Fractal patterns in language

Ym~l&m ,n Pyhol. Vol. 11. No. I, pp 105%109. lYY3 Printed in Great Britam

FRACTAL

PATTERNS

IN LANGUAGE

BENNY SHANON Department of Psychology, The Hebrew University, Jerusalem,

Israel

Abstract-Fractal structures are noted in natural language. Specifically, it is pointed out that descriptions in language are invariant under scale. In other words, the same terms may be used to describe states of affairs at different levels of resolution. Thus, in themselves, linguistic expressions are not indicative of the level being entertained. The functional advantage of this state of affairs is noted. Fractals are very much at the focus of contemporary scientific research. (For the original presentation see Mandelbrot, 1977; for a popular review see Gleick, 1988.) Fractal patterns have been noted in various disciplines of the physical sciences, in the life sciences and even in economics. Recently, ideas fractals and to chaos theory have started to appear in the psychological

related to literature

as well. By and large, these are encountered in investigations of a physiological orientation. In the main, these investigations consist either of perceptual models couched in brain structure and dynamics [especially to be noted is the study of olfactory perception by Freeman (1991) and Skarda and Freeman (1987)], or of broad theoretical works relating mind and brain (see Vandervert, 1991). note I draw attention to fractal patterns manifested in human cognition

In this and its

expression in natural language. By way of introduction, let us consider the-by now classical-example of a fractal pattern: the pattern is presented by coast lines as they are depicted in an atlas. The line of the coast is an alternating sequence of land dentures and water bays. What is of significance here is that when the resolution of the map is increased the same pattern of dentures and bays will be observed. A more detailed map may reveal that what had looked ( in the less finer) map as a smooth curve of either a denture or a bay is, in fact, not smooth. In the finer resolution, the seemingly smooth line will itself appear to consist of an alternating sequence of land dentures and water bays. Thus, even though more detail has been revealed the overall configurational pattern of the coast has not changed: It is an alternating sequence of land dentures and bays. Indeed, however fine the resolution of the map, the configurational pattern will always be the same. In other words, the coastal pattern is invariant under scale. In itself, then, an inspection of the configuration on the map cannot allow one to determine what the level of resolution at hand is. This invariance is one of the defining characteristics of fractals. It is this characteristic that we shall examine in the cognitive domain. The following discussion presents fractal patterns manifested by linguistic expressions as they are employed to describe states of affairs in different levels of resolution or detail. By no means is the discussion meant to be conclusive. Its 105

IOfi

1%. Stl;llloll

ottly aim is lo point out th;tt pttet-tts (lotiiaiti at‘r da0 tlotcd in httgttagc.

sitiiihr to those cticotttitetwl It is tiopett tttat the tii;tt.litlg

itt tttr ptt\sic,:tl of’ttlc. :tti:tto,gies

that ti)tlow nilI open tiew lines of’ in\ t3tigatiott itt both httguage and c~ogtiiliott. I>et tth cotisickt;I titoutit;iiti ;ttitl tti;kt ttigt~ct~tiiost t-egiott of‘ it, that t7frtwtl to a4 “the top.” We shall fi)ctts on 1his tq$otl. ‘Ihis wgioti too has ;I pit3 \vtiicti lvilt tw 1-eterred to 21s “tI1r top.” ;\tttl so on. ;tticl so fill-ttt. ;\t cxli atrco3ai\~e Iv\-cl of‘ rcsolutioti, ;i segtiwtit Iwt‘t of. ttlc titoutitaiti.

tiia~. lw sclectetl wtiiclt will tw t-cftyretl to ;ts the upprrtnost All tttesr. ttighcst pirts of’ the tlif’f’ct7ttt stypetlts of’ tttc.

tiiotttilaiti will t)e rcf’kt-twl to ;IS “top.” I;ut-ttict-mot-c-, the setttctices tlcsct-ibitig ;tIt ttlrsr srgtltetlts will att he of ttw satiic t\‘pe. As iti the C;ISC of’thc* maps of’tlif‘f’~t~~trt t-rsottrtiotts itt tttc ;itlas, ;tti itispectioti of‘ ttw stwtenw itself \vilt not rr\y;tl bvttic.lt lwcl of’ t-esotution is twittg c.titciXtitiecl. Ititleetl, ttic scypetits iti ;tIt tcvcls of’ t~csolutioti might t)c clew-iktl In tiiwtts 01’ ;I SctltC’tl(‘t’ kll( I1 ;I5 ( I ): (1) ‘l‘tir top is steep. ‘1.k vu-y smllc pattrtm tttay tlcsct-the trtripot-al logoits

to thr

(2)

‘l‘tie

clif‘f’cwmt tqjtitiitig

is ettcoiitttct-ed witti tetnpot-al cxptwsiotis. Sctttettcc. (2) stt-ctclics of tlif’fk-rttt t-esolutiotis, ;lII c-otllpltwl~~ ;111;1tops of the tiioittlt;titi dew-itwtl I)!, (I ): 1~2s ;I Itzippy time.

