Algebraic and Geometric Logic

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Algebraic and Geometric LogicAuthor(s): Ter Ellingson-WaughReviewed work(s):Source: Philosophy East and West, Vol. 24, No. 1 (Jan., 1974), pp. 23-40Published by: University of Hawai'i PressStable URL: http://www.jstor.org/stable/1397600 .Accessed: 14/09/2012 06:09

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The logic of ethics would be neither pure logic nor ethics; but it might bemore useful for some purposes than either illogical ethics or unethical logic.

The outcome of comparative efforts along these lines would be, adaptingWittgenstein's term, a Wohltemperierte Vorstellungsklavier-a Well-tem-

pered Representational Keyboard -that would be adequate for communica-tion between all the various disciplines and that hopefully would allow for the

possibility of transposition into the mode(s) of everyday discourse as well.This goal may or may not be achievable. If such a harmonizing of separate dis-cursive modes were possible, the possibility would almost certainly have torest on a basis of a shared cultural heritage and environment.

For intermodal dissonance must become greater as we try to communicateacross cultural boundaries.3 The structures of our discursive modes (whichmay be either the images or the vehicles of our cognitive structures) vary cul-

turally: Christian and Buddhist ethics, Hindu and Napoleonic jurisprudence,

vary in their structures as well as in their contents. Doing ethics, while takingaccount of both Christian and Buddhist definitions, is like trying to perform

music simultaneously in the scales of Rag Todi and C major. If anythingcomes out at all, it must sound very strange to adherents of both systems, or

totally alien to one. The second of these results achieves nothing new. Thefirst, if both sides can tune in after the initial shock, might provide a usefulbasis for mutual expression and evaluation.

When I speak here of Western and Indo-Tibetan logic, Iexpect

thediscussion to grate on the ears of both Western and Tibetan logicians.

Western logic is now usually defined as something like the principles ofvalid inference, 4 or the science of necessary inference. 5 By such definitions,the system I am comparing is not really logic at all, since it involves neithersequential inference nor principles of validity. But I am using in this discus-sion a quite different conception of Western logic. Rather than defining logicby its intentions and goals, as in the definitions quoted, I attempt to character-ize it in terms of how it appears from the outside -to describe its practicalapplications rather than to define its theoretical essence. Accordingly, lookingat current logic textbooks and the use of logic in current philosophical writings,it seems possible to describe Western logic-not its theory or history, but mostof its current practice-as the symbolic transposition of semantic contents intoa mathematical ramework.

8 Of course, intercultural dissonance can be useful for some purposes, as when some formsof Oriental dress or thought are adopted by Western counterculture roups. Dissonancebecomes, as it were, a feature of a certain aesthetic style.4 William and Martha Kneale, The Development of Logic (London: Oxford

UniversityPress, 1971), p. 1.5 Willard V. O. Quine, Elementary Logic, rev. ed. (New York: Harper Torchbooks,1965), p. 1.


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This is also the sense in which I define the Tibetan system discussed hereas logic. It is not so characterized by Tibetan philosophers. Their logic(Sanskrit pramana, Tibetan tshad ma), like ours, is concerned with inference,validity, and argumentation, and is closely related theoretically to epistemologyand historically to the practice of dialectic and debate. The system I describe

belongs, by contrast, primarily to the ritual rather than the philosophical mode.Thus the entire system of Buddhist formal logic and all the current and

historical exceptions to the currently dominant Western usage are left out ofthe picture. Hopefully the new comparative point of view achieved by thismethod will justify the amount of dissonance created by these omissions. Mygoal is to expand the Western concept of logic into a broader and more flexi-ble form; I have left the Tibetan concept of tshad ma totally untampered with.

My characterization of logic, of course, could be dismissed as dealing onlywith the techniques of logic. Likewise, the philosophical analysis of language

could be rejected as dealing not with the substance of philosophical thoughtbut merely with the technical devices by which it is expressed. To those in-clined to such rejections, this discussion will be of little use; but I hope it willhold some interest for others.

