemiacy of the vecto ) = (qualitative detemiacy of the vecto F ). I geeal case, howeve, the dimesios of vecto legths ae diffeet. Theefoe, these vectos ca ot have a commo poit, ad the multiplicatio ca ot be pefomed (i.e. the multiplicatio has o sese). This fact is expessed fomal-logical law of absece of cotadictio: (qualitative detemiacy of the vecto ) (qualitative detemiacy of the vecto F ). Cosequetly, the mathematical opeatio of scala poduct of two vectos epesets a fomallogical eo: a violatio of the law of absece of cotadictio. 4. As is kow, the coss-poduct of two vectos ad F is defied as follows: (a) oe bigs the iitial poits of vectos i coicidece with each othe (i.e., the iitial poits ae coected); (b) oe postulates the elatio 5

6 H F = h F si ϕ whee the coss betwee the symbols of vectos deotes the opeatio of vecto multiplicatio of vectos, ϕ is agle betwee the vectos, H is vecto which is omal to the plae fomed by the vectos ad F ; h is uit vecto which is omal to the plae. Ude the established ageemet, the diectio of vectos H ad h is detemied by the ight-had scew ule. I my opiio, the stadad defiitio of the coss-poduct of vectos is ot fee fom objectio. The objectio is that the stadad defiitio is cotay to the fomal-logical laws. Really, the coicidece of the iitial poits of the thee vectos meas that the dimesios of legths (i.e., the qualitative detemiacy) of the vectos, F, ad H ae idetical: (qualitative detemiatio of the vecto ) = (qualitative detemiatio of the vecto F ) = (qualitative detemiatio of the vecto H ). I geeal case, howeve, the dimesios of the legths of the vectos ae diffeet. Theefoe, these vectos ca ot have a commo poit, ad the opeatio of vecto multiplicatio ca ot be pefomed (i.e., the opeatio of multiplicatio has o sese). This fact is expessed fomallogical law of absece of cotadictio: (qualitative detemiacy of the vecto ) (qualitative detemiacy of the vecto F ) (qualitative detemiacy of the vecto H ). Cosequetly, the mathematical opeatio of the coss-poduct of vectos is a fomal-logical eo: a violatio of the law of absece of cotadictio. Discussio 1. As is kow, the cofidece i the scietific method of eseach ad i atioal thikig eplaced all othe ways of cogitio i the 20th cetuy. Ratioal thikig epesets the geatest achievemet of makid. Ratioalizatio of thikig ad of sciece is dialectical impeative of ou time. The developmet of atioal thikig i the 21st cetuy leads to citical aalysis, ecosideatio, ad atioalizatio of the geeally accepted theoies ceated by the classics of sciece (fo example, N. Boh, E. Schödige, W. Heisebeg, A. Eistei, I. Newto, G. Leibiz, L. Eule, J. Lagage, A. Cauchy, W.R. Hamilto, J.W. Gibbs, O. Heaviside, etc.). Ratioalizatio ad citical aalysis of sciece ae two side pieces (compoet factos) i pogess of sciece. Citical aalysis ad atioalizatio of theoies ae based o fomal-logical aalysis of scietific cocepts, of the completeess of cocepts, of the completeess of a system of cocepts because oly the completeess leads to claity (Cofucius). Recetly, idepedet eseaches give attetio to citical aalysis of theoetical physics, mathematics, biology, etc. (see, fo example, I the pocess of citical aalysis ad of itepetatio of scietific theoies,...we ca hadly ely o ay of the old piciples eve if they ae vey commo. The oly madatoy equiemet is the absece of logical cotadictios. (N. Boh). Logical cosistecy of theoies is achieved with use of he fomal-logical laws. Ad a atual-scietific itepetatio of theoies is based o the use of atioal dialectics. The system of uivesal (geeal-scietific) cocepts ad laws i.e., sciece of he geeal laws of developmet of the Natue, huma society, ad coect thikig is the 6

