cog TE- E JOURNAL OF TFtE LEARNEti J SC]ENC -S 6.11. 3-40 Capynght 0 1997,Lawrence Erlbaum AsseciareS, Inc. Analogical Reasoning and Conceptual Change : A Case Study of Johannes Kepler Dedre Gentner and Sarah Breit DepartmentofPsychology Northwestern University Ronald W. Ferguson Department ofComputer Science Northwestern University Arthur B. Markman DepartmentofPsychology Columbia University Bjorn B. Levidow ConnectSoft, Inc_ Seattle, Washington Phillip Wolff Department ofPsychology Northwestern University Kenneth D. Forbus Department ofComputer Science Northwestern University The work of Johannes Kepler offers clear examples of conceptual change . In this article, using Kepler's wrirk as a case study, we argue that analogical reasoning Requests for reprints should he sent to Dedre Georner, Department or Psychology, Northwestern Universal y, 2029 Sheridan Road, Evanston, IL 60208-2710. Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED 1997 2. REPORT TYPE 00-00-1997 to 00-00-1997 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Analogical Reasoning and Conceptual Change: A Case Study of Johannes 5b. GRANT NUMBER Kepler 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Northwestern University ,Qualitative Reasoning Group, Department of REPORT NUMBER Computer Science,1890 Maple Avenue,Evanston,IL,60201 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE 38 unclassified unclassified unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 4 GEN7NER Er AL. 3 facilitates change of knowledge in four ways: (a) highlighting, (b) projection, (c0 irrept-escalation, and (d) restructuring.We present these fourmechanismswithin the context of structure-mapping theory and its computational implementation. the structure-mapping engine- We exemplify these mechanisms using the extended analogies Keplerused in developing a causal theory of planetary motion. The roads by which men arrive attheir insights into ceEesdal matters seem to me almost as worthy of wonder as those matters in themselves, —Johannes Kepler (as cited in Koestier, 1963. p. 261) Analogy is an important mechanism of change of know] edge- Researchers studying transfer of learning have shown that analogies to prior know [edge can foster insight into new material (Bassok, 1990 ; Bassok & Holy oak, 1989; Carrambone & Hoivoak, 1989; Dunbar, 1994; Forbus, Gerstner, & Law, 1995: Gerstner & Gent-her, 1983; Gentrier,Rattciivann, & Forbes, 19.93; Gick & Holyaak,1980, 1983; Holyoak, Juno, & Billrnan, 1984; Holyoak & Thagard, 1989 ; Keane, 1998; Novick & Holyoak, 1991; Novick & Tversky, 1987; Ross, 1987; Spellman & Ho]yoak, 1993)- These laboratory results are supported by direct and indirect observations of the scientific process. The journals of Boyle, Carnot,Darwin, Faraday, and Maxwell (and Kepler) contain many examples of generative uses of analogy (Darden, 1992 ; Gcntner, 1982; Gerstner & 1eziorski, 1993; Itiersessian, 1985, 1986, 1992; Nersessian & Resnick. 1989; Ranney & Thagard, 1988; Thagard, 1999; Tweney, 1991: Wiser, 1986; Wiser & Carey, 1983). Modern scientists like Oppenheimer (1956) and Glashow (1980) havecommented explicitly on the usefulness ofanalogy in their work. Nersessian's (1992) detailed analyses of the analogies used by Faraday and Maxwell provide evidence that analogy was useful in the development of electromagnetic field theory. Finally, direct field observations of molecular biologists at work demonstrate that analogy is frequently used in the everyday practice of science (Dunbar, 1944). Our goal in this article is to show how analogy promotes conceptual change. We first lay out four theoretically driven specific mechanisms by which analogy can act tocreatechangesin knowledge and consider the sorts of changes these processes can bring about- En particular, we ask whether analogical mechanisms can bring about changes in concepts as well as changes in the theoretical structure relating the concepts- - We draw on the works of Kepler (1571–1630) to illustrate our points. The goal of modeling the thought processes of a mind like Kepler's is daunting, to say the least. We make no claim to have captured anythingclose to Kepler's full cognitive processes- Yet, weconsider Kepler apatticularly apt subject for the study of analogy and conceptual change. First, his work spanned and contributed to a period of immense change in