‘l‘tic tqititiing ma!. he ttic first part of.2 cwtaiti thy (s~\cr;tl tiottt-s, ~2)). the f’itxt IMtI of’ tht part (dtt ttoitr, sa)), ttir first pit3 of‘ that patI (i.c., sevri-al titititttcs), etc. ~l‘he tetllpol~;ll sttXTc.t1 Illa) he stiiallei~ ;ttitl stiialtct~, but iti all (xx3 ftir esl)rcssioti “the Iwgititiitig” uill apply. ;\ sititilat- pittcwi is noted wittt ~wl~s. Iti the foltowitig “st;lrt” retet-s to petiotls of’ totally tlitt.et-etit tetiipor;it (3) hl;itt .s/~r~/d to clotiiesticatc ;ttiitriala.

setttmccs tltc satnt clitiicttsiotis:

‘l‘tie trip \/uI./M/ iti Itttlia. ‘Ilie excut-sioti .\/odd with ;I \.isit to the iLlusekttii. fin. that tiiattn-) Likewise, act,jecti\w struti as “big” (or “stnalt,”

vet+)

niav IK usctl

clescrihe htli ttiittgs which arc (in ati aholttte setise) of’ large dint&ions things wltich arc’ stltalt. ‘l‘he vitvrt. of‘ the Himalayas \vill point antI (1) Loolk at this big tttoittitaiu antI at ttic stidl otie to its Iclt.

to

mid S;IV:

But the s;ttne setttetlccs tttay also he used by the %wwof‘tttr (hlileatn Hills that 1-eact1 110 IllOI‘c‘ that1 1000 Ill. I kt that hatI ;ct1ts used tlatltra1 tatlgu~lge tl1ey cottI(t ha\32 used that \w‘v smic sctitetice to tlescrik the piles of‘wtid they It-atispot-t in their fhragitig hikes. I ~ippeci;ttr that tttr pattetns tiotccl may he distiiissrcl as It-i\,i;tl. Agaitist tttis, tet tile point out tttat states of af‘t’ait-s cati \w’!’ \vett be et~vkiged iti wtiictt ati invat-iaticc ot‘ terms tloes ttot tiottl. Iii such mses differcnt tatigwgcs, ctialccts or coding spstctiis woutd hc etiiplo\ecl itt tlift’krettt contexts or cotitetititat domtitis. (k~tiseqtictitl~, tlic mere itletitif&tioti of’ ttte lattguag:6, clialcc.t or coding system woi~ltl etiatde one to detetmiitie the curltext or cotitctlt at hand. ]cbvs in the Ihspot-a, fi)r instance, eniplo~td H&t-e\\ only itt sact-cd cotitcxts; iti dad) pithrice ttic local \~cwiawt;ir ~I‘tiits, words which ;I dictiottq~ in differart, tioti-o~el-lal)l~iti~ Iti point of fact, the ‘l‘altiiutl

(ot- ;I s~~ecif‘icalty ,]cwisti cliatcct of‘ it) ~~2s used. cvot~ltl chat-actet-ize as sytioti) tiis were eniployetl cotttexts, hence iti effect hatI clifttwit tiiwtiitigs. specifies contents which citt ty cspt-essetl ottt!. itt

