Louis H. Kauffman- Reflexivity and Eigenform: The Shape of Process

Constructivist Foundations 121http://www.univie.ac.at/constructivism/journal/

Reflexivity and Eigenform

The Shape of Process

1. Introduction

“Reflexive” is a term that refers to the pres-ence of a relationship between an entity anditself. One can be aware of one’s ownthoughts. An organism produces itselfthrough its own action and its own produc-tions. A market or a system of finance is com-posed of actions and individuals, and theactions of those individuals influence themarket just as the global information fromthe market influences the actions of the indi-viduals. Here it is the self-relations of themarket through its own structure and the

structure of its individuals that moves its evo-lution forward. Nowhere is there a way to cutan individual participant from the marketeffectively and make him into an objectiveobserver. His action in the market is concom-itant to his being reflexively linked with thatmarket. It is just so for theorists of the mar-ket, for their theories, if communicated,become part of the action and decision-mak-ing of the market. Social systems partake ofthis same reflexivity, and so does apparentlyobjective science and mathematics. In orderto see the reflexivity of the practice of physicalscience or mathematics, one must leave theidea of an objective domain of investigation

in brackets and see the enterprise as a wide-ranging conversation among a group ofinvestigators. Then, at once, the process isseen to be a reflexive interaction among themembers of this group. Mathematical results,like all technical inventions, have a certainstability over time that gives them an air ofpermanence, but the process that producesthese novelties is every bit as fraught with cir-cularity and mutual influence as any otherconversation or social interaction.

How then, shall we describe a reflexivedomain? It is the purpose of this paper to givea very abstract definition that neverthelesscaptures what I believe to be the main con-ceptual feature of reflexivity. We then imme-diately prove that eigenforms, fixed points oftransformations, are present for all transfor-mations of the reflexive domain. This willencourage us and will give us pause to thinkfurther about the relationship of reflexivityand eigenform.

The existence of eigenforms will encour-age us, for we have previously studied themwith the notion that “objects are tokens foreigenbehavior.” Eigenforms are the naturalemergence of those tokens by way of recur-sion. So to find the eigenforms dictated by alarger concept is pleasing. The existence offixed points for arbitrary transformationsshows us that the domain we have postulatedis indeed very wide. It is not an objectivelyexisting domain. It is a clearing in whichstructures can arise and new structures canarise. A reflexive domain is not an already-existing structure. To be what it claims to be,a reflexive domain must be a combination ofan existing structure and an invitation to cre-ate new structures and new concepts. Thenew will become platforms from which fur-ther flights of creativity can be made. Thus inthe course of examining the concept of reflex-ivity we will find that the essence of the mat-ter is an opening into creativity; and that willbecome the actual theme of this paper.

Louis H. Kauffman

A

University of Illinois at Chicago <[email protected]>

Mathematical

concepts in second-order cybernetics

R

Purpose:

The paper discusses the concept of a reflexive domain, an arena where the apparent objects as entities of the domain are actually processes and transformations of the domain as a whole. Human actions in the world partake of the patterns of reflexivity, and the productions of human beings, including science and mathematics, can be seen in this light.

R

Methodology

– Simple mathematical models are used to make conceptual points. Context – The paper begins with a review of the author’s previous work on eigenforms – objects as tokens for eigenbehaviors, the study of recursions and fixed points of recursions. The paper also studies eigenforms in the Boolean reflexive models of Vladimir Lefebvre.

R

Findings

–The paper gives a mathematical definition of a reflexive domain and proves that every transformation of such a domain has a fixed point. (This point of view has been taken by William Lawvere in the context of logic and category theory.) Thus eigenforms exist in reflexive domains. We discuss a related concept called a “magma.” A magma is composed entirely of its own structure-preserving transformations. Thus a magma can be regarded as a model of reflexivity and we call a magma “reflexive” if it encompasses all of its structure-preserving transformations (plus a side condition explained in the paper). We prove a fixed point theorem for reflexive magmas. We then show how magmas are related to knot theory and to an extension of set theory using knot diagrammatic topology. This work brings formalisms for self-reference into a wider arena of process algebra, combinatorics, non-standard set theory and topology. The paper then discusses how these findings are related to lambda calculus, set theory and models for self-reference. The last section of the paper is an account of a computer experiment with a variant of the Life cellular automaton of John H. Conway. In this variant, 7-Life, the recursions lead to self-sustaining processes with very long evolutionary patterns. We show how examples of novel phenomena arise in these patterns over the course of large time scales.

R

Value

– The paper provides a wider context and mathematical conceptual tools for the cybernetic study of reflexivity and circularity in systems.

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Key words

– Reflexive, eigenform, cybernetics, Boolean algebra, knots, magma, Russell paradox, cellular automata.

4/3/121.kauffmanVolume 4 · Number 3 · July 2009

122 Constructivist Foundations


Louis H. Kauffman

This essay begins with a discussion of thenotion of “eigenform” as pioneered by Heinzvon Foerster in his papers (Foerster 1981a–c)and explored in papers by the author (Kauff-man 1987, 2003, 2005). We include some ofthe material from (Kauffman 2005) in thispaper for the sake of completeness. In (Foer-ster 1981a) the familiar objects of our exist-ence can be seen to be nothing more thantokens for the behaviors of the organism, cre-ating apparently stable forms.

In this view, the object is

both

an elementof a world

and

a token or symbol for the pro-cess of its production/observation.

An object, in itself, is a symbolic entity,participating in a network of interactions,taking on its apparent solidity and stabilityfrom these interactions. We ourselves aresuch objects: we, as human beings, are “signsfor ourselves,” a concept originally developedby the American philosopher, Charles S.Peirce (Kauffman 2001). Eigenforms aremathematical companions to Peirce’s work.

In an observing system, what is observedis not distinct from the system itself, nor canone make a complete separation between theobserver and the observed. The observer andthe observed stand together in a coalescenceof perception. From the stance of the observ-ing system, all objects are non-local, depend-ing upon the presence of the system as awhole. It is within that paradigm that thesemodels begin to live, act and enter into con-versation with us.