theory- He inherited from Copernicus a conception of the solar system in which the planets moved in perfect circles at uniform speed . By the end KEP! ER`S CONCEPTUAL CHANGE 5 of his career, hehad abandoned thissimple and beautiful view for a model in which theplanets travel in elliptical paths at nonuniform speed,with the Sun as the cause of their motion. Second, Kepler was a prolific analogizer. In his books, journals, and [criers he constantly used analogies, some only fleetingly and others with tenacious persistence. In some cases. he returned toan analogy repeatedly across different works, extending and analyzing it further on successive bouts, Third, Kepler's writings are unusually rich in descriptions of his thought processes, including fulsome descriptions of his blind alleys and mistakes_ The candor and detail of Kepler's writings helps to mitigate the problems inherent in inferring thought processes after the fact from written records. At least part of Kepler's inclusiveness seems to have stemmed from a fascination with the mental paths that led to his conceptual shifts, as evidenced by the quote at the beginning of this article. In this article, we trace Kepler's extended analogy between light and the vis marrix(a precursorofgravity) and also his further analogy between magnetism and the s'is mortis_ Our goal is to characterize the processes by which these analogies led to changes of knowledge, using structure-mapping theory as a framework . We first describe the basic theory. Then we discuss four mechanisms by which analogy brings about change of beliefs, Finally, we. apply this framework to Kepler's analogies_ STRUCTURE-MAPPING THEORY S tructtre-mapping theory (SMT; 6entner, 1983, 1989) is based on the assumption that analogy involves a process of alignment and projection . Assertions in a base (or source) domain are placed into correspondence with assertions in a target domain, and further assertions true of the base domain are then inferred to be potentially true of the target. For example, when (as we later discuss) Kepler compared the target domain of the Sun and planet to the base domain of two lodestones, he inferred that if the Sun and planet also have polarity, they may alternately attract and repel one another. depending on whether their "friendly" or "unfriendly"polesare proximate. This illustratesthepower of an analogy to provide a whole system of inferences about a novel domain_ But a mechanism for inferring new knowledge must be constrained_ To be cognitively plausible, a theory of analogical mapping must provide some natural limit to what will be inferred based on the mapping_ It must also explain the fact that some analogies and some interpretations of a given analogy are preferred over others, even when no differ- ences in factual accuracy are at stake- SMT (Gencner, 1983, 1989) and its computational counterpart, the structure- mapping engine (.ME; Falkenhainer, Forbus, & Gentner, 1989) meet this need by making strong assumptions about the nature of cognitive representation and how 6 GENTAlER ET. ._ it is used in the mapping process_ Structure mapping assumes that domain knowl- edge is in the fora, of symbolic structural descriptions that include objects, relations between objects, and higher order relations among whole propositions . On this view, the analogical process is one of structural alignment between two mental representations to find the maximal structurally consistent match between them . A strucl irafiy consistent match is one that satisfies the constraints ofparallel con- nerriviry and one-to-one mapping (Falkenhainer et al., 1989; Gentner, 1983, 1989; Gerstner & Markman, 19.93, in press; Hafford, 1993; Holyoak & Thagard, 1989; Keane, I988: Markman & Gaither, I993a, 19936; Medin, Goldstone, & Gerstner, 1993). Parallelconnectivity says that if two predicates are matched then their arguments must also match_ For example, if the predicate HEAVIER(a,b) matches die predicate. HE. A VIER(x,y) then a most match x and b must match y. One-to-cite mapping requites that each element in one representation corresponds to at most one element in the other representation. To explain why some analogiesare better than others, structure mapping uses the principle of systematicity: a preference for mappings that arc highly interconnected and contain deep chains ofhigher order