introductory comments to francisco varela

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INTRODUCTORY COMMENTS TO FRANCISCO VARELA' S

CALCULUS FOR SELF-REFERENCERICHARD HERBERT HOWE

a& HEINZ VON FOERSTER

a

aBiological Computer Laboratory, University of Illinois, Urbana, Illinois, U.S.A

Version of record first published: 31 May 2007.

To cite this article: RICHARD HERBERT HOWE & HEINZ VON FOERSTER (1975): INTRODUCTORY COMMENTS TO FRANCISCO

VARELA' S CALCULUS FOR SELF-REFERENCE, International Journal of General Systems, 2:1, 1-3

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Int. I . General Syxtems1975, Vol. 2, pp . 1-3.Q Gordon and Breach Science Publishers Ltd.Printed in G reat Britain

INTRODUCTORY COMMENTS TO FRANCISCO VARELA'SCALCULUS FOR SELF-REFERENCERICHARD HERBERT HOWE and HEINZ VON FOERSTER

Biological Computer Laboratory, University of Illinois, Urbana, Illinois, U.S.A.Etymologically speaking, correct opinion is ortho-dox; paradox, however, lies beyond opinion.Unfortunately, orthodox attempts to establish theorthodoxy of the orthodox result in paradox, and,conversely, the appearance of paradox within theorthodox puts an end to the orthodoxy of theorthodox. In other words, paradox is the apostle ofsedition i n the kingdom of the orthodox. This is aheadache, an d has been ever since the Cretanphilosopher Epirnenides put forth the propositionthat "All Cretans are liars". The paradoxical pro-position is seditious . because it maintains anundesirable autonomy vis-a-vis any orthodoxattempt to, apprehend i t: when apprehended astrue, it turns out to be false; when apprehended asfalse, it turns out to be true.As long as was possibJe, logical orthodoxyattempted to treat such sed~tious ntrusions just aswould any other orthodoxy, that is, to dismiss themas cranks, as (syntactic) pathologies, (semantic)freaks, in short, as aberrations (of thought). Para-dox survived in the kingdom of the orthodox onlyby virtue of the asylum granted it as an entertain-ment'for the learned, much as in happier times theinsane were displayed for the amusement of thecurious.This comfortable state of affairs, however,abruptly changed when Bertrand RusseH dis-covered that paradox is central-and not peripheral-to all logical inquiry, that is, it affects the generalvalidity of logical formalism per se , and must bedealt with at the outset in any logical theory.Consequently, already in the introductory 'chap-ters of the Principia Mathematical Whitehead andRussell (W & R) address themselves to logicalproblems that arise with "certain contradictions"(i.e., the paradoxes) and to remedies that bringabout their solution: "An analysis of the paradoxesto be -avoided shows that they all result from acertain kind of vicious circle."' In order for theirreaders to appreciate the similarity of these para-

doxes, W & R describe seven of the most popularones, including Epimenides remark and, of course,Russell's class of all classes that are not members ofthemselves."In all the ab ove contradictions (which are merely ~t lc ct ion sfrom an infinite number) there is a com mon characteristic,which we may describe as self-reference or reflexiveness. Theremark of Ep ime nid s must include itself in its ow n scop e.If all classes, provided they arc not members of themselves,are members of w this must also apply to w ; and similarlyfor the analogous relational contradictions".'

Now that the seditious spirit corrupting ortho-doxy in logic was identified as Self-Reference,W & R could go about exorcising it. The method,known as the Theory of Logical Types, by whichthe two liberal gentlemen sumssfully performedthis operation, was simply to prohibit self-referential utterances, statements, propositions,descriptions, etc.This prohibition not only eliminates the potenti,alfor formulating paradoxes of the above kind, it alsoeliminates the potential for contaminating utter-ances, statements, propositions, descriptions, etc.with the properties of those who utter, state,propose, describe, etc. In other words, implicit inthe Theory of Types is the proviso that is theultimate protector of the Claim to Objectivity: '"The properties of the observer shall not enter intothe description of his observations."*The logic of our Western industrial corporatesociety (with limited liability) is unidirectional,deductive, competitive, and hierarchical, and thekeystones of its paradigms are the Claim toObjectivity and the Theory of Types, which excludein principle the autonomy of paradox and of theindividual. In the scientific revolution that we nowcreate and experience, however, we perceive a shiftfrom causal unidirectional to mutualistic systemicthin~ung,~.' from a preoccupation with theproperties of the observed to the study of th e