After this journey into objects and eigen-forms, we take a wider stance and considerthe structure of spaces and domains that par-take of the reflexivity of object and process. InSection 6 we give a definition of a

reflexivedomain

. Our definition populates a space(domain) with entities that could be con-strued as objects, and we assume that eachobject acts as a transformation on the space.Essentially this means that given entities

A

and

B

, there is a new entity

C

that is the resultof

A

and

B

acting together in the order

AB

(sothat one can say that “

A

acts on

B

” for

AB

and“

B

acts on

A

” for

BA

). This means that thereflexive space is endowed with a non-com-mutative and non-associative algebraicstructure. The reflexive space is expandablein the sense that whenever we define a pro-cess, using entities that have already beenconstructed or defined, then that process cantake a name, becoming a new entity/transfor-

mation of a space that is expanded to includeitself. Reflexive spaces are open to evolutionover time as new processes are invented andnew forms emerge from their interaction.

Remarkably, reflexive spaces always haveeigenforms for every element/transforma-tion/entity in the space! The proof is simplebut requires discussion.

Just as promised, in a reflexive domain,every entity has an eigenform. From thisstandpoint, one should start with the conceptof reflexivity and see that from it emergeeigenforms. Are we satisfied with thisapproach? We are not. In order to start withreflexivity, we need to posit objects and pro-cesses. As we have already argued in this essay,objects are tokens for eigenbehaviors. And acorrect or natural beginning is a processwhere objects are seen as tokens of processes.

By now the reader begins to see that thestory we have to tell is a circular one. We givea way to understand this circularity in our lastsection, where we discuss creativity in recur-sive processes and the emergence of novelty.

The paper continues in Section 6 bystudying an allied concept that we call a

magma

. A magma is a domain with a binaryoperation * that allows one to combine ele-ments

a

and

b

of the domain to form a newelement

a

*

b

of that domain. In the magmaeach element

a

is also a mapping of thedomain to itself via left combination:

x

→

a

*

x

. We assume that each such trans-formation preserves the structure of thecombinatory operation. Magmas are veryclose in concept to reflexive domains. Wedefine the notion of a reflexive magma andshow that such magmas satisfy a fixed pointtheorem and so contain eigenforms. In Sec-tion 7 we show how magmas arise naturallyin the context of knot theory and a theory ofknot sets. Sections 8 and 9 discuss the rela-tionships of reflexivity with the lambda cal-culus of Church and Curry and with Cantor’sdiagonal argument and the Russell paradox.Section 10 is a minimalist discussion of self-reference and reflexivity in relation to theconceptualization of a universe that comes toobserve itself. Section 11 is an account of a

computer experiment with a variant of theLife cellular automaton of John H. Conway.In this variant that we have discovered, 7-Life, the recursions lead to self-sustainingprocesses with very long evolutionary pat-terns. We show how examples of novel phe-nomena arise over the course of large timescales. This example will be a later spring-board for the discussion of the emergence ofnovelty from deterministic processes. Here, itis an example showing how the course of aprocess is just as important as its eigenformor infinite concatenation.

The paper ends with a discussion of thewider context of reflexivity. We are acutelyaware that this paper about reflexivity onlygives certain conceptual tools and does not yetaddress the actuality of the reflexive conditionof persons and observers who are inextricablypart of the universes that they hope to study.In so doing they will adopt points of view andthese very points of view will create patterns,new forms, objects of study and will act as aveil over the original intent. It is only throughworking with many points of view and manyinvestigations that the particularities of singlelenses will begin to fall away and a widerunderstanding will emerge.

2. Objects as tokens for eigenbehaviors

In his paper

Objects as Tokens for Eigenbehav-iors

, von Foerster (1981a) suggests that wethink seriously about the mathematical struc-ture behind the constructivist doctrine that

perceived worlds are worlds created by theobserver.

At first glance such a statementappears to be nothing more than solipsism. Atsecond glance, the statement appears to be atautology, for who else can create the rich sub-jectivity of the immediate impression of thesenses? At third glance, something more isneeded. In that paper he suggests that thefamiliar objects of our experience are the fixedpoints of operators. These operators

are

thestructure of our perception. To the extent thatthe operators are shared, there is no solipsismin this point of view. It is the beginning of amathematics of second order cybernetics.

Consider the relationship between anobserver

O

and an “object”

A

. The key pointabout the observer and the object is that “theobject remains in constant form with respect

Given

F

in a reflexive domain.Define

G

by

Gx

=

F

(

xx

).

Then

GG

=

F

(

GG

) and so

GG

is aneigenform for

F

.

Volume 4 · Number 3 · July 2009 123

concepts in second-order cybernetics

Mathematical

to the observer.” This constancy of form doesnot preclude motion or change of shape.Form is more malleable than the geometry ofEuclid. In fact, ultimately, the form of an“object” is the form of the distinction that “it”makes in the space of our perception. In anyattempt to speak absolutely about the natureof form we take the form of distinction for theform (paraphrasing Spencer-Brown 1969). Itis the form of distinction that remains con-stant and produces an apparent object for theobserver. How can you write an equation forthis? The simplest route is to write

O

(

A

) =

A

.

The object

A

is a fixed point for theobserver

O

. The object is an eigenform. Wemust emphasize that this is the most schemat-ically possible description of the condition ofthe observer in relation to an object

A

. Weonly record that the observer as an actor(operator) manages through his acting toleave the (form of) the object unchanged.This can be a recognition of the symmetry ofthe object but it also can be a description ofhow the observer, searching for an object,makes that object up (like a good fairy tale)from the very ingredients that are theobserver herself. This is the situation thatHeinz von Foerster has been most interestedin studying. As he puts it, if you give a personan undecideable problem, then the answerthat he gives you is a description of himself.And so, by working on hard and undecideableproblems we go deeply into the discovery ofwho we really are. All this is symbolized in thelittle equation

O

(

A

) =

A

.And what about this matter of the object as

a token for eigenbehavior? This is the crucialstep. We forget about the object and focus onthe observer. We attempt to “solve” the equa-tion

O

(

A

) =

A

with

A

as the unknown. Notonly do we admit that the “inner” structure ofthe object is unknown, we adhere to whateverknowledge we have of the observer andattempt to find what such an observer couldobserve based upon that structure.