I

Most of Western philosophy has closely adhered to a traditional methodologyof

linear,discursive

presentation.In recent decades we have become aware

that underlying this methodology there functions a logical system whose basicoperations are those of algebraic quantification and negation, and that ourlinguistic formulations can thus be readily translated into sequences of mathe-matical symbols. This discovery seems to reinforce the prejudice that ourmethods are scientific, although perhaps not yet fully perfected, while othersystems used in other cultures are primitive, imprecise, and in Walt Kelly'sterms, mythillogical.

Some items, of even our own experience, seem to clash with our familiarlogical structures, such as the constancy of the speed of light, or some findingsof particle physics. And, on the other side, some critical observers have helpedreveal to us the logic of foreign views and ways. Anthropologists, particularly,have been active in trying to expand our consciousness of possible alternatemodes of thought, from Frazer's famous characterization n The Golden Boughof magic as false, primitive science, to Levi-Strauss' exploration in TheSavage Mind of the science of the concrete. There is also increased philo-sophical investigation of non-Western thought systems, particularly those of

the Oriental high cultures. Yet the impression persists that because our logicis fundamentally mathematical, it is also fundamentally superior.But we make a serious mistake if we assume that the only mathematical logic


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possible is our traditional algebraic type. Centuries ago, another kind of sys-tem was invented and elaborated within the tradition of the Tantric religionsof India and Tibet, which can only be described as a geometrical logic. Our

algebraic system utilizes sequential techniques of quantification and negation.Neither of these is possible in the Indo-Tibetan geometric system, which in-stead demonstrates configurational relationships of similarity (symmetry) and

congruence. Equivalence can be shown in both systems; but in the algebraic,it is a quantitative equivalence, while in the geometric, it is a qualitative kind.

And, while we have recently learned to present our formulations as constructsof abstract symbols in algebraic equations, the Indians and Tibetans have

traditionally presented their formulations in pictorial symbols structuredwithin geometric constructs known as mandalas.

II

The basic form of the mandala (Fig. 1) is that of a circle enclosing a squarewhose diagonals are its diameters. Since the mandala represents simultaneouslya cosmogram, a psychogram, and a purified ritual site where religious powers(dbang) can be obtained, the geometric elements of circle and square havevarious meanings.

The circle represents the cycle of samsara, of worldly existence and re-births, and may contain pictorial representations of secular scenes (Fig. 2).Its connotations are of

unstructuredness, endlessness,and

intolerablyunbroken

regularity. At the same time, it is a boundary which sets off and defines by con-trast the special character of the structured system within it. Its outer ring isthe fire of Enlightenment which burns away misconceptions. Its middle ringis of vajra (rdo rje), showing the diamondlike sharpness, clarity, and indes-tructible solidity of Enlightenment. Next, there may be (as in Fig. 2) an in-side structured view of the worldly cycle, according to the Buddha's systemof the chain of Dependent Origination (pratityasamutpada). Finally, the innercircle is formed of the petals of the divine lotus upon which Enlightened rebirth

takes place. Logically, the circle serves to enclose (bracket) and define a sys-tem which is contiguous with itself, but which, because of its structured nature,is of a fundamentally different quality.

The square, by contrast, is a highly regular structure. Its diagonals sub-divide it into four congruent isosceles, right triangles. Visually it is a palace,its walls hung with jewels and topped with royal parasols; its gates facingthe four quarters of the world, crowned with pairs of unicorn deer (bse ru),

listening with one-pointed concentration to the wheel representing the Bud-dha's

teaching,the true

gatewayto the

palace. Logicallyit is a

structure intowhich symbols may be inserted to postulate relationships of similarity andequivalence.