relations (Forbus & Gerstner, 1989; Forbes et al., 1995; Gentner, 1983, 1989; Creamer et al., 1993). Thus, the probability that an individualmatch willbe included in the final interpretation ofacomparison is greater if it is connected by higher order relations to a common system of predicates (B owdle & Gerstner, 1996: Clement & Gerstner, 1991; Gentner & Bowdle, 1994), We focus on two predictions that derive from this framework_ First, the correspondences mandated by a comparison are governed not only by local similarity but also by the degree to which the elements play the same roles in the common higher order structure (e.g., Clement & Gentner, 1991; &entner, 1988; Gerstner & Clement, 1988; Spellman & Halyoak, 1993). Relational commonalities thus tend to outweigh object commonalities in determining the interpretation of a comparison . Second, because comparison promotes a structural alignment, differences relevant to the common structure are also highlighted by a comparison (Gerstner & Markman, 1994 ; Mark- man & Gentler, 1993a, 1993b, 1996, in press). Thus, paradoxically, comparisons can illuminate differences as well as commonalities, SME sitnu Fatesthe comparisonprocess (Falkenhainet at ale 1989; Falkenhainer, Forbus, & Gentler, 1986). To capture the necessary structural distinctions we use an nth-order typed predicate calculus.. Entities-stand for the objects or reified concepts in the domain (e.g., planet, orbit)_ Attributes arc unary predicates used to describe independent descriptive properties of objects (e .g., HEAVY(planet)). Functions' are used primarily to state dimensional properties (e.g_, BRIGHT- unlike attributes and relatiens, dv cot lake truth values but rather map objectsonto other objects or values. For brevity, we sometiines use the term pp-militatew refer to a!l twee categories: relations,atttibuies, and funaians. KEPLER'SCONCEPTUAL CHANGE 7 NFSS(Planet)), Relations me muftiplace predicates that represent links between two or more entities, attributes, functions, or relations (e.g., REPELS(lodestone-1. lodestone-2); using magnetism as the domain). To represent beliefs about physical domains, we use the qualitative process (QP) theory as a representation language (Forbus, 1984r 1990; Forbus & Gentner, 1986; Forbus, Nielsen, & Faltings, 1991; sec Forbus. 1984, for a full description of the QP language and its model building capabilities). QP them), allows the repre- sentation of qualitative proportionalities between quantities and relations_ For example, the statement QPROP + (a, b) expresses a positive qualitative relation between the quantities a and b: That that a is a monotonic positive function of (at ]east) b. (PROP — (a, b) expresses a negative qualitative.relation_ Relations can hold between expressions as well as entities. Such higher order relations allow the construction of large representational structures that can de- scribe, for example, the relation between magnetism and lodestone attraction L'vIPLIES(AND(I AGNETIC(lodcstone-1), COMPOSED-OF{filing-1 . iron)), ATTRACTS(lodestane-1, Ti ii ng-1)) It is the presence of structurally interconnected representations that is the key to implementing structure mapping_ Given two representations in working memory, SME operatesina local-to-global manner to find one or a few structurally consistent matches, In the first stage, SME proposes matches between al[ identical predicates at any level (attribute. relation, higher order relation, etc,) in the tworepresentations. At this stage, there may be many mutually inconsistent matches . In the next stage, these local correspondences arc coalesced into large mappings, called kernels, by enforcing structural consistency (one-to-one mapping and parallel connectivity). SlvfE allows correspondences between nonidentical entities and dimensions (rep- resented as functions), in accordance with the principle that lower order information need not match identically_ However, relations must match identically, reflecting the principle that comparison is implicitly directed toward finding structural commonality. For example, ATTRAC I'S(Sun, planet) may map to AT- TRACTS(magnet, nail), but can never map to COMPOSED-OF(nail, iron). In the next step, SME gathers these structurally consistent clusters into one or a few global interpretations. At this point, it projects candidate inferences into