We can start anew from the dictum thatthe perceiver and the perceived arise togetherin the condition of observation. This is astance that insists on mutuality (neither per-ceiver nor the perceived causes the other). Adistinction has emerged and with it a worldwith an observer and an observed. The dis-tinction is itself an eigenform.

3. Compresence and coalescence

We identify the world in terms of how weshape it. We shape the world in response tohow it changes us. We change the world andthe world changes us. Objects arise as tokensof a behavior that leads to seeminglyunchanging forms. Forms are seen to beunchanging through their invariance underour attempts to change, to shape them.

For an observer there are two primarymodes of perception –

compresence

and

coa-lescence

. Compresence connotes the coexist-ence of separate entities together in oneincluding space. Coalescence connotes theone space holding, in perception, the observerand the observed, inseparable in an unbrokenwholeness. Coalescence is the constant condi-tion of our awareness. Coalescence is theworld taken in simplicity; compresence is theworld taken in apparent multiplicity.

This distinction between compresenceand coalescence, drawn by Henri Bortoft(1971), can act as a compass in traversing thedomains of object and reference.

Eigenform isa first step towards a mathematical descriptionof coalescence

. In the world of eigenform, theobserver and the observed are one in a processthat recursively gives rise to each.

4. The eigenform model

We have seen how the concept of an object hasevolved to make what we call objects (and theobjective world), processes that are interde-pendent with the actions of observers. Thenotion of a fixed object has become a notionof a process that produces the apparent stabil-ity of the object. This process can be simpli-fied in a model to become a recursive processwhere a rule or rules are applied time and timeagain. The resulting object of such a process isthe

eigenform

of the process, and the processitself is the

eigenbehavior.

In this way we have a model for thinking

about object as token for eigenbehavior. Thismodel examines the result of a simple recur-sive process carried to its limit. For example,suppose that

That is, each step in the process enclosesthe results of the previous step within a box.Here is an illustration of the first few steps ofthe process applied to an empty box,

X

:

If we continue this process, then successivenests of boxes resemble one another, and inthe limit of infinitely many boxes, we find that

The infinite nest of boxes is invariantunder the addition of one more surroundingbox. Hence this infinite nest of boxes is a fixedpoint for the recursion. In other words, if

X

denotes the infinite nest of boxes, then

X

=

F

(

X

).

This equation is a description of a state ofaffairs. The form of an infinite nest of boxes isinvariant under the operation of adding onemore surrounding box. The infinite nest ofboxes is one of the simplest eigenforms.

Remark. On reading the above descriptionof the limiting process

X

→

F

(

X

)

→

F

(

F

(

X

))

→

…

the reader may find herself thinking along thefollowing lines: “Doesn’t he mean to put thosethree dots in the nested boxes on the

outside

ofthe boxes rather than on the inside? After all,

F(X) =

F(X)

F(F(X)) F(F(F(X)))

X

X = F(F(F(…))) = …

F(X) = … = X



Louis H. Kauffman

the operation

F

surrounds

X

with a square, soat each stage, a square is added from the out-side. Shouldn’t the picture then be like thisone below?”

“I have illustrated the new picture withthe three dots on the left, the right, the topand the bottom to show how in this way ofthinking the nest of boxes grows outward andconsequently it grows in all these directions.If we take this construction to infinity, then itwill either fill the plane with boxes, or thewidths between successive boxes will have togrow smaller and smaller, just as, with thethree dots inside, you had to make the boxessmaller and smaller. But really, this secondpicture is quite different from the first pic-ture. In fact if we

do

make the second picture,and imagine that it is a solution to the equa-tion F(X) = X, it does not seem to be a solu-tion! Look at the picture below.”

“Now I have put a box around the out-wardly growing infinite nest of boxes, butthis means that I have allowed an infinitenumber of boxes to grow there (going out butstaying in a finite amount of space by crowd-ing one next to another) and then I put onemore box around all of them. The result is notthe same! This is a new form of boxes.

So with the outward growth, I make newinfinities, but I do not solve the equation

X = F(X). Now I see what you were doingwith the inward nest of boxes. You let it growinwardly and obtained a limit form that didnot see the one box more that you put aroundthe outside. I had to try this other method inorder to see what you were doing. And I amsure that other readers will have to experi-ment in this way and in new ways to reallyunderstand this construction of eigenforms.”

Comment on the remark. Indeed the patientreader was right that there is more than oneway to go to infinity. A simpler example canbe seen in the equation x = ax where we solveit by letting x = aaa…, an infinite repetition ofa’s going off to the right.

Then ax = a(aaa…) = aaaa… = aaa… = x.But if we do it in the other order and take

xa, we find that

xa = aaaa…a

which means an infinite row of a’s followed byone more a.

And we see that in this way of thinking xais not equal to ax.

Similarly, y = …aaa is not a solution toay = y but it is a solution to ya = a. This mayseem a bit strange and abstract, so it is betterto think with the boxes (I think). But in ordi-nary mathematics we use this same sort ofinfinite construction. For example, we write

x = 1 + a + aa + aaa + aaaa + …

and rewrite it in the form

x = 1 + a(1 + a + aa + aaa + …) = 1 + ax

and conclude that

x – ax = 1, x(1 – a) = 1,

x = 1 / (1 – a).

Hence

1 / (1 – a) = 1 + a + aa + aaa + aaaa + …

Here we are using aa for the product of awith itself, so these can be numbers. And onecan verify that indeed if a is a number and theabsolute value of a is less than one, then thisformula is true. For example, if a = 1 / 2, then1 / (1 – (1 / 2)) = 2 and the formula asserts that

2 = 1 + 1/2 + 1/4 + 1/8 + 1/16 + …

This is true, and the reader should ask her-self how she knows that it is true! The readerwill also be interested in seeing what happens

when a is bigger than or equal to 1 in absolutevalue. For example, if a = 2, then our formulawould seem to say that

–1 = 1 + 2 + 4 + 8 + 16 + 32 + …

Is there some truth in this absurdity?We can see what has actually happened by

making a closer analysis.Let X = 1 + 2 + 4 + … + 2N where 2N means

2 multiplied by itself N times. Then we have

X = 1 + 2(1 + 2 +… + 2N–1),

X = 1 + 2(1 + 2 +… + 2N–1 + 2N) – 2(2N),

X = 1 + 2X – 2N + 1,

So we have

X – 2X = 1 – 2N + 1,

which is the same as saying

X = –1 + 2N + 1.