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FIG. 1. Mandala of the Four Guardian Kings. Lokesh Chandra and Raghu

Vira. A New Tibeto-Mongol Pantheon, vol. 14. New Delhi, The InternationalAcademy of Indian Culture, 1964, p. 33. (Courtesy: Dr. Lokesh Chandra)


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FIG. 2. Mandala of Sri Cakrasamvara. Lokesh Chandra and Raghu Vira.A New Tibeto-Mongol Pantheon, vol. 14. New Delhi, The International Acad-

emy of Indian Culture, 1964, p. 63. (Courtesy: Dr. Lokesh Chandra)


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We have said that the geometric logic expressed in the mandala is, like

algebraic logic, mathematical; it is also symbolic. But its symbols, being notabstract but pictorial, are also not single-valued but multivalent. This featureis crucial to understanding the workings of the system. By use of multivalent

symbolization, a number of sets of symbolic equivalence can be simultaneously

diagrammed and, subsequently, interpreted on various levels as the needarises. Criteria of abstract simplicity are here abandoned, while symbols areselected for their richness and complexity. Biological elements (plants and

animals), human figures (Buddhas), and cultural objects (royal robes and

ornaments, weapons musical instruments, etc.) are characteristic of the sym-bols used. Their combination produces symbolic composites which are religi-ously, psychologically, and culturally highly evocative, such as the weapons-carrying, bull-headed fierce Buddha known as The Diamond Terrorizerand The Slayer of Death (Vajrabhairava/rdo rje 'jigs byed, Yamantaka/

gshin rje gshed, Fig. 3).These are not arbitrary creations. Images, as objects of contemplation to

purify the body, mind, and senses have to be created in wrathful as well aspeaceful aspects, and sometimes with multiple heads and hands, so that theysuit the physical, mental, and sensuous capacities of different individualsstriving for the final goal.6

So the multivalence of the symbols used is both intentional and functional.What about their systematic usage in the mandala?

Take the case ofFigure

1.According

to a structural convention, the four

quarters of the central square are assigned to the four directions of the com-

pass and given their associated colors:

WestRed

South Center NorthYellow White Green

EastBlue

Superimposed upon this stylized cosmic map we see a planetary configurationof five figures surrounding a larger central figure. The central figure is apeaceful form of Vajrapani(phyag na rdo rje), patron of Tantra and a symbolof the fierce, protective aspect of Buddhism. His satellites here are the fourgreat Guardian Kings of the Four Quarters, who also represent this fierce,protective aspect in a lower, more worldly form. We thus have here thesymbolic, diagrammatic subdivision of a general concept into its (worldly

6 Dalai Lama XIV Bstan 'dzin rgya mtsho, An Introduction to Buddhism (New Delhi:Tibet House, 2509 B.E.).


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FIG. 3. Vajrabhairava Rdo rje 'jigs byed). Collection f Field Museum,Chicago.


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directional) concretely structured components. Formally, two kinds of rela-

tionships are postulated:(1) equality and similarity of the satellite figures positioned in the four

quarters of the square, and (2) subordination of the planetary figures to the

central, implying a holistic, integrated reality which can be seen outwardlyas a system of structured components. Geometrically, the use of concentriccircles implies the equivalence of symbols pictured at three different levels of

organization. These are: (a) the circle outside the square, symbolizing realityas cyclic, undifferentiated, and chaotic; (b) the circle within the square,structured by the geometric subdivisions of the square, picturing reality as astructured composite made up of related component parts; and (c) the centralcircle with the Buddha image, representing an undifferentiated, holistic, inte-

grated view. These are successively higher ways of regarding the same

reality, which might be called experiential, analytic, and Enlightened, respec-

tively. Each is valid for its own level; we might think of them as threedifferent lenses for viewing the same picture from different perspectives.Geometrically and logically, repetition of the circular form and concentricityindicate equivalence of what is symbolized at the various levels, while their

position inside or outside of the structured square indicates the analytic levelat which they are to be taken. The central circle actually is to be considered adimensionless point, at the intersection of the circles' radii and the square'sdiagonals.