die target. It dims this by adding to the target representation any predicatesthat currently belong to the common structure in the base but are not yet present in the target_ These predicates function as possible new inferences imported from the base to the target. The mappings are given a structural evaluation, reflecting the size and depth of the system of matches. SIwfEhas many useful properties for modeling conceptual change. First, the final interpretation preserves large-scale connected structure. Second, this global inter- a QZN''rNER Fr AL. pretation does notneed to be explicit at the outset. Theassertions that willconstitute the final point of the analogy need not be present initially in the target and need not have been extracted as a separable "goat structure" or `problem-solution structure" in the base before the comparison processes begins . SMEbeginsblindly, using only local matches, and the final global interpretation emerges via the pull toward connectivity and systematicity in the later stages of the process, Third, SIvIEmakes spontaneous, structurally consistent inferences from its comparison process, unlike many other models of analogy (cf. Holyoak & Thagard, 1989; I arkrnan, 1996). Finally, this mode] of the analogy process allows us to delineate four specific subprocesses that can change conceptual structure : highlighting, projection, re- representation, and restructuring (Gentner & Wolff, in press). The FourAnalogical Processes of Conceptual Change Pa's Highlighting. first result is a matching system of predicates between the base and target. This models the psychological assumption that the process of alignment causes the matching aspects of the domains to become more salient (Elio & Anderson, 1981, 1984; Gerstner & Wolff, in press; sick & Holyoak, 1980, 1983; Markman & Gerstner, 1993a, 1993b; Medin et a]., 1993; Miller, 1979; Ortony, 1979). This process of highlighting is important because human representations, we suggest, are typically large, rich, and thickly interwoven nets of concepts_ In particular, early representations tend to be conser vative, in the sense that they retain many specific details of the context of learning: They are particularistic and contextually embedded (e.g., Brown_ Collins, & Duguid, 1989; Forbes & Gemmel., 1986; Medin & Ross,. 1989), High] ighting cari create a focus on a manageable subset of relevant information_ Moreover, the relational identity constraint, combined with rerepresentation processes, means that the output of an analogy may reveal hitherto unnoticed relational commonalities, There is considerable psychological evidence that comparison can reveal nonobvious features (Gerstner & Clement 1988; Gent- ner & lmai, 1995; Markman & Gentner, 1993a; Medin et al., 1993; Ortony, Vondruska, Foss, & Jones,1985;Tourangcau& Rips, 1991) and that highlighting of common information can influence category formation (Elio & Anderson, 1981, 1984; Medin & Ross, 1989; Ross, 1984, 1989 Skorstad, Gerstner, & Medin, 1988). Projectionof candidate inferences. As previously described, SME pro- jects candidate inferences from the base to the target domain_ These projected inferences, if accepted, addto the knowledge in the target domain_ However, not all inferences made by SME will be correct Postrnapping processes, such as the application of semantic and pragmatic constraints, arenecessary to ensure the KEPLER'S t]0NCEPTUAL CHANGE 9 correctness of the inferences (Falkenhainer, 1990; Kass, 1994: Kolodner, 1992, 1993; Novick & Holyaak, 1991)- Rereprese nation. In rerepresentation, the representation of either or both domains ischangedto improve the match. Typically, this involves akind of tinkering in order that two initially mismatching predicates can be adjusted to match. For example,suppose an analogy matches well but fora mismatch between BRJGHTE .R- THAN(x,y) and FASTER-T'HAN(a,b) (as in Kepler's analogy that is discussed later), These relations can be rerepresented as GREATER-THAN(B1 IGHT- NE'SS(4, BRIGHTNESS(y)), and GREATER-THAN(SPEEf(a), SPEED(b)) to allow comparison. This involves a kind of decomposition that separates the GREATER-THAN magnitude relation (which is common to both) from the specific dimension of increase (which is distinctive). Studiesof the development of children's comparison abilities support the psychological validity of such