Do you see what has happened? We areinterested in finding out what happens when Ngoes to infinity. But here if we ignore the term2N+1 we will get the wildly wrong answer of –1.You have to take infinity with a grain of salt aswell as looking at it as the vastness of all thegrains of sand on the beach. End of comment.

A further comment. Perhaps you thought thatwe showed that the equation: –1 = 1 + 2 + 4 +… is wrong. There is a point of view in whichit is right! Consider that in binary arithmeticwe represent 1 by 1, 2 by 10, 4 by 100, 8 by 1000and so on. Then 1 + 2 + 4 + 8 is represented inbinary by 1111, and when you add 1 to 1111you find a series of carrys taking you to theanswer 10000. Suppose you had a computerthat could only handle binary numbers up tofour bits. Then when you added 1 to 1111 youwould get 0000, since the computer wouldthrow away the last bit. In this sense 1111 rep-resents –1 in such a limited computer, and inthe same way the infinite sum 1 + 2 + 4 + 8 +… represents –1 in an infinite computer thatis not prepared to have bits beyond the firstinfinity!

In the process of observation, we interactwith ourselves and with the world to producestabilities that become the objects of our per-ception. These objects, like the infinite nest ofboxes, may go beyond the specific properties


concepts in second-order cyberneticsMathematical

of the world in which we operate. They attaintheir stability through the limiting processthat goes outside the immediate world ofindividual actions. We make an imaginativeleap to complete such objects to becometokens for eigenbehaviors. It is impossible tomake an infinite nest of boxes. We do notmake it. We imagine it. And in imagining thatinfinite nest of boxes, we arrive at the eigen-form.

The leap of imagination to the infiniteeigenform is a model of the human ability tocreate signs and symbols. In the case of theeigenform X with X = F(X), X can be regardedas the name of the process itself or as the nameof the limiting process. Note that if you aretold that

X = F(X),

then, substituting F(X) for X, you can write

X = F(F(X)).

Substituting again and again, you have

X = F(F(F(X))) = F(F(F(F(X)))) = F(F(F(F(F(X))))) = …

The process arises from the symbolicexpression of its eigenform. In this view, theeigenform is an implicate order for the processthat generates it. (Here we refer to implicateorder in the sense of David Bohm 1980).

Sometimes one stylizes the structure byindicating where the eigenform X reenters itsown indicational space with an arrow or othergraphical device. See the picture below for thecase of the nested boxes.

Does the infinite nest of boxes exist? Cer-tainly it does not exist on this page or any-where in the physical world with which we arefamiliar. The infinite nest of boxes exists in theimagination. It is a symbolic entity.

The eigenform is the imagined boundaryin the reciprocal relationship of the object(the “It”) and the process leading to the object(the process leading to “It”). In the diagram

below we have indicated these relationshipswith respect to the eigenform of nested boxes.Note that the “It” is illustrated as a finiteapproximation (to the infinite limit) that issufficient to allow an observer to infer/per-ceive the generating process that underlies it.

Just so, an object in the world (cognitive,physical, ideal, etc.) provides a conceptualcenter for the exploration of a skein of rela-tionships related to its context and to theprocesses that generate it. An object can havevarying degrees of reality, just as an eigenformdoes. If we take the suggestion to heart thatobjects are tokens for eigenbehaviors, then anobject in itself is an entity, participating in anetwork of interactions, taking on its appar-ent solidity and stability from theseinteractions.

An object is an amphibian between thesymbolic and imaginary world of the mindand the complex world of personal experi-ence. The object, when viewed as a process, isa dialogue between these worlds. The object,when seen as a sign for itself, or in and of itself,is imaginary.

Why are objects apparently solid? Ofcourse you cannot walk through a brick walleven if you think about it differently. I do notmean apparent in the sense of thought alone.I mean apparent in the sense of appearance.The wall appears solid to me because of the

actions that I can perform. The wall is quitetransparent to a neutrino, and will not even bean eigenform for that neutrino.

This example shows quite sharply how thenature of an object is entailed in the proper-ties of its observer.

The eigenform model can be expressed inquite abstract and general terms. Supposethat we are given a recursion (not necessarilynumerical) with the equation

X(t + 1) = F(X(t)).

Here X(t) denotes the condition of obser-vation at time t. X(t) could be as simple as aset of nested boxes, or as complex as the entireconfiguration of your body in relation to theknown universe at time t. Then F(X(t))denotes the result of applying the operationssymbolized by F to the condition at time t. Youcould, for simplicity, assume that F is inde-pendent of time. Time independence of therecursion F will give us simple answers and wecan later discuss what will happen if theactions depend upon the time. In the time-independent case we can write

J = F(F(F(…)))

– the infinite concatenation of F upon itself.Then

F(J) = J

since adding one more F to the concatenationchanges nothing.

Thus J, the infinite concatenation of theoperation upon itself leads to a fixed point forF. J is said to be the eigenform for the recur-sion F. We see that every recursion has aneigenform. Every recursion has an (imagi-nary) fixed point.

We end this section with one more exam-ple. This is the eigenform of the Koch fractal(Mandelbrot 1982). In this case one can writesymbolically the eigenform equation

K = K { K K } K

to indicate that the Koch Fractal reenters itsown indicational space four times (that is, it ismade up of four copies of itself, each one-third the size of the original. The curly brack-ets in the center of this equation refer to thefact that the two middle copies within thefractal are inclined with respect to oneanother and with respect to the two outercopies. In the figure below we show the geo-metric configuration of the reentry.

… =

The It

The Process leading to It


Reflexivity and EigenformLouis H. Kauffman

In the geometric recursion, each line seg-ment at a given stage is replaced by four linesegments of one third of its length, arrangedaccording to the pattern of reentry as shownin the figure above.