The equivalence of these symbols is a different kind of equivalence fromthat postulated in algebraic formulations. To be sure, the symbols in the inter-mediate, quartered circle seem equivalent in a normal way, as the mytho-logical guardians of the western, northern, eastern, and southern quarters aresaid to be equal in power and importance: w =n = e'= s. Furthermore,the set of four is equal to some sort of protective aspect or quality of Bud-dhism: p = (w + n + e + s). And this protective aspect is also equivalentto the central symbol, the Buddha Vajrapani: p = V. But then: V = (w +n + e + s), which is false, since Vajrapani represents a qualitatively differ-

ent level of symbolization than the Four Guardians. This qualitative shift isnot represented within the algebraic system and has to be taken as a failureof the rule of equivalence, like the failures encountered in quotational contexts.We could, perhaps, accommodate the difference by considering one symbol,(V), to be metalinguistic or metasystemic. However, there is no difficulty inaccommodating the qualitative shift in the geometric formulation of themandala, because qualitative equivalence can be easily expressed by geometri-cally similar shapes in concentric relation.

Furthermore,this

particularinstance

representsa

very simpleuse

of thegeometry of the mandala. Its real usefulness becomes apparent when we con-sider a more complex case which exploits the multivalence of symbols to


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present simultaneous relationships of categories on several qualitativelydifferent levels, as in the often-mentioned Mandala of the Five Tathagatas.7

The Tathagatas, or Jina (rgyal ba), are a group of Buddhas found in the

top order of the pantheon. They are deployed in a mandala with the layoutgiven above:

West-RedAmitabha

South-Yellow Center-White North-GreenRatnasambhava Vairocana Amoghasiddhi

East-Blue

Aksobhya

In this case, the center is taken as a fifth direction.8 Depending on the systemfollowed, the central figure might be taken either as one member among

equals, or, as both one member and at the same time the apotheosized embodi-ment of the whole set of five.9

The Tathagatas collectively embody a large number of sets of symbolicassociations. Some of these are shown in Table 1. Besides being associatedwith specific directions of the compass and colors of the spectrum, they also

represent the systematic divisions of the physical elements, the constituents ofthe human personality (skandha/phung po), the passions, the senses, differ-ent families (rigs) of followers whose personal characteristics and inclina-

tions correspondwith the

qualities represented bya

specificindividual

Tathagata, and several other sets of associations. Thus, the geometric struc-ture in which they are situated affords a means of simultaneously diagram-ming, through symbols, corresponding patterns of relationships between setsof categories where comparison would ordinarily be quite difficult even be-tween individual members of different sets.

The geometry of the mandala furnishes a basis for the structural comparisonof different types of systems; and, it is in this structural comparison of entire

systems that its superiority to the algebraic equation becomes evident. Levelsof meaning can be projected outward through an indefinite number of con-centric circles, as in Figure 2. And each symbolic component of a mandalacan be structurally subdefined by a subordinate mandala; or, viewed in another

way mandalas can be arranged into an encompassing mandala, so that it is

7 The fullest discussion is found in G. Tucci, The Theory and Practice of the Mandala(New York: Weiser, 1970).8 The list of directions can be expanded to include up, down, and those intermediate be-tween the four compass points.9 Cumulative enumeration, ounting the sum of a set as one of its members, s a commonTibetan technique. Anthropologist R. E. Miller considers this type of enumeration o con-stitute a persvasive pattern in Tibetan culture (private communication).


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Table 1 Symbolism of the Five Tathagatas

AmoghasiVairocana Ratnasambhava Amitabha Don yod 'Rnam par Rin chen 'Od dpag med pa

snang mdzad 'byung nas Boundless UnfailTathagata Illuminator Jewel-born Light succes

Direction Center South West North

Color White Yellow Red GreenElement Space Earth Fire WaterPassion Ignorance Pride Passion JealousySkandha Form Feeling Perception ActionSense Sight Hearing Smell Taste

Family* Tathagata Jewel Lotus Extensive

Symbol Wheel Jewel Lotus Crossed vAnimal Lion Horse Peacock Winged d

Adapted from G. Tucci, Tibetan Painted Scrolls, Vol. I. Rome, 1949, pp. 238-240.* Families grouped according to system of Anuttara-yoga-tantra


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possible to generate mandalas of mandalas (Fig. 4). The structure is infinitelyreplicable at all levels of description, and every aspect of the system under con-sideration can be simultaneously diagrammed in symbolic form within thestructural model.