rerepresentation in learning: Cbi]drenare better ableto match cross-dimensional analogies when they have been induced to rerepresent the two situations to permit noticing the common magnitude increase (Gentner& Rattermann, 1991; Genmer, Rattermann, Mark-man, & Kotovsky, 1995 ; Kotovsky & Gentler. in press). We discuss SME'simplemen- tation of rerepresentation later in this article. Restructuring. Restructuring is the process of Large-scale rearrangement of elements of the target domain to form a new coherent explanation. This rearrange- ment can take the form of adding or deleting causal links in the target domain as well as altering specific concepts . It should perhaps be considered separately from the other three processes or possibly as arising from a combination of the other three. For example, when little is known about a target domain, a mapping from the base can providecasual linkages that significantly alter the connectivity in the target. However, on this account, there must besome minimal alignment as a basis for inference; even if no initial relational match exists, there must be at least a partial object mapping (which could be suggested by local similarities or pragmatically stipulated; Farbus & Oblinger, 1990; Holyoak & Thagard, 1989; Winston, 1980)- We conjecture that substantial restructuring during a single mapping iscompara- tively rare because normally the candidate inferences projected from the base domain will be at least compatible with the existing target structure, Furthermore, as Nersessian (1992) pointed out, massive restructuring from a single base can be dangerous: She noted that Faraday's modeling of magnetic fields by analogy with the concrete lines of iron filings created by magnets led to an overly concrete, partly erroneous model of the fields. In general, we suspect that most restructuring occurs as a result of multiple analogies iteratively applied as well as other processes. With these tools in hand, we now return to Kepler. We begin with some historical background, 10 GENTNtR ET A4. KEPLER AND THE SOLAR SYSTEM Kepler' (1571–1630) is best known today for his three laws of pfanetary motion,' His far more important contributions in changing our conception of the solar system aredifficult to appreciate. Ironically, this is in part because of his very success. The conceptual structure that existed prior to Kepler's work is now almost impossible for us to call forth. Medieval cosmology differed from our own not only in the specific conceptual structure but also in the character of its explanations; They sought to find mathematical regularities, not causal mechanisms_ It is here that Kepler's contribution lies. As Caspar (1993) put it_ "It is Kepler's greatest service that he substituted a dynamic system for the formal schemes of the earlier astrono- mers, the law of nature for mathematical rule . and causal explanation for the mathematics] description of motion" (p. 136), Holton (1973) went further: "Ke- pler's genius lies in his early search for a physics of the solar system, He is the first to look for a universal physical law based on terrestrial mechanics to comprehend the whole universe in its quantitative details" (p. 71). To understand the magnitude of the conceptual change involved, an account of the prior state of belief is necessary-' Western cosmology in the 16th century, c ontinuing the tradition laid down bythe Greeks. stated the laws ofplanetary motion in purely mathematical terms, It postulated a universe with the Earth at the center, 'Opinionson Kepicr'sstandingbavc'+aricd. School children are Eaugirtthat he was s mathematician who made his discoveries by trying all possible mathematical corrrbinatecris, much in the Mauler ad Langley, Bradshaw, and5i.T=l's (1983) Bacoo program- Kor-suer (1963)portrayed him as a ne0-PLa- tOnrc mystic, a "sleepwalker" who stumbled mar his discoveries by accident . Many of hisCOrniT ritarors consider that he ranks among the great scientists (e.g., Caspar, 1993; Cingericlt, 1993; Holton, 1973; Koyre, 1973; Tattlmin & Goodfiekt, 1961). Furthermore, as we make clear, he proceeded not by mechanical applicationofformulae but bythebold application ofanalogies andcausal ptincip)cs. This discussion of Kepler's work was compiled from a variety of sources_ Barker(199i, 1993), Barkerand Goldstein (1994), Basirti rdt (1952), Buutnieid (1957), Caspar (1993), Gthgerich (1993), Hansom (1958), Holton (1973), Kaestter (1963). Koyre (1973), Kuhn (1957), Layer (1984), Mason (1962), Stephenson (t994a, 19946), Toulmin andGoodfield (1461),and Vickers (1984) 3The first law stars that theorbits ofthe planets are elliptical withtheSun at one focus.The second law (chrorwlrgica]lythe fast) statesthan the equal areas are swept in equal firm by a Lineconnect-mg a planet and the Sun. The third law stales that the p•vdpct ofdabsquare of the period of a planet's revolution andthe cube or its reran dislaooc from the Sunis canstaaL `Thus account is taken chiefly from Butterfield (1957) . Hanson (1958), Koyre (1573), Kuhn (1957), Layer (t9&4), Masan (19621, Sambursk, (1975), and Taulmin and Goodfie]d{1961}, It is necessarily much abbreviated and oversimplified. There were disscneets, both among the t.',rpef ~otalyly Arista:, chus of Samos (310–230 B.C.), called 'the Copernicusof Antiquin" for his heliocentric theory (Kuhn, 1957----and in the Western scholastic tradition—ncluding William of Ockharn (1295–1349), who argued that postulating a spinning earth would rnnpbfy the explanations (an insane of Ockharri s razor), 1 uridan (c. 1497-135aJ, Albert of Saxony (e- 1360), Oame (C. 1323?-1392), and Niaatas of Curs (]405-]4s4i. Hoswever, even solid= willing tO postulate arotating earth did not generally countenance an earth that revolved arormd the Sun. KEPLER'S cor1cEFtruAI CHANGE 11 around which revolved crystalline spheres containing the heavenly bodies_ The set of beliefs laid down by Plato and Aristotle and culminating in Ptolemy's system of the 2nd century A.D. was roughly as follows: 3 , The Earth is at the center of the universe and is itself unmoving_ 2. The Earth is surrounded by physically real crystalline spheres, ' containing the heavenly bodies, which revolve around the Earth_ 3. The heavenly bodies move in perfect circles at uniform velocity, as befits incorruptible bodies. (Epicycles and eccentrically positioned circles were admitted into the system to account for the observed motions.) 4. Al] motion requires a mover,The outermostsphere, containing the fixed stars, i$ moved byan"unmoved mover," the Primurn Mobile. Each sphere imparts motion to the next one in: in the Aristotelian universe, there is no action-at-a-distance. In addition, each sphere is controlled by its own spirit that mediates its motion .' (The heavenly bodies were known not to move in synchrony,.) 5. Celestial phenomena must be explained in entirely different terms from earthly phenomena, Indeed, heavenly bodies and their spheres are made of different matter altogether. They are composed not of the four terrestrial elements—earth, air, fire, and water—but instead of a fifth element (the quintessence), crystalline aether (pure, unalterable, transparent, and weightless). The further from Earth, the purer the sphere_ This Aristotelian-Ptolemaic system was integrated with Catholic theology, largely by Magnus (1206-1280) and Aquinas (1225-1274), Angelic spirits were assigned to the celestial spheres in order of rank; The outermost sphere, that of the Primum Mobile, belongedto the Seraphim; next inward, the Cherubim controlled the sphere of the fixed stars: next came Thrones, Dominations, Virtues, Powers, Principalities, Archangels, and finally Angels, who controlled the sphere of the moon. The resulting conceptual scheme, dominant until the lath century, was one of extreme clarity, intricacy, and cohesion. Thirteen centuries after Ptolemy's mode], Copernicus (1473-1543) published (in 1543, the year of his death) De Revalulite ibus OrTiwn Cdestir rn. proposing the revolutionaryideathat the Earth and other planets moved rather than theSun.' ~[l~ere were variations en this haste saw= with diffetent numbers of&phrses .AYis[otle's (354—322) system contained 55 spheres. However. elite the Ort k system was merged with Christian theeLngy, the resulting system had 9 (or 10, depending on what is counted) spiritually signi Omit spheres. ~In , ode's theory of motion, a hamogGneous body required an external mover_There was a kind of analogy of d i e form spier !planet I I tout I body I I anew i moved. 'Copernicus's theory was only partly hctioeentric_ For mathematical reasons, he placed the center citric solar system al the center of the Earth's orbit, rather than at the Sun itself_