The recursion corresponding to the Kocheigenform is illustrated in the next figure.Here we see the sequence of approximationsleading to the infinite self-reflecting eigen-form that is known as the Koch snowflakefractal.

Five stages of recursion are shown. To theeye, the last stage vividly illustrates how theideal fractal form contains four copies ofitself, each one-third the size of the whole.The abstract schema

K = K { K K } K

for this fractal can itself be iterated to producea “skeleton” of the geometric recursion:

K = K { K K } K = K { K K } K { K { K K } K K { K K } K } K {

K K } K = …

We have only performed one line of thisskeletal recursion. There are sixteen K’s in thissecond expression, just as there are sixteenline segments in the second stage of the geo-

metric recursion. Comparison with this sym-bolic recursion shows how geometry aids theintuition. The interaction of eigenforms withthe geometry of physical, mental, symbolicand spiritual landscapes is an entire subjectthat is in need of deep exploration.

It is usually thought that the miracle of rec-ognition of an object arises in some simpleway from the assumed existence of the objectand the action of our perceiving systems. Thisis fine tuning to the point where the action ofthe perceiver and the perception of the objectare indistinguishable. Such tuning requires anintermixing of the perceiver and the perceivedthat goes beyond description. Yet at the math-ematical levels, such as number or fractal pat-tern, part of the process is slowed down to thepoint where we can begin to apprehend theprocess. There is a stability in the comparison,in the correspondence that is a process hap-pening at once in the present time. The closedloop of perception occurs in the eternity ofpresent individual time. Each such processdepends upon linked and ongoing eigenbe-haviors and yet is seen as simple by the per-ceiving mind. The perceiving mind is itself aneigenform.

Mirror-mirrorIn the next figure we illustrate how an eigen-form can arise from a process of mutualreflection. The figure shows a circle with anarrow pointing to a rectangle and a rectanglewith an arrow pointing toward a circle. Forthis example, we take the rule that an arrowbetween two entities (P → Q) means that thesecond entity will create an internal image ofthe first entity (Q will make an image of P). IfP → Q and Q → P, then each entity makes animage of the other. A recursion will ensue.Each of P and Q generates eigenforms in thismutuality.

In this example we can denote the initialforms by C (for circle) and B (for box). Wehave C → B and B → C. The rule of imagingis (symbolically):

If P → Q then P → QP.If P ← Q, then PQ ← Q.

We start with the mutual reference C ↔ B.This condition of mutual mirroring can be

described by two operators C and B:

C(P) = CP corresponds to C → P.B(Q) = BQ corresponds to Q ← B.

Solving the eigenform equations

C(Y) = X, B(X) = Y,

we have the mirror-mirror solution

X = BCBCBCBC…, Y = CBCBCBCB…,

just as in the figure.We are quite familiar with this form of

mutual mirroring in the physical realm whereone can have two facing mirrors, and in therealm of human relations where the complex-ity of exchange (mutual mirroring) betweentwo individuals leads to the eigenform of theirrelationship.

5. Boolean self-reference and the work of Vladimir Lefebvre

Vladimir Lefebvre (1982) models ethical situ-ations involving multiple reflections (I thinkabout your thoughts about me, while youthink about my thoughts about you.) usingBoolean algebra and a graphical formalism.

The crux of this endeavor begins withexamining self-referential equations in theBoolean context. In this context we have thearithmetic of 0 and 1 with 1 + 1 = 1 and oth-erwise 0 and 1 behaving as in ordinary arith-metic, with complementation (a → a´) inter-changing them: 0´ = 1 and 1´ = 0.

In interpreting Boolean algebra for logic,we take a + b to mean “a or b” and ab to mean“a and b.”

Thus we take 1 as T (True) and 0 as F(False).

Note that a > b (a implies b) is representedby a´ + b in this system.

K = K { KK } K

then

then

then



A very simple form of Boolean self-refer-ence is the equation

x = x.

This just says that x is equal to x. It is likethe biblical “I am that I am.”

A diabolical form of Boolean self referenceis the equation

x = x´.

This says that x is equal to not x, and canbe interpreted as the statement of the liar whoasserts that he is lying, “I am a liar.” I like tothink of the solution to this equation as an os-cillation between 0 and 1.

After all, if x = 0, then x = x´ = 0´ = 1 and ifx = 1, then x = x´ = 1´ = 0.

So x oscillates just like a buzzer of a door-bell.

The simplest general form of a Booleanself-referential equation is

x = ax + bx´.

What are the possibilities?We can have

[ x = x + x´ = 1 (constantly true), [ x = 1x + 0x´ = x (self-affirming), [ x = 0x + 1x´ = x´ (self-denying), [ x = 0x + 0x´ = 0 (just false).

Thus there does not seem to be a lot ofstructure in this simplest version of self-refer-ence. However, we should think a bit furtherand realize that a and b can be propositionsthat have relative truth values and we may notneed to know the actual truth values of a andb. Consider the equation

x = (b´ + c)x + cx´.

If x = 0, then we have 0 = c.So we conclude that for x = 0 to be a solu-

tion, this equation reduces to x = x or x = 0.However, if x = 1 is a solution, then we have 1= b´ + c, and the equation will have a solutionjust so long as b implies c is true. In either casethe equation has a non-oscillatory solution.This is the form of the self-referential equa-tion at the base of Lefebvre’s analysis of ethicsand reflectivity.

The next thing to notice is that (a > b) > c= (a´ + b)´ + c = ab´ + c = (b´ + c)a + ca´.

Thus we have

a = (a > b) > c

as an allowable self referential Boolean equa-tion.

Lefebrve interprets the right hand side ofthis equation as “c thinks of b thinking of a.”Thus the self-reference is “a is thinking of cthinking of b thinking of a.”

Lefebvre takes ba as notation for “a impliesb.” Thus

ba = b + a´

and ba stands for a > b, which is interpreted as“b is thinking about a” or “b has an internalimage of a.”

Thus our self-referential equation be-comes a = cx where x = ba.