III

Western thought does not seem to have evolved a geometric logical system of a

complexity and sophistication comparable to that which we have described inthe mandala. Some theoretical recognition of this mode of thinking does exist,and, particularly in the social sciences, there is a growing practice of struc-

tural, diagrammatic modes of analysis. We will briefly consider both of thesetheoretical and practical parallels.

A close approach to the distinction made here between geometric and alge-braic logic is the standard psychoanalytic distinction between primary and

secondary mental processes. Secondary process is ordinary logical dis-

cursive, linear thought, which is unambiguous, verbal, cognizant of reality,contradiction, truth or falsity, and so forth. Primary process is the mode of

pictorial symbolic representation characteristic of unconscious thought, whichis especially visible in dreams:

They reproduce logical connection by approximation in time and space.... A causal relation between two thoughts is either left unrepresented or is

replaced by a sequence. . . . The alternative either-or is never expressedin dreams, both of the alternatives being inserted in the text of the dream asthough they were equally valid. . . . Ideas which are contraries are by pref-ference expressed in dreams by one and the same element. . . . similarity,consonance, the possession of common attributes is very highly favored by themechanism of dream formation. The dream work makes use of such cases. . by bringing together everything that shows an agreement of this kind into

a new unity.'0

This new unity is achieved through the mechanism of overdetermination ofdream symbols; that is, symbols are selected by multiple causal processes,which combine with each other at various symbolic levels, so that each symbolis associated with several different thoughts and each thought is expressedin several different symbols. Thus, The fact that the meanings of dreams are

arranged in superimposed layers is one of the most delicate, though also oneof the most interesting, problems of dream-interpretation. 1 All of these fea-tures of primary process thought-spatial presentation of relationship, con-struction of multivalent symbols by combining pictorial elements, presentationof systems structurally rather than sequentially, and superimposition of mean-

10 S. Freud, On Dreams (1901) (New York: Norton, 1952), 64-66.11 S. Freud, The Interpretation of Dreams (1900). (Standard Ed.) (New York: BasicBooks, 1956), p. 219.


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FIG. 4. Mandala of mandalas. Lokesh Chandra nd Raghu Vira. A New

Tibeto-Mongol Pantheon, vol. 14. New Delhi, The International Academy ofIndian Culture, 1964, p. 24. (Courtesy: Dr. Lokesh Chandra)


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ing layers-we have found in the geometric logic of the mandala. We mightadd that the feature of combining contradictory elements into a single symbolis also used there, as in the fire which both destroys and enlightens, or the

sexually and aggressively active Buddha who has transcended the worldlypassions.

A further emphasis appears in Freud's later work; The function of judg-ment is concerned ultimately with two sorts of decision. It may assert or denythat a thing has a particular property; or it may affirm or dispute that a par-ticular image or presentation exists in reality. l2That is, secondary process functions (judges) by predication and existential

quantification, and by negation and falsification. In the primary process func-

tioning of the unconscious id, however, there is nothing that could be com-

pared with negation; . . . there is nothing that corresponds to the idea of time;there is no recognition of the passage of time and . . . the id of course

knows no judgments of value: no good and evil, no morality. 13 Again,by comparison, the mandala does not admit negation. There is no wayof presenting a negation within a diagrammatic system; the nearest possibilityis simply to exclude an element from the diagram. Also, time is representedcyclically rather than historically; that is, its passages literally not recog-nized, since what comes after also went before and is now present.14 Andmuch of the symbolism presented is overtly morally ambiguous, with itsidealized violence and lust.