Fractal patterns

in language

107

“the holy language” and contrasts them with ones which can he expressed in the various spoken languages. One might argue, for instance, that prayers or divine and hence may only he articulated in one utterances have intrinsic sanctity, privileged code and never be translated. An analogous phenomenon is the social stratification of language. In _Japanese, for instance, totally different linguistic forms are used in different levels of social interaction. Scientific coding also reveals context specificity. In the United States the old British system is employed in the measurement of quantities in everyday life hut in scientific measurements the metric system is employed. Lastly, let me note a personal example. Corning from Israel where the temperature rarely falls below freezing point and having lived for a while in Boston and Ithaca I refer to (and think of) temperatures above freezing in degrees centigrade whereas to those I)elow it in Fahrenheit. ‘I‘he terms in which I specify degrees, then, are sufficient to determine the temperature range at hand. That fractal patterns are encountered in human language and that theil occurrence is not trivial is, of course, just the beginning of‘the story. ‘l‘h? present note is no more than an invitation. Fractals open the door for a host of issues that merit further cognitive-linguistic investigation. First, the contrast between the fractal patterns in (l)-(4) and the domain-specific terms noted afterwards indicate that scale-invariance does not hold universally XTOSS language. Further, the scale- (or domain-) invariance of individual words may be graded. Words may exhibit invariance in some orders of‘ niagnitude or domains of discourse, yet fail to exhibit it in others. Further still, worcls may he naturallv employed in SOIIW domains or levels of resolution, but in other ones their enil~loynient may he peculiar or even unacceptable. Variations of this kind many perhaps indicate that fractal scale-in~,al-i~trlce in language ~‘a11 he treated, like so nlay other in terms of fuzzy operators (Zadeh, 196.‘). phenomena in language, ‘l‘aking ;I more general perspective, some possible functional benefits of the patterns noted here may be suggested. ‘l‘he patterns at hand seen1 to be the attestations of a simple yet very important fact: Hun~n cognition categori/.es the world and natural language describes it by nleans of a system whose structure does not depend on the particular domain being entertained. ‘I‘he advantage of‘ this state of affairs cannot be overstated. ‘I‘he contexts of‘ life-both of the individual and of the species at large--are unbounded and tlot known beforehand. A language not maintaining in\m-iance across domains \vo~~ld belike the coding system of Barges’s famous f;rrrlr.\ rho :Llrt//o,Yorr,\ (see I~org”“, 1979)--of practically no use. Context is, of course, a subject matter that has received milch attention in conteniporary cognitive literature. In psychological and linguistic discussions the unboundedness of contexts is visually invoked in conjunction with the context dependence of linguistic ternls. Standardly. it is pointed out that due to the unbounded variation of contexts, fixed definitions of word nleanings art not feasible. As a rule, words exhibit an extreme context dependency, one which enables speakers to use them in utterly novel senses and yet, to be-on nlost occasions-very well understood (for demonstrative examples, see (Clark Xc Clark, 1979). In the present discussion, context variation was linked to scale-

10X

H. Shmon

invariance. On first glance, this may wem curious: (:o~itext-~le},cricIt.ncc‘ is ;I symptom of variance; hence, it may seeni incongruent for it to IX ~wmciateti with invariance. Further- reflection, however, reveals that this ncctl not he the C;ISC. Intieeti, the s~aie-iIlv~u_iaIIce of cvords nuy IK I-egwded as ;I striking m;mif&t;ltion of‘ how naturidly language (anti cognition in general) acc.otii~i~o~l~~tes to ail c‘ontexts. Likewise, it is their pertrct ~~c~orn~notl~~tioll with contexts that ;tllo~vs words to Miave well in ail senimtic domains mid at ail levels of’ resolution. One tiianif’~station of this ~xtgula~- I)eiia\401 at ail levels ib the f‘ractai-like scaicinvariance we iiavc noted here. For further discussion 01’ tile l~~x~ide~n of context ill impige mti cognition the I-edcr is ~~te~-l-ed to Siiiinoii (19H8). hawing tob2rd the end of’ this discussion, let me Iiote that invaria1ic.c ;ICI‘OSS iewls and sc-ales is just one feature exhibittd by f’ractal lmtterns. Ml, ii;l\~iIlg cxl~101-etl only this featuw si~o~~iti Ilot IX taken to ‘imply that it ia the 011ly 011c to I)e et~counteretl in ia~iguagr. Kelated f’6aturt.s that readily conic to n;iIicl ;II-c wpeated nesting anti self-reference. ‘l‘iiese f’wturcs. in f;ict, &tine some of’ tiiu most importmt strut-turd characteristics of tiic syntax of natural higuage, ;111(1 thev also have senmntic ni~lniftstations. Furthc, self~refbwlce is a kcv f’ctt urc of the l~iienoriienon 0f‘conscioiIsness. Selt-rrt’erc~~ce in its tlifftirent nI;lnifest;ltiorla have I-ecei\~ed extensive trwtnient in piiiiosoph~~, logic. linguistics, ;ctitl ps\,chology. On the one iiatiti, the piie~~onie~ia in question nrecl not neccw;irily call fi)r ;I f’nctai analysis; on the otiiel- iimti. their ;iIialysis Imcra 011 a host of c~~rnple?i issues in syntactic mid semantic theory whose consitientio~~ is c+31-1\~ outsi& t iir scope of this iimitetl discussion. 1%~contrast. the patterns suryed here 21-e Iiot t heoi.~-depenti~tit. ‘I‘iiey are hsd 011 l~iieno~nc~ioio~i~~~il olw13 ;itions ;IIKI the\i~resuppos~ neitiicr ;I specific tiieoretic~il stance in linguistics Iior a111 l~i~oct~lu~~;~l ps~ciioiogicai ~issllni},tioIis.“’ III closing, irt IIW take a lm~le~~ perslwcti\~e anti co1nmr111 OII tilr cluestic) of’ causes and origiw. Why is it that iiu~iia~i cognition mid lmiguag~ rxiiibita f’ractal l)attel-ns? Kecentiy, ;I few sciiol~ii-s have sptnhteti ahout this question. Some prol~osed that chaotic. beiuvior iI1 l~s~ci~ology is due to t lie st ruct urc 01‘ t iir hr;iin, ~vhich itself exhibits chaotic dynamics (see Skal-da & Freemen, 19H’i). oti1e13 1i;ive gone so far 2s to suggest that the mcountering of similar patterns in mind, brain mid warid reflect the interactiw dynamic c-o-evolution of all these sysletna (such ;I proposai is m;itle 1)) the ~iei~ral positivisrii of‘ V;i~itlerwI-t; see. fi)r