Using Laws of FormHere is a second take on this theme, usingLaws of Form (Spencer 1969) bracket nota-tion. In the Laws of Form notation, we takea´ = ⟨a⟩ and ab stands for a + b while the con-junction ab in Boolean algebra becomes⟨⟨a⟩⟨b⟩⟩ in accordance with DeMorgan’s Law.We also have 0 as the void state in Laws ofForm and 1 = ⟨ ⟩, the marked state, a singlecrossing from the void. Then the Booleanarithmetic of 0 and 1 corresponds to the Lawsof Calling ⟨ ⟩ ⟨ ⟩ = ⟨ ⟩ and Crossing ⟨⟨ ⟩⟩ = “void.”

In Laws of Form notation, “a implies b” iswritten as

⟨a⟩b = b⟨a⟩.

In Lefebvre’s notation this is the same as

b⟨a⟩ = ba.

Thus Laws of Form is a useful alternateformalism for this theory.

We can interpret b⟨a⟩ as “b thinks of a.”Consider the self-referential equation

a = ⟨⟨a⟩b⟩c.

“a is thinking of c thinking of b who thinksof a.”

This is a self-reference that can be madeinside two-valued primary arithmetic, since itnever oscillates like a = ⟨a⟩. You can think ofthis fixed point in the form of the infinitereentry:

a = ⟨⟨⟨⟨⟨⟨⟨⟨⟨⟨…⟩b⟩c⟩b⟩c⟩b⟩c⟩b⟩c⟩b⟩c.

It is amusing to write this in ordinaryBoolean form as a = c + da where d = ⟨b⟩ andxy = ⟨⟨x⟩⟨y⟩⟩ and x + y replaces x y (LOF jux-taposition). Then we get

a = c + da

a = c + d(c + da) = c + dc + d2a

and so on,

a = c + d2c + d3c + d4c + …

a = “c / (1 – d).”

The infinite reentry expressions in LOFbecome an infinite power series in Booleanalgebra. This brings us closer to classicalmathematics and its role in producing imagi-nary values.

Lefebvre (1982) in his “Algebra of Con-science” models structures such as

a = ⟨b⟩a

b = ⟨a⟩b

as “a thinking about himself with an image ofb” and “b thinking about himself with animage of a.”

We can use the LOF notation to representthe self-referential algebra of Lefebvre, and itis useful to do this.

It is important to see how fixed point equa-tions and reflexivity are intertwined in theBoolean structure. One might think thatthese concepts would not live in the Booleancontext, but of course we do manage to dis-cuss them in the Boolean context of our ownthought. So Lefebvre’s model is a microcosmof our condition, and of course this is exactlythe point!

This section is just a small introduction toLefebrvre’s theory of reflexivity. It is worthpointing out that he uses the Boolean back-ground skillfully when it is required, but usesthe symbolism of reflection on the surface ina way that corresponds to nested linguisticstatements. For example, a⟨a⟨a⟩⟩ represents “athinking about a, who has a self-image thatcorresponds to the true (external) a.” Whenwe evaluate this expression we find

a⟨a⟨a⟩⟩ = a⟨⟨ ⟩⟩ = a.

Thus the non-self-doubting a is simplyhimself.

On the other hand, a⟨a⟨ ⟨a⟩ ⟩⟩ represents awith an image of himself whose image of him-self is false (⟨a⟩). Evaluating this expression,we find

a⟨a⟨⟨a⟩⟩⟩ = a⟨a a⟩ = a ⟨a⟩ = ⟨ ⟩.

Thus the individual with a doubting selfimage receives a marked value for his skepti-



cism. What about an individual who directlydoubts himself? Then we have a ⟨ ⟨a⟩ ⟩ = a a =a. He is in the same boat as the individualwith a self-image who doubts. From theseexamples, we see that the Lefebvre systemneeds to be examined carefully for its internalmeanings. This will be the subject of anotherpaper.

6. Reflexive domains and the magmaA reflexive domain D is an arena whereactions and processes that transform thedomain can also be seen as the elements thatcompose the domain. Every element of thedomain can be seen as a transformation ofthe domain to itself.

In actual practice, an element of a domainmay be a person or company (collective ofpersons) or a physical object or mechanismthat is seen to be in action. In actual practicewe must note that what are regarded asobjects or entities depends upon the way inwhich observers inside or outside the domaindivide their worlds.

It is very difficult to make a detailed math-ematical model of such situations. Each actoris an actor in more than one play. His actionsundergo separate but related interpretations,depending upon the others with whom heinteracts. Mutual feedback of a multiplicityof ongoing processes is not easily described inthe Platonic terms of pure mathematics.

Nevertheless, we take as a general princi-ple for a mathematical model that D is a cer-tain set (possibly evolving in time), and we let[D, D] denote a selected collection of map-pings from D to D. An element F of [D, D] isa mapping F: D → D.

We shall assume that there is a one-to-onecorrespondence

I: D → [D, D].

This is the assumption of reflexivity. Everyelement of the reflexive domain is a transfor-mation of that domain. Each denizen of thereflexive domain has a dual role of actor andactant.

Given an element g in D, I(g): D → D is amapping from D to D, and for every mappingF: D → D, there is an element g in D such thatI(g) = F. The reflexive domain embodies aperfect correspondence between actions and

entities that are the recipients of theseactions. See Dana Scott (1980) for a specificconstruction of reflexive domains relevant tocomputer science and logic.

An important precursor to this notion ofreflexive domain in mathematics is thenotion of Gödel numbering of texts. Onechooses a method to encode a text as a spe-cific natural number (a certain product ofprime powers). Then texts that speak aboutnumbers can, in principle, speak about othertexts and even about themselves. If a text isseen as a transformation on the field of num-bers, then that text is itself a number (itsGödelian code) and so can be transformingitself. The precision of this idea enabledGödel to construct mathematical systemsthat could talk about their own propertieswithout contradiction and he showed that allsufficiently rich mathematical systems havethis property. In this way, these systemsbecome self-limiting due to the possibility ofstatements whose coded meaning becomes“This statement has no proof in the system ofmathematics in which it is written, “while thesurface meaning of the same statement is adiscussion of the properties of certainnumerical relations. The domain of numeri-cal relations appears innocuous, and yet itsows the seeds of its own limitations throughthis ability to reflect itself through the mirrorof the Gödel coding.