And, remarkably,the

symbolicmode of

thoughtin the

primary processis

presented as something primitive, archaic, and infantile. 15 Remarkably,because this assessment accords so well with our ethnocentric bias againstsymbolic modes of thought in primitive cultures. The relative prominence of

primary process in infantile and psychotic thought, art, and religion is veryfrequently called to our attention.

Is pictorial, diagrammatic representation then simply a primitive or patho-logical substitute for normal, logical thought processes ? Perhaps; but, if so, then

only in the spontaneous, unelaborated forms produced by individual infants,artists, and dreamers. The difference between these forms and the mandalais that the latter incorporates a logical, mathematical structure by which the

pictorial symbols are integrated into a geometric system and understood ac-

cording to an elaborate, sophisticated set of conventions. Between a product ofthis system and a private fantasy, there is a difference just as great as between

12 S. Freud, Negation, Standard Edition Vol. XIX, p. 236.13 S. Freud, New Introductory Lectures on Psychoanalysis (1933). In The Complete In-troductory Lectures (Standard Ed.) (New York: Norton, 1966), p. 538.14 On

cyclicaland historical time, see Mircea Eliade, The Sacred and the Profane

(NewYork: Harper Torchbooks, 1961), pp. 104 ff.15 S. Freud, Introductory Lectures on Psychoanalysis, Standard Ed., in Complete Intro-ductory Lectures (New York: Norton, 1966), chapter 13.


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a set of algebraic logical equations and a baby's first sentences. Because themandala organizes its symbols by a culturally elaborated structure, it is thebasis of a logical system; and its geometric system shares with our algebraicthe common status of a mathematical symbolic system.

As for the applicability of geometric systems outside of mythological andritual contexts, we are at present witnessing a considerable growth of efforts

to arrive at diagrammatic understandings of social structures by social scien-tists. These efforts have so far utilized forms similar to the following paradig-matic diagram of the basic structuralist hypothesis, as adapted by Leachfrom Lane.16

capacity

Deep

Surface

Speech, Myths Patterns of

Deep

discourse marriage and family

relations

< Cultural metaphors >

(shifts of register)We have here the mandala's methodology of diagrammatic representation ofa basic structure, which can be replicated simultaneously at different levels andwith different groups of referents, and combined into an overall structure at ametasystemic level. The efficiency of this approach becomes apparent when onetries to devise a similarly economical verbal formulation of structural relation-ships between language, myth, kinship, and the structure of structure itself.

16 Edmund Leach, The Influence of Cultural Context on Non-Verbal Communication nMan, in R. Hirde, ed., Non-Verbal Communication Cambridge: Cambridge UniversityPress, 1972), p. 332.

Surface


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By such diagrams, one can present systemic, metasystemic, and subsystemicformulations relating to several different systems simultaneously.

Of course, this presentation is relatively crude when compared with that ofthe mandala. There is some rudimentary usage of geometric similarity, paral-lel, and convergence. But the precision of structural configuration and the com-

plexity of symbolic expression found in the mandala are simply not possible inthese elementary diagrams. We will have to advance further in our under-

standing of both structural relationships and of diagrammatic symbolizationbefore more highly sophisticated forms of presentation become useful to us.

The impetus to develop this kind of understanding does exist. Leach says ofthe structuralist approach:

At the heart of the argument is the thesis that (a) the phenomenon of me-dium transfer by which we are able to express the spoken word and music inwritten signs, and (b) the integrative capacity whereby simultaneous signalsreceived through the senses of seeing, hearing, tasting, touching, smelling, etc.are felt to constitute a single rather than a multiple experience. Both implythat, at some level of the mind, we are endowed with an innate structuringcapacity which is most easily conceived of in algebraic terms-the algebrabeing the structure which is common to all the diverse cultural manifestationsin which the operations of the mind may be observed.17