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2 COMMENTS TO VARELA'S CALCULUSproperties of the obser~er.~he initiator of thisshift was Kant, who placed the autonomy of theobserver at the center of his philosophy, thusmaking this autonomy responsible for the pro-perties of the observed.' It can bc no coincidencethat the realization of this shift in our contemporaryscientific paradigm takes place just as the relation-ship of (individual) autonomy to (social) responsi-bility has become intensely problematical. Onemanifestation of this shift is our rapid recovery ofthe significance of both paradox and self-reference,and of their intrinsic relationship.For instance, in psychiatry the significance ofthe relationship paradolr-self-reference, which inone context may have destructive, but in anothercontext constructive consequences has been pointedout again and again by George Bateson.'? In thedestructive (pathological) case, a paradoxicalinterpersonal (e.g., mother-daughter) relationexists, th e "double bind", in which autonomy ofone partner (daughter) is encouraged by the other(mother) on one level of discourse, but denied onanother (say, the interpretive) level; the (con-trolled) breakdown of the "metalogue" causes thevictim to withdraw affectively, and other schizo-phrenic symptoms develop."On the other hand, stress experienced throughthe irresolvability of paradox in known domains(c.g., two incongruent flat retinal images of the" s k e wscene) necessitates creating new dimensions("depth")."A parndox is a contradiction in which you take sides-both sides. Each half of the parad ox p r o m h e othe r . . .if y ou s w a t out one of these paradom YO U embark. . . n avoynge, which may include hallucinations and tram . . .But you come out knowing something you didn't knowbefore, something about the nature of where you arc in theuniverse"."

Even in the context of inquiries into the structureof logical form it became evident that dogmaticprohibitions as expressed in the Theory of LogicalTypes are untenable in a general theory of logicalfonns. For instance, according toCurryand Feys:' '"We can no longer 'explain' a paradox by runningaway from it; we must stand and took it in the eye."And, indeed, these authors not only looked para-doxes in the eye, but also constructed a whole classof operators, tl ~e paradoxical combinators," oneof which, Y, called "the paradoxical combinator,"may be used to construct logico-mathematicalobjects of a more or less paradoxical nature. Forinstance, Y may be used to construct Russell's

paradox and " .. the famous argument of Godelmay, evidently, be thought of as an application ofy.PPl.4The immediate precursor of Francisco Varela'sCalculus for Self-Reference(CRS) nd, most likely,a necessary prerequisite for understanding it, is.of course, G. Spencer Brown's Laws of Form.15Although a skeleton of the form of these laws isgiven by Vareia in the appendix to his paper, thoseunfamiliar with this formalism should get hold ofLaws of Form for the sake of enjoying an amazingbook and for relishing Varela's Calculus.The train of thoughts leading to CRS s initiatedby G . Spencer Brown's Calculus of Indications(CI), which is implemented with ultimate parsimonyby a single operator marked 1 a "distinctor"),which does several things at one time. Since we

cannot make an indication without drawing adistinction, when this mark is taken as a token forindicating the state distinguished by the distinctor,then 1 s zn "indicator" (for the state so marked isnow the marked state); a "signal" (signallingdistinction); and an "intentor" (since use of anysignal is intent). The state not marked with themark1 s the unmarked state.Rules for concatenating this operator to give aprimary arithmetic are determined by two axioms(no other ones are needed):Axiom 1. The aw of calling.The value of a call made again is the value of the call.That is to say, if a name is called and then is calledagain, the value indicated by the two calls takentogether is the value indicated by one of them.That is to say, for any name, to recall is to call.(In notation:

the "form of condensation.")Axiom 2. The law of crossing.The value of a crossing made again is not the ualue ofthe crossing.That is to say, if it is intended to cross a boundaryand then it is intended to cross it again, the valueindicated by the two intentions taken together is thevalue indicated by none of them.That is to say, for an y boundary, to recross is notto cross.(In notation:

7sthe "form of cancellation.")


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COMMENTS TO VARELA'S CALCULUS 3

With two initialsInitial 1. Position

- =Initial 2. Transposition' r n l=mIthe primary algebra is established.Letf(X) be the form of an algebraic expression,then expressions of arbitrary length can be estab-lished recursively (where Xi = f ( X i+ l ) and, e .g.,f 2 ) ( x i ) f(f(Xi)) etc.):

y = f "'(X,).For n -+ these become recursive expressionsof indefinite length, an d because of the identity

y = lirn f("-')(X,-,) = lim f'"'(Xn)n -m n -m

the function may re-enter its own scope to give,when s o collapsed

the formal equivalent to Russell's paradoxf =A

Thanks to the ingenious notation, the bi-stablenature of such expressions becomes transparent,forcing a new logical dim en sio k which Spencer-Brown interprets as Time.Starting from precisely this point, Varella goesthrough Russell's argument backwards, interpretingthis bi-stability as indicating indication: self-indication, self-reference, autonomy, which heindicates by the stylized symbol of the snake eatingits own tail:

The calculus now being developed from thecalculus of indication augmented by the state ofself-indica:ion or otttono_my has not only logicalbut also epistemological significance. In placingthe autonomy of the observer at the center of hisphilosophy, Kant's intention was not to effect a

shift from objectivity to subjectivity but rather toinitiate an ethics, for he clearly saw that withoutautonomy there could be no responsibility andhence no ethics.' Ethics-and not subjectivity-isthe com plem ent of objectivity. Lying-and no tobjectivity-is the problem and the force of theparadox of the Cretan liar. With his calculus ofthe paradoxical, the sel f-refe rential, the autono-mous, Varela has opened for the first time thepossibility of a Calculus of Responsibility.

REFERENCESI. A. N. whitehead and B. Russell, Principio Mathematico,Second Edition, University Press, Cambridge, 1925.2. lbid, p. 37.3. lbid, p. 61.4. R . Abramovitz er 01.. Cybernetics of Cyb erne tics,B.C.L. Report No. 73.38, Biological Computer Labora-tory, University of Illinois, Urbana, 1974, p. 374.5. T. Kuhn, Tlre Structrrre of Sciertti/ic Revolurions,University of Chicago Press, Chicago, 1961.6. W. R. Ashby, "Systems and their informationalrncasures." In: Tretld~ rr Cerreral Sysrems T heor y,edited by G . J. Kl ir , John Wiley, New York, 1972,pp. 78-97.7. S. Beer, Decbiorr uric1 Conrrol, John Wiley, New York,1964.8. H. R. Maturana, Bio/ogy oJ~Cognitio)t,B.C.L. ReportNo. 9.0, Biological Computer Laboratory, Universityo f l l l i ~ ~ o i i ,rbana, 1970.9. 1. Kant, Kririlik k r reir~e n Vernurtfr, Kiiniglich Preus-sische Akal lemie der Wissenschaften, Berlin, 1903.10. C. aleson, Steps to or1 Ecolofy o j Min d, Ballantine,

Ncw York, 1972.1 1 . G. ateson, D. D. Jackson, J. Haley, and J. H. Weak-land, "Toward a theory of schizophrenia," Behavior01Scietrre, 1, No. 4 , 1956.12. S. Brand, "Both sides of the necessary paradox (Con-veru tions with Gregory Bateson)." I n I1 CyberrteticFrorltiers, by S. Brand, Random House, New York,1974, pp. 9-36.13. H. B. Curry and R. Feys, Combinarory Logic, NorthHolland, Amsterdam, p. 5.14. Ibid, p. 178.15. G. Spencer-Brown, Lrr~ns j Form, First edition :GeorgeAllen and Unwin. London, 1969. Second edition:Jul ian Press, New York, 1972.16. 1. Kant, Grwrdlegutrg zur Meta pllysik der Sirten.Koniglich Preussische Akadernie dcr Wissenschaften,Berlin, 1903.

introductory comments to francisco varela

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