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ERNANGALILEANMcikWLLIN*IDEALIZATIONReally powerful explanatory laws of the sort found in theoretical physics do notstate the truth . . . We have detailed expertise for testing the claim of physics aboutwhat happens in concrete situations. When we look to the real implications of ourfundamental laws, they do not meet these ordinary standards . . . We explain byceterisparibus laws, by composition of causes, and by approximations that improveon what the fundamental laws dictate. In all of these cases, the fundamental lawspatently do not get the facts right.’IN GALILEO’Sdialogue, The New Sciences, Simplicio, the spokesman for theAristotelian tradition, objects strongly to the techniques of idealization thatunderlie the proposed ‘new science’ of mechanics. He urges that they tend tofalsify the real world which is not neat and regular, as the idealized laws wouldmake it seem, but complicated and messy. In a provocatively titled recentbook, Nancy Cartwright argues a similar thesis, although on the basis of verydifferent arguments to those of Simplicio. Her theme is that the theoreticallaws of physics, despite their claims to be fundamental truths about theuniverse, are in fact false. They do have broad explanatory power, and thereinlies their utility. But explanatory power (in Cartwright’s view) has nothing todo with truth; indeed, the two tend to exclude one another. Idealization inphysics, though permissible on pragmatic grounds, is thus not (as the Galileantradition has uniformly assumed) truth-producing.In this essay, I plan to review some of the characteristic techniques of whatmay broadly be called ‘Galilean idealization’, and to inquire briefly into theirepistemic implications in the natural sciences. I will leave the issues raised bythe connected topic of the composition of causes to another paper, in theeffort to get straight first what sorts of idealization the ‘new science’ did usherin. My approach will be conceptual-historical.I will make use of texts,mainly from Galileo, in order to clarify the various sorts of ‘idealizing’ moves.Although I will be concerned on occasion to assign historical responsibilitiesfor these moves, my main intent is not the historical one of inquiring into theorigins of the ‘idealizing’ technique. This would bring us back through thelong story of the methods of analysis and synthesis, as these were employed inthe Renaissance and Middle Ages, to the abstractive theories of Aristotle and*University of Notre Dame, Program in the History and PhilosophyIndiana 46556, U.S.A.’ N. Cartwright,How the Laws o/PhysicsLie (Oxford: ClarendonStud. Hist. Phil. Sci., Vol. 16. No. 3 pp. 247-273,Printed in Great Britain.2471985.of Science, Notre Dame,Press,1983), p. 3.0039-368118553.00 0.000 1985 PergamonPress Ltd.

Studies in History and Philosophy of Science2-Mto Plato’snotiondiscoveringdecidingof Form.My aims ratherwhat the techniquesuhetherare, first, the systematicwere and, second,they need be inimicalthe epistemologicalto the truth-likenessone ofone ofof science.The term, ‘idealization’,itself is a rather loose one. I shall take it to signify adeliberate simplifyingof something complicated(a situation,a concept, etc.)with a view to achieving at least at a partial understandingof that thing. It mayinvolve a distortionof the original or it can simply mean a leaving aside ofsome componentsin a complex in order to focus the better on the remainingones. The point of idealization(but here is the great divide between thePlatonicand Aristoteliantraditions)is not simply to escape from theintractableirregularity of the real world into the intelligible order of Form, butto make use of this order in an attempt to grasp the real world from which theidealizationtakes its origin.I have includedseveral techniquesbelow that are not usually called‘idealization’,but which do, nevertheless,qualify under the definitionsabove.By labelling them ‘Galilean’,I do not mean to imply that Galileo invented allof them or even that he had a major responsibilityfor all of them. The firstone, as we shall see, is much older than Galileo. Nonetheless,I think it islegitimate to group them together as a unit under his name, because each ofthem playeda distinctivepart in shaping1. MathematicalWhen Galileowas trying to establishthe ‘new science’at its origins.Idealizationthe Copernicandoctrineof the motionof the earth in hisSwoChief World Systems, the most serious objection hefaced is clearly and honestly stated in the Second Day of the dialogue. Sinceobjects placed near the edge of a spinning horizontal wheel tend to fly off on atangent, why don’t objects on the earth do the same if (as Copernicus holds) itis spinning on its axis? Galileo’s spokesman,Salviati, in response employs anelaborate geometricalanalysis of the ‘horn angle’ between tangent and curvein an attempt to show that the initial departure of the curve from the tangentline is so small that even the slightest tendency on the part of objects to falltowards the center of the earth will be sufficient to ‘bend’ their path into thecircle required to keep them safely anchored to the earth’s surface, no matterhow fast the earth may be rotating. The argumentis, as we know, fallaciousand its conclusionwrong. Galileo did not have the dynamic concepts nor thetechniquesof differentiationhe needed to resolve the challenge.Our interest here, however, is in Simplicio’s response. (Recall that Simpliciois enunciatingwhat Galileo took to be the objectionsthe Aristoteliansof hisday would be likely to express.)After all, Salviati, these mathematical subtleties do very well in the abstract, butthey do not work out when applied to sensible and physical matters. For instance,