By algebraic Leach appears to mean the use of symbolic variables-for whichdifferent referents (values) can be substituted-within the basic mathematicalstructure. However, as we have see, it is also possible to interchange symbols

within a geometric logical framework. In the case of the mandala, this involvedpictorial, multivalent symbols rather than abstract, single-valued ones.18 More-over, we have seen that the mandala excels at precisely those two operationswhich Leach emphasizes: (a) medium transfer (also describable as meta-

systemic interchange of elements)19 at a highly complex level, by use ofmultivalent symbols; and (b) structural integration of these symbols into geo-metrically patterned symbol-systems which replicate the structural patterns ofthe referent systems, and which, because of symbolic multivalence, simulta-

neously present divergentreferent

systemsas a structural

unity.It would therefore seem that our innate structuring capacity -or at least,the common structure that underlies structures-can best be conceived not inalgebraic, but in geometric terms. Leach himself admits, . . . any inferenceswhich are made about the 'algebraic structures of the mind' are mere guessesabout the mechanism of a black box which is inaccessible to inspection. ...Even so, structuralists fully appreciate that the sorts of guesses which they

17Ibid.18

However, mandalas f abstract ymbols lso occur.19See Ellingson, Classifying Musical Notation Systems, paper read at Society forEthnomusicology, oronto, 972.


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have so far been able to put forward are quite rudimentary. 20 There are prob-lems in applying conventional truth-testing criteria to structural analyses.What exactly are structures, or in what sense do they exist ? How arestructural analyses verifiable or falsifiable? Some structuralists realize that thelevel of meaning of their formulations is rather different from that of ordinarydescriptive discourse, as when Levi-Strauss begins his study of myths with

. . . it would not be wrong to consider this book itself as a myth: it is, as it

were, the myth of mythology. 21 And both Levi-Strauss and Leach have been

compelled to look outside the range of normal algebraic logical structures, par-ticularly into the area of musical forms, to furnish structural analogues for theirformulations.22 t would therefore not be entirely fantastic to assume that theelaboration of geometric-logical structures would be a useful area of investiga-tion for social scientific theorists. The growing use of diagrammatic-symbolicpresentation seems to indicate a tendency in this direction already.

No such tendency seems to exist in philosophy itself. There are, of course,the Venn diagrams, which utilize only the geometric devices of inclusion and

congruence, and which function mainly as an adjunct of more elaborate alge-braic formulations, as a pedagogical device. Geometry will probably enter

philosophical logic as algebra did, by the back door of applied and theoreticalscience.

But far from being primitive or imprecise, geometric logic is complementaryto algebraic-when its system is sufficiently elaborated-and can be used in

conjunctionwith it.

Despitetheir

differences,the two

systemsare to a

degreemutually translatable, at least at the level of individual elements or individual

relationships. A mandala can be described in verbal discourse, although itwould be far less efficient to do so than to diagram it. And because of this

capacity for translation, I have refrained from characterizing geometric logicas nonlinguistic. Some linguistic formulations, like poetry, seem to follow a

logical plan of organization, which is more geometric than algebraic. EvenLevi-Strauss' book, mentioned above, follows a plan which at times is seen asmusic and at other times as myth.

It remains to work out in detail the grammar of geometric structures. Wewould have to specify the linguistic meanings of the geometric conventionsused in structures like the mandala: similarity, congruence, concentricity, bi-

section, quartering, subdivision, inclusion, radiation or projection, tangency,parallelism, perpendicularity, and the rest. We would also have to reach a bet-ter understanding of the uses of multivalent and pictorial symbolism. Only then

20 Leach, op. cit., p. 333.21 C. Levi-Strauss, The Raw and the Cooked (New York: Harper Bros., 1969), p. 12.22 Levi-Strauss, op. cit., 14 ff.; Leach, op. cit., 318 ff.


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could we explore the uses of geometric logic with that of algebraic and legi-timately compare its usefulness.

At present, we can at least recognize the existence of geometry as the legiti-mate basis for a mathematical logic and recognize that its application has beenelaborated in the context of some Asian

philosophiesto a

degreeof

sophistica-tion the extent of which we are still unable to grasp. This realization could bethe ground for an impartially comparative view of other philosophies, and thebase on which we build new understandings of the structures of our thoughts.

Algebraic and Geometric Logic

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