Fractal

instance,

Vandervert,

patterns

199 1). Cognizant

in language of the various

109

epistemological,

ontologi-

cal and metaphysical issues that may be raised in the marking of crossdisciplinary analogies, I have chosen to eschew these issues. What I attempted to do is just draw attention to patterns that, to my knowledge, have not been previously discussed in the literature. Given the fruitfulness of fractal analyses in other domains, I hope this may open the door for new lines of investigation in cognition and language.*

REFERENCES Barges, J. L. (1’.)7<)). Funes the Memorious. In Lcr/,?rtY~~~h.~ (pp. X7-95). Ha1-lrlo~lrls\vo~.tl~, U.K.: Penguin Books. Clark, E. V., 8r (Xark, H. H. (1979). When II~LIII~ surbtce as verbs. Larrgrqr, 85, 7!)7-81 1. Freeman, W. J. ( 19’31). The physiology of‘ perception. Scirnlrfic AwtCm, 264, 7X-X5. Gleick, J. (l!IXX). Clwos: Mokiug (I wu~ JC~PIKC.Har~nondsworth, U.K.: Penguin Hooks. Goldberg, A. L., R@ey, D. R., 8c West, B. J. (l!K)O). (Chaos and fractals in human physiology. Scim/lf~ Avwricm, 262, Y4-49. Mandelbrot, B. R. (15177). b‘rcrc.lnl.\: Fat-ttt,dtn ttcp cruddttttmsiott. San Francisco, CA: W. H. Freeman. Shanon, B. (198H). Sentantic representation of‘ nreaning: A critique. f’.\~c/rob@cct/ Mtr/lc~/in, 104, 70-83. Shanon, B. (1X1 1). Cognitive psychology and nrodern physics: Some analogies. E:Io.otputt ,~otttxcd u/’ Cognitiw P.~~~c/~lf~~q, 3, 201-234. Skarda, C. A., 8c Freeman, W. J. (l(387). H ow brains make chaos in order to nuke sense 01’ the world. H~hvioml citd HI.& S&tm~, 10, 16 t-195. Vantlervert, 1,. R. (It)!) 1). A measurable and testable brainbased emergent interactionism: An alternative to Sperry’s rnentalist emergent interactionisni.,lorrr-t/ctl otfhlitd crrtd BP/mJio)., 12, 20 l-2 lfl. Zadeh, L. (1965). Fuzzy sets. Itlfot-tttrt~iott crtd r.otr/d, 8, 33X-353.

*For a general, comprehensive discussion of the nature and status of analogies between the psychological and the physical domains the reader is referred to Shanon (1991); there, a typology of analogies and of their possible sources is presented and the more cautionary, neutral orientation advocated here is argued for.