Galilean Ideakarion249mathematicians may prove well enough in theory that a sphere touches a plane at asingle point, a proposition similar to the one at hand; but when it comes to matter,things happen otherwise. What I mean about these angles of contact and ratios isthat they all go by the board for material and sensible things.’Simpliciois challengingGalileo’splan!o make geometrythe languageofphysics. In his view, the Book of Nature is nof written in the language ofmathematics,as Galileo had ebullientlyasserted in The Assayer ten yearsbefore.’ Geometryis an abstraction,an idealization.It leaves aside thequalitative detail that constitutesthe physical singular as physical. How thencan it serve as the language of a science of nature? This is a fundamentalobjection to the entire Galilean program. And it has been voiced again andagain since Galileo’stime. Bergson and Husserl are only two of thedistinguishedphilosopherswho have pressed it in our own day.It may be worth asking in parenthesiswhether this objection would havebeen voiced in quite the same way by Aristotle himself. He did, of course,separate mathematicsquite sharply from physics, partly on the basis of thedegree of abstraction(or idealization)characteristicof each. Physics abstractsonly from the singularityof the changingconcrete object; mathematicsabstracts in addition from qualitativeaccidents and change.” A physics thatborrows its principlesfrom mathematicsis thus inevitablyincompleteUSphysics, because it has left aside the qualitative richness of Nature. But it is noton that account distortive, as far as it goes.Mathematics,in Aristotle’s eyes, is a science of real quantity; it is a scienceof the quantitativeaspect of the real word, not just of a postulated constructrealm.s This is why he can use it so freely in his physics, in his discussions ofthe continuum,of falling motion, of planetary motions, to mention only someof the more obvious examples. Even in the disorderly sublunaryrealm, it stillfurnishesa reliableMeteorologymodeof analysis,takes for granted.as the analysisFurthermore,Aristotleof the rainbowhas no objections,in hison’ Drake translation(Berkeley: University of CaliforniaPress, 1967), p. 203; Opere, 7, p. 229.The translationsare in some cases slightly modified.’ The Assayer, in The Controversy on fhe Comers of 1618, S. Drake (translation)(Philadelphia:University of PennsylvaniaPress), p. 183; Opere 6, p. 232. Neither Plato norAristotle could have allowed this aphorismto pass, but for quite different reasons. Aristotle’sreason (apart from a likely hesitation over the theological implicationsof the ‘Book’ metaphor)would not be that the language of mathematicsis inappropriateto the ‘Book’, rather that it is notthe only (nor the primary) language in which the Book is written.’ Aristotle, Metaphysics VI, 1; Physics, II, 2.’ Some of the formulae he uses are still Platonic in overtone and imply the characteristicPlatonic separationbetween mathematicsand the sensible order. But it seems fair to say that hismatureview is that mathematicsdeals not with separableForms but with quantitativecharacteristicswhich exist only as embodied in matter. For a full discussion,see A. Mansion,Introducrion ci la physique aristotelicienne (Louvain: Nauwelaerts,2nd ed. 1946, ch. 5).

250Studiestruth-grounds,harmonics.of theto theorderembodiment‘mixedThey are not physics,of Nature.in matterButsciences’in Historyand Philosophyof astronomy,of course;it wasthey cannotPlato,notwas a bar to the properoptics,of Sciencemechanics,give a full explanationAristotle,realizationwhosaidthatof geometricalconcepts.The long separationbetweennaturalphilosophersand the exponentsof the‘mixed sciences’ through the Middle Ages and Renaissancecontributedto ashift in the original Aristotelianposition, one which may have been augmentedby the influenceof neo-Platonism.One notes a growing distrust amongnatural philosophersof broadly Aristoteliansympathiesfor those who usemathematicsin the context of physical problems.6 The story is a complicatedone; the only reason to allude to it here is to suggest that Galileo was notwrongin attributingto Simpliciothescepticismconcerningtheroleofmathematicsin science of nature that did, in fact, characterizemany of theAristoteliansof that day.’Salviati’s response to Simplicio is to point to an ambiguity in his objection.If he means that matter is such that when a sphere is realized in it, it may toucha plane at more than one point, this is demonstrablyfalse. If on the otherhand, he means that perfect spheres are never, in fact, realized in Nature, or asSimplicio puts it, that ‘a metallic sphere being placed upon a plane, its ownweight would press down so that the plane would yield somewhat’,s then thismay well be true. But from this it does not follow that ifsuch a sphere were tobe realized in Nature, it would not have the properties that geometry demandsof it. Matter cannot alter those properties;it merely makes them difficult toreproduce exactly. Just as the businessmanmust allow for boxes and packingsin computingthe real from the observedweights of his wares, so the‘geometricalphilosopher’:’ See J. Weisheipl, The Developmentof Physical Theory in the Middle Ages, London: Sheedand Ward, 1959.’ In his book, Explanatory Sfructures (London: Harvester, 1978), Stephen Gaukroger arguesthat for Aristotle“mathematicsis simply not applicableto reality”(p. 202), thus makingSimplicio’s objection a properly ‘Aristotelian’one. He goes on to propose this as the fundamentalcharacteristicof the old and the newpoint of separationbetween the “explanatorystructures”sciences. (Yet he himself recalls in some detail Aristotle’s use of mathematicswhen treating of thespeed of fall, p. 211). The roots of his misunderstandingmay lie in his characterizationof‘abstraction’,which he somehow sees as being contrastedwith ‘reality’. “If we are merely dealingwith abstractions,we are dealing with a situation which is not ‘real”’ (p. 222). This leads him tocontrast concept formationin Aristotle’s physics (abstractive)with that in Galileo’s physics (nonabstractive).Indeed the latter “is specified in terms of state-variableswhich may be quite alien toeveryday experience”(p. 221). Yet he does not show that the two concepts, distance and time, interms of which Galileo formulateshis laws, are different from the corresponding‘abstractive’concepts of Aristotle.He thinks they have to be, or else they will not grasp the ‘real’. But forAristotle, they did grasp the real, even though only the quantitativeaspect of it.’ Dialogue. p. 206; Opere, 1, 233.

Galileanwhen251Ideali:ationhe wantsto recognizein the concretethe effectswhichhe has provedin thefor the impediments of matter, and if he is able to do so, Iassure you that things are in no less agreement than are arithmetical computations.The errors lie, then, not in the abstractness or concreteness, not in geometry orabstract,mustallowphysics as such,accounts.’but in a calculatorwhodoesnot knowhow to keep properThis is a good response. An ‘impediment’is not something which prevents,or lessens the force of, the applicationof mathematicsto nature. Rather, itindicatesa practicaldifficultyin realizingthe simple relationsof themathematicalsystem within the complexity of the material order.” How arewe to know to what extent they are realized?By ‘allowingfor theimpediments’:the assumptionis that impedimentsof this sort can be allowedfor, that is, that their effects can be calculated.The groundsfor thisassumptionare inductive. We can see whether in practice the science we baseon it can be made to work for complex real situations.This response couldhave been given as easily by Aristotle as by Salviati.”Salviati, it must be admitted,is not entirely single-mindedin maintainingthis view. He sometimes lapses back into a Platonic pessimism about “theimperfectionsof matter, which is subject to many variations and defects”, andis “capableof contaminatingthe purest mathematicaldemonstrations”.”But if this were the case, the Book of Nature would not really be written in thelanguage of mathematics,or would, at least, be poorly written. Salviati muchmore commonlyseems to take for granted that realizationin matter is not abarrier to intelligiblZFyin geometricterms, and that the c

GALILEAN IDEALIZATION ... *University of Notre Dame, Program in the History and Philosophy of Science, Notre Dame, Indiana 46556, U.S.A. ’ N. Cartwright, How the Laws o/Physics Lie (Oxford: Clarendon Press, 1983), p. 3. Stud. Hist. Phil. Sci., Vol. 16. No. 3 pp. 247-273, 1985. Printed in Great Britain. 247 0039-3681185 53.00 + 0.00 0 1985 Pergamon Press Ltd. 2-M Studies in History and ...

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