The Gödelian example is not just a pieceof mathematics. It is a reflection with mathe-matical precision of the condition of our lan-guage, thought and action. We are alwaysequipped to comment on our own doingsand in so doing to create new language aboutour old language and new language aboutour worlds. All our apparent well-thought-out and directed actions in worlds that seemto extend outward from us in an objectiveway are fraught with the circularity not justof our meta-comments, but also with the cir-cular return of the consequences of thoseactions and the influence of our very theoriesof the world on the properties of that worlditself.

We now prove a fundamental theoremabout reflexive domains.

We show that every mapping F: D → Dhas a fixed point p, an element p in D suchthat F(p) = p. What does this mean? It meansthat there is another way, in a reflexivedomain, to associate a point to a transforma-

tion. The point can be seen as the fixed pointof a transformation and in that way, thepoints of the domain disappear into the self-referential nature of the transformations.

Let me tender persuasions. Suppose thatp = F(p). Then we can regard this equation asan expression of p in terms of F and itself andwrite

p = F(p)= F(F(p)= F(F(F(p)))= F(F(F(F(p))))

and continue in this fashion until the appear-ance of p on the right hand side is lost in thedepths of the composition of F upon itself.

p = F(F(F(F(F(F(F(F(F(F(…)))))))).

The infinite composition of F upon itselfis invariant under one more compositionwith F and so F(p) = p is consistent with thisprocess.

To show that an entity p is a fixed point fora process F is to show that p can be formallyidentified with the infinite concatenation ofF upon itself. This is an image of the wayobjects become tokens for eigenbehaviors, inthe language of Heinz von Foerster.

Here we show that eigenforms exist inreflexive domains without an infinite limit.The interested reader should compare thisargument with the work of William Lawvere(1972). Lawvere proves a more general resultin the context of Cartesian closed categories.We have taken his argument and shaped itparticularly for this discussion of reflexivity.

Fixed Point Theorem. Let D be a reflexivedomain with 1–1 correspondence F: D → [D,D]. Then every F in [D, D] has a fixed point.That is, there exists a p in D such that F(p) = p.

Proof. Define G: D → D by the equation Gx =F(I(x)x) for each x in D.

Since I:D → [D, D] is a 1 – 1 correspon-dence, we know that G = I(g) for some g in D.

Hence Gx = I(g)x = F(I(x)x) for all x in D. Therefore, letting x = g, I(g)g = F(I(g)g)

and so p = I(g)g is a fixed point for F.

Q.E.D.

We shall discuss this proof and its mean-ing right now in a series of remarks, and laterin the paper in regard to examples that will beconstructed.



Remark 1Suppose that we reduce the notational com-plexity of our description of the reflexivedomain by simply saying that for any two enti-ties g and x in the domain there is a new entitygx that is the result of the interaction of g andx. (We think of gx as I(g)x = I(g) applied to x.)

In mathematical terms, we define

gx = I(g)x.Then the proof of the fixed point theorem

appears in a simpler form: we define Gx =F(xx) and note that GG = F(GG).

Thus GG is the fixed point for F!I like to call G “F’s Gremlin” (Kauffman

2001).1

This is an apt description of our G. At firstG looks quite harmless. Applying G to any Awe just apply A to itself and apply F to theresult. GA = F(AA). The dangerous mixturecomes when it is possible to apply G to itself!Then GG = F(GG), and GG is sitting right inthere surrounded by F and you cannot stopthe action. Off goes the recursionGG = F(GG)

= F(F(GG))= F(F(F(F(GG))))= F(F(F(F(F(F(F(F(GG))))))))The diabolical nature of the Gremlin is

that he represents a process that once started,is hard to stop.

Gremlins seem innocent. They just dupli-cate entities that they meet, and set up anoperation of the duplicate on the duplicand.But when you let a gremlin meet a gremlinthen strange things can happen. It is a bit likethe story of the sorcerer’s apprentice. A recur-sion may happen whether you like it or not.

Such are the processes by which we makethe world into a field of tokens and symbolsand forget the behaviors and processes andreflexive spaces from which they came. Fixedpoints and self-references are the unavoidablefruits of reflexivity, and reflexivity is the natu-ral condition in a universe where there is nocomplete separation of part from the whole.

Remark 2A reflexive domain is a place where actionsand events coincide. An action is a mappingof the whole space because there is no intrin-sic separation of the local and the global.Feedback is an attempt to handle the lack ofseparation of part and whole by describingtheir mutual influence.

When we define a new element g of D viagx = F(x) for any mapping F: D → D, and wehave a notion of the combination of elementsof D: a, b → ab, then we can define gx = F(xx)and so get gg = F(gg). Here we have not madea big separation between the elements of Dand the mappings, since each element g of Dgives the mapping I(g)x = gx. But in fact, wecould define ab = I(a)b in a reflexive domain.

Whenever anyone comes up with a trans-formation, we make that transformation intoan element of the domain by the definition gx= F(x). We transmute verbs to nouns. Thereflexive domain evolves.

The space is not given a priori. The spaceevolves in relation to actions and definitions.The road unfolds before us as we travel.

Remark 3We create languages for evolving concepts.The outer reaches of set theory (and categorytheory) lead to clear concepts, but these con-cepts are not themselves sets or categories. Agood example is the famous Russellian con-cept of sets that are not members of them-selves. Russell’s concept is not a set. Anotherexample is the concept of set itself. There is noset that is the set of all sets.

This very limitation on the notion of a setis its opening. It shows us that set theory canbe an evolving language. Language and con-cepts expand in time.

Here is a transformation on sets: F(X) ={X}. The transform of a set X is the singletonset whose member is X. If X is not a memberof itself, then F(X) is also not a member ofitself. But a fixed point of the transformationF is an entity U such that {U} = U. We haveshown that within the domain of sets that arenot members of themselves, there is no fixedpoint for the transformation X → {X}. Thisfragment of set theory (sets that are not mem-bers of themselves) is not yet a reflexivedomain. We shall allow sets that are membersof themselves if we wish to have a set theorywith reflexivity.