7 uity of fomal logic ad atioal dialectics. This uity is ot oly coect methodological basis of sciece but also the coect methodological basis fo a citical aalysis of theoies. 2. The oigi of vecto calculus is closely elated to the eeds of mechaics ad physics: the idea of motio, the cocepts of pocess, velocity, acceleatio, displacemet, foce, ad vecto wee itoduced ito mathematics i the 17-18th cetuies. The mode meaig of the wod vecto epesets geealizatio of its pevious (out-of-date) meaig i astoomy, whee, i 18th cetuy, a vecto is called a imagiay staight lie segmet coectig the plaet to the cete (focus) of the motio. At peset, vecto calculus is a bach of mathematics i which oe studies the popeties of opeatios o vectos. But the mathematical fomalism does ot cotai motio, mathematical pocess. A mathematical pocess is caied out oly i computes. (This is why cotiual mathematics must be eplaced by discete mathematics compute mathematics). I specific scietific poblem, oe cosides the quatities of the vaious atue. These quatities have diffeet dimesios: legth, aea, volume, weight, tempeatue, speed, stegth, etc.). If oe selects a (defie, explicit, appoited) detemied uit, the each value of the quatity must be expessed by deomiate umbe. But mathematics does ot coside the specific quatities: the mathematical popositios ad laws ae fomulated, abstactig fom the specific atue of the quatities, takig ito cosideatio oly thei umeical values. I lie with this, mathematics cosides the quatity i geeal, the vecto i geeal, ad so o, eglectig the atual-scietific meaig of the quatity. Abstact mathematical popositios, theoies, ad models ca ot be tested ad used i the atual scieces. Fom the poit of view of fomal logic ad of atioal dialectics, i ode to test ad use mathematical popositios, theoies, ad models i pactice, it is ecessay to defie the atual-scietific (pactical) meaig of mathematical cocepts (objects) ad elatios, i.e., to coside ot a quatity i geeal, a umbe i geeal, vecto i geeal, but to coside the atue (i.e., dimesios) of quatities (legth, aea, volume, weight, tempeatue, speed, acceleatio, displacemet, foce, etc.). Fom this poit of view, the stadad vecto calculus does ot have a atual-scietific meaig because the stadad vecto calculus is based o the cocept of vecto i geeal. Claificatio of atual-scietific meaig of cocept of vecto ad a logical aalysis of opeatios o the physical vectos show that the stadad popositios of vecto calculus, elatig to the physical vectos, ae cotay to fomal logic. 3. Thee ae two opiios about the existece of logical eos i geeally accepted theoies (fo example, i physics ad mathematics). The fist opiio is that, although a theoy (fo example, the special theoy of elativity) cotais logical eos, it woks well (Gead 't Hooft). The secod opiio is that the system of fou fudametal fomal-logical laws is icomplete ad isufficiet fo a pachesto (i.e., fo complete explaatio) ad mathematical desciptio of eality. I essece, these opiios ae idetical. Howeve, i my opiio, these views ae ot fee fom objectio. The objectio is as follows. If oe will discove additioal fomal logic laws, the the complete system of laws should ot be cotadictoy: the fou basic laws will etai its place ad impotace i a ew, complete system (i othe wods, the fou basic laws will ot be efuted). I this case, the theoies that ae eoeous i icomplete logical system will also be eoeous i the complete logical system. Ad the theoies that cotai logical eos ae false i essece. But the followig questios will always emai ope: Why devices that ae based o false scietific theoies (ideas) wok? Why do the false scietific theoies cotibute to the developmet of makid? Whee is the limit of developmet based o false theoies? What is the dage of developmet based o false theoies? What ae the essece ad pedestiatio of developmet? Coclusio 7

8 Thus, the fomal-logical ad dialectical aalysis of the foudatios of vecto calculus leads to the followig mai esults: the stadad vecto calculus is icoect theoy because (a) it is ot based o the coect methodological basis: the uity of fomal logic ad of atioal dialectics; (b) it does ot cotai the coect defiitios of cocepts of movemet, diectio, ad vecto ; (c) it does ot take ito cosideatio the dimesios of physical quatities (i.e., umbe ames, deomiate umbes, cocete umbes), chaacteizig the cocept of physical vecto, ad, theefoe, it has o atual-scietific meaig; (d) opeatios o physical vectos ad the theoetical popositios of the stadad vecto calculus, elatig to the physical vectos, ae cotay to fomal logic. Refeeces [1] T. Apostol. Calculus, ol. 1: Oe-aiable Calculus with a Itoductio to Liea Algeba. Joh Wiley ad Sos. ISBN , (1967). [2] T. Apostol. Calculus, ol. 2: Multi-aiable Calculus ad Liea Algeba with Applicatios. Joh Wiley ad Sos. ISBN , (1969). [3] Kiyosi Ito. Ecyclopedic Dictioay of Mathematics (2d ed.), MIT Pess, ISBN , (1993). [4] A.B. Ivaov. "ecto, geometic", i Hazewikel, Michiel, Ecyclopedia of Mathematics, Spige, ISBN , (2001). [5] D. Pedoe. Geomety: A compehesive couse. Dove. ISBN , (1988). [6] R. Ais. ectos, Tesos ad the Basic Equatios of Fluid Mechaics. Dove. ISBN , (1990). [7] R. Feyma, R. Leighto, ad M. Sads. "Chapte 11". The Feyma Lectues o Physics, olume I (2d ed ed.). Addiso Wesley. ISBN , (2005). [8] Mechaics. Bekeley physics couse.. 1. McGaw-Hill book compay, (1964). [9] T.Z. Kalaov. The Citical Aalysis of the Foudatios of Theoetical Physics.Cisis i Theoetical Physics: The Poblem of Scietific Tuth. LAP Lambet Academic Publishig. ISBN , (2010). 8