Achot T. Grigoryan (U.S.S.R.)
MEASURE, PROPORTION AND MATHEMATICAL STRUCTURE
/OF GALILEO’S MECHANICS
The subject of m easure and proportion as an object of scientific anal ysis in Galileo’s works is closely connected w ith th e broadest and most general problems of th e history of th e 17th-century science. W hen did the classical science arise? W hat concepts an d notions designated its origin? In this respect the concepts of uniform motion of a body le ft to its own resources and of uniform ly accelerated motion of a freely fal ling body, on the one hand, and methods of experim ental an d q u an tita tive m athem atical investigation of nature, on th e other, a re usually re garded as th e m ost characteristic. The synthesis of th e concepts m en tioned is most typical of th e 17th century. These components of th e new science did exist earlier, b u t th e synthesis changed th e character of both altogether. The concepts of uniform and uniform -difform m otion w ere used by the Paris nominalists. In th e 17th century and, first of all, in th e Discorsi (partly in th e Dialogo already) they became quantitative characteristics of m otion and m ade th e quantitative m athem atical a p proach to it possible in principle. Bacon announced th e decisive p a rt of experim ental science, experim ent had m uch success w ith G ilbert and a long story in the 15th and 16th centuries. B ut only w ith Galileo it became a quantitative experim ent answ ering n ot only th e question “is it so?”, “is it not so?” and “w hat’s th e cause of this o r th a t phenom e non?” but most often th e question of “equal—unequal?”, “equal to w hat?”, “how m uch is this?” With Galileo m athem atics had n o t yet become an apparatus of science b u t all th e necessary prerequisites for it had been created.
We are going to discuss some peculiarities of the m athem atical struc tu re of Galileo’s mechanics in connection With this fundam ental problem of the history of th e origin of classical science.
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not only connected w ith th e aim of th e most im portant experim ents of Galileo. It was also connected w ith his basic idea. For A ristotle a n d his followers to define motion was to rep ly to the question: is a body in its n atu ral place (in case of “n atu ra l” motion) or had it been pushed forw ard (in case of “forced motion)? Galileo defines motion by its veloc ity which characterized w ith A ristotle not the motion itself b u t rath e r th e form of a moving body and th e medium in w hich it was moving. T hat is w hy the m easurem ent of velocity is of fundam ental im portance to- Galileo in th e study of motion. In his Dialogo he speaks mostly ab o u t th e unchangeable absolute velocity, about the uniform motion of cos mic bodies participating in th e daily m otion of th e Earth. The velocity of uniform motion becomes an object of the analyses. The ques tion here is th e proportionality of the w ay covered and tim e pas sed. This concept of direct (and inverse) proportionality was fa m iliar to Galileo’s predecessors. B ut Galileo introduced a princi pally new thing as com pared w ith all his forerunners; in fact Galileo was the first to introduce quite distinctly (and opposing the new no tion to the traditional definitions) the concept of the proportionality for all values of time. In th e Discorsi th e re is a definition of uniform m otion (distances covered in an y equal periods of tim e are equal) and the following addition to it: “Visam est addere veteri difinitioni (quae sim pliciter appelat m otum aequabilem dum tem poribus aequalibus tra n - siu n tu r spatia) particulam quibuscunque, hoc est omnibus tem poribus aequalibus: fieri enim potest, u t tem poribus aliquibus aequalibus mo bile p ertran siat spatia aequalia, dum tam en spacia transaeta in parti'bus eorundem tem porum minoribus, licet aequalibus aequalia non sin t” <(G. G., VIII, 191).
In such a definition of uniform motion a m ore general concept o f accelerated motion is anticipated. One can say, to use modern term s, th a t th e mom entaneous constant velocity is introduced in connection w ith th e possibility of changing velocity.
As soon as a certain change of velocity comes into play, a uniform velocity becomes th e object of changes, th e object undergoing changes, i.e. the object of analysis. This signifies a great advance: before Galileo the object of search for active causes had been the change of co-ordinates. W ith Galileo velocity became such an object, th a t is to say, proportions of tim e and space.
Galileo does not seek a physical cause of the change of velocity, but neither does h e dismiss th e concept of such a cause for uniform mo tion as it was done in th e Dialogo. He discovers the law determ ining th e change of velocity and the change of th e way. The velocity of a freely falling body is in proportion to the time, th e w ay to th e square of time. This square-law immediately enlarges the num ber of physically valid abstract formulas of proportionality. Before Galileo such w ere the
direct and inverse proportionalities only. N either Galileo nor N ewton or the mechanicists of th e 18th century introduced th e concept of ac celeration—it appeared as late as th e beginning of th e 19th century. T h e velocity is th e object of changes for Galileo. The law of such a change under uniform ly accelerated motion is th e law of a change w hich has not y e t gained the form of costancy of a certain proportion. The propor tionality of force and acceleration became the fundam ental thesis in post-N ew tonian classical mechanics. This constancy of proportion or, in o th er words, constancy of mass seemed to be a n absolute tru th u n til Einstein. H ere an unalterable proportion becomes an unalterable m eas ure—a dynam ic variable.
The ratio of th e covered space and passed tim e has already become a n independent quantity—co n stan t (as in the Dialogo) o r variable (as in th e Discorsi). One can say th a t velocity has become w ith Galileo already m easure (before Galileo only space and tim e b u t n o t th e ir proportion w ere measured), and acceleration still rem ained proportion and had n o t y e t become an independent local predicate characterizing a body. That is w hy Galileo spoke of a n unalterable proportion in case of uniform mo tion b u t did not speak of it in case of uniform ly accelerated motion. W e discover two basic ideas of proportionality in Galileo’s mechanics:
1) A moving body left to its ow n resources covers portions of space
which are in proportion to time.
2) A body undergoing “n atu ra l” movement, i.e. freely, covers por
tions of space w hich a re in proportion to square of time.
The first idea is characterized w ith converting proportion into m eas ure: the proportion of space and tim e becomes an independent variable— an object of causally explained alterations. The second idea had not y et acquired this character: th e proportion of space and square of tim e had n ot so fa r been converted into a certain m easure—acceleration. Ga lileo does n ot exam ine th e alterations of this acceleration. T hat is w hy acceleration as a subject of changes does not figure in Galileo’s m echan ics.
Later on different kinds of functional dependence lost their specific physical sense and w ere used in an ab stract way. B ut w ith Galileo th e m athem atical categories figure as definitions of motion. In this concrete form we can clearly see the connection betw een m easure and propor tion. Preservation of a certain qu an tity of a certain m easure (of veloc ity—explained in th e Dialogo, of acceleration—im plied in the Discorsi) is form ulated as an im m ediate resu lt of preservation of proportionality expressing a physical law.
A physical law is expressed in a one-to-one correspondence of th e tw o sets: th e set of positions of a body and th e set of instants. Since motion is continuous, we have one-to-one correspondence of the tw o in fin ite sets. The finite trajectory of a particle becomes a n infinite set of
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points an d th e finite tim e of motion becomes an infinite set of instants. Laws of mechanics give us a one-to-one correspondence betw een the elements of these sets.
Obscure as they m ay appear, these internal m athem atical an d logi cal peculiarities of Galileo’s dynamics enabled him to express some very profound ideas about infinite sets.
Galileo devoted m any pages in th e Discorsi to infinity (infinity as a result of an infinite division of a finite quantity). One of the most im portant ideas here is th e assertion of th e impossibility to apply con cepts discovered in the study of finite sets to infinite ones.
Simplicio pays atten tio n in the Discorsi to a paradoxical featu re of segments composed of a n infinite num ber of points: one infinite set can be bigger th an another. Salvaiti’s answ er is: “Q ueste son quelle difficol- tä che derivano dal disoorrer ehe noi facciamo, col nostro intellecto fi- nito intorno a gPinfiniti, dandogli quelli attrib u ti che noi cLiamo alle cose finite e term inate...” (G. G., VIII, 77-78).
Galileo did not know th a t w hen th e notions “bigger” and “sm aller” a re generalized and th e notion of M ächtigkeit introduced, th e problem of com parison of infinite num bers can be solved n o t only in the nega tive (impossibility to apply th e logic of finite num bers) b u t also in th e positive. B ut he saw th e beginning of the way leading later in the works of Cantor to the positive solution. Salviati gives an extrem ely in terest ing exam ple of such idea. The set square is sm aller than the set of all num bers (not all th e num bers are squares), b u t is equal to th e set of roots (each square has its root), and th e set of roots is equal to th e set of all num bers (each num ber can be a root). Here it can be seen that Galileo does not give a positive solution of th e problem b u t sees th e ini tial w ay to such a solution. He passes from vain searches for an actual infinity as a calculated innum erable set, which had been fairly freq u en t before Galileo, to the comparison of sets according to th e one-to-one correspondence of th e ir elements (“each square has its root”, “each num ber can be a ro o t”).
Coming back to Galileo’s dynamics we discover physical prototypes of a sim ilar trend. A physical variable is measure, w hich had not broken aw ay w ith proportion. M easure itself became an independent object of analysis (velocity) or, explicit, stays w ithin limits of proportion (accele ration). B ut in all th e cases one deals w ith the comparison of sets: in
stan ts and positions, instants and velocities. r
-The ultim ate physical m eaning of Galileo’s ideas about infinity is th e fact th a t they preserve the “birthm ark^” of th eir origin and had n o t so far become so ab stract as th ey did later, in the explanation of one more characteristic point. We find out in Galileo’s ideas some very vague conjecture about infinitesim al quantity as a variable— about “un terzo medio termine, che ä il rispondere ad ogni segnato num ero.” In
other words, it is a qu an tity w hich can be equal to an y p art of th e whole, to its infinitesim al p a rt as well.
Proportion is a n early pseudonym to the la ter simple functional de pendence, expressing th e principal absolute trustw orthiness of scientific statem ents about th e position of a particle in this or th a t point at the given moment. This trustw orthiness is the basis of classical notions about nature. W ith Galileo m athem atics is n ot only an instrum ent of science b u t also a criterion of its trustw orthiness.
In th e w ide-know n lines of the Dialogo devoted to th e extensive an d intensive trustw orthiness of scientific knowledge Galileo says: “L’in- tellecto umano ne intende alcune cosi perfettam ente, e n e ha eosi assolu- ta certezza, quanto se n ’abbia l’istessa n atu ra; e ta li sono le scienze m atem atiche pure, cioe la geom etria e l’aritm etica” (G. G., VII, 128-129). M athem atical cognition, Galileo proceeds, “arriv a a com prendere la ne- cessitia, sopra la quale non p a r che possa esser sicudezza m aggiore.”
Necessity is a physical law ; it is expressed in th e form of constancy of some proportions and measures w hile oth er ones do alter. Thus, th e concepts of Galileo m entioned above engendered, to some extent, th e idea of a univocal determ ination of all th e processes in nature.