Remark 4: TranscendenceThe leap to infinity via self-reference, the pro-duction of the finite base of a new level ofinfinity, the completion of an incompletion,the emergence of eternity from the world oftime – all these metaphors are intimatelyrelated to the going back and forth between aprocess and its eigenform.

How then is observation different fromaction?

If observation is a form of recursion cou-pled with the production of the finite base ofthe limiting form, then observation is a tran-scendence to a new level. The model of obser-vation as a simple eigen-vector must beshifted to a model of observation as the act ofproducing an eigenform.

It is not enough to produce an eigenform.The fixed point is itself an active element andcan itself engage in transformation.

In the creation of spaces of conversationfor human beings, we partake of a reflexivityof action and apparent object, where it is seenthat every local manifestation of process,every seemingly fixed entity in a movingworld is an indicator of global transforma-tion. The local and the global intertwine in areflexive and cybernetic unity.

Retuning (returning/tuning/retuning) tothoughts of reflexivity, one creates by goingoutside oneself, but the creation returns in theform of a conversation with one’s self. Thereis a feedback loop between the person/designer and the world that she makes.

Each one acts in the creation of the other.Priorities may be assigned, but it is the loopthat interests us, and the possibility of the sta-bility (or at least temporal persistence) ofwhat is created in that loop.

Remark 5: The magma as reflexive domainA magma is an algebraic system with a binaryoperation a * b that is left-distributive: a * (b * c) = (a * b) * (a * c). This means thatevery element of the magma is a structure pre-serving mapping of the magma to itself (vialeft multiplication). A magma is composed ofits own symmetries.

It may help the reader to see how elementsof a magma become mappings of the magmato itself, preserving the combinational struc-ture. Let A(x) = a * x for a given element a ina magma M.

Then A: M → M and A(x * y) = a * (x * y)= (a * x) * (a * y) = A(x) * A(y).

Thus for all x and y in M we have A(x * y)= A(x) * A(y). Each element of the magmagives rise, by left multiplication, to a struc-ture-preserving mapping of the magma toitself.

Here is an example of a magma. Let TRI ={a, b, c} be a set with three elements a, b and c.



Define a * a = a, b * b = b and c * c = c.And define a * b = c = b * a, a * c = b = c * aand b * c = a = c * b.

In other words, each element combineswith itself to produce itself, and any pair ofdistinct elements combine to produce theremaining element that is different fromeither of them. The reader can verify that TRIis indeed a magma. For example,

a * (b * c) = a * (a ) = a(a * b) * (a * c) = (c) * (b) = a.

Note also that the multiplication in thismagma is not associative:

a * (a * b) = a * c = b(a * a) * b = a * b = c.

We will return to this magma in the nextsection and see that TRI is intimately relatedto the simplest knot, the trefoil knot.

Another example to think about is OM,the free magma generated by one element J.Here we consider all possible expressions andways that b can combine with itself and withother elements generated from itself.Remarkably, the free magma is an infinitelycomplex structure. For example, note the fol-lowing consequences of the distributive law(here using XY instead of X * Y):

J(JJ) = ((JJ)(JJ)) = ((JJ)J)((JJ)J)

= (((JJ)J)(JJ))(((JJ)J)J)).

In the free magma an infinite structure isgenerated from one element and all its pat-terns of self-interaction.

Suppose further that we assume that everystructure-preserving mapping of the magmaM is represented by an element of the magmaM. This will place us in the position of creat-ing from the magma something like a reflex-ive domain.

In the next section we shall see that mag-mas arise very naturally in the topology ofknots and links in three-dimensional space.This is an excellent way to think about them,and it provides a way to think about reflexiv-ity in terms of topology. Here we take anabstract point of view and see when the struc-ture-preserving nature of elements of amagma leads to the analog of a reflexivedomain.

I shall call a magma M reflexive if it has theproperty that every structure-preservingmapping of the algebra is realized by an ele-

ment of the algebra and (x * x) * z = x * z forall x and z in M.

A special case of this last property wouldbe where x * x = x for all x in M. We shall seethis property come up in the knot theoreticinterpretations of the next section.

Suppose that M is a reflexive magma. DoesM satisfy the fixed point theorem? We findthat the answer is, yes.

Fixed Point Theorem for Reflexive Magmas.Let M be a reflexive magma. Let F: M → M bea structure-preserving mapping of M to itself.Then there exists an element b in M such thatF(p) = p.

Proof. Let F: M → M be any structure-pre-serving mapping of the magma M to itself.This means that we assume that F(x * y) =F(x) * F(y) for all x and y in M.

Define G(x) = F(x * x) and regard G: M →M. Is G structure preserving? We must com-pare G(x * y) = F((x * y) * (x * y)) = F(x * (y *y)) with G(x) * G(y) = F(x * x) * F(y * y) =F((x * x) * (y * y)).

Since (x * x) * z = x * z for all x and z in M,we conclude that G(x * y) = G(x) * G(y) for allx and y in M.

Thus G is structure preserving and hencethere is an element g of M such that G(x) =g * x for all x in M. Therefore we have g * x =F(x * x), whence g * g = F(g * g). For p = g * g,we have p = F(p). This completes the proof.

Q.E.D.This analysis shows that the concept of a

magma is very close to our notion of a reflex-ive domain. The examples of magmas relatedto knot theory, given in the previous section,show that magmas are not just abstract struc-tures, but are related directly to the propertiesof space and topology in the worlds of com-munication and perception in which we live.

7. Knot sets, topological eigenforms and the left-distributive magma

We shall use knot and link diagrams to repre-sent sets. More about this point of view can befound in the author’s paper “Knot Logic”(Kauffman 1995). In this notation theeigenset Ω satisfying the equation

Ω = {Ω}

is a topological curl. If you travel along thecurl you can start as a member and find thatafter a while you have become the container.

Further travel takes you back to being amember in an infinite round. In the topolog-ical realm, Ω does not have any associatedparadox. This section is intended as an intro-duction to the idea of topological eigenforms, asubject that we shall develop more fully else-where.

Set theory is about an asymmetric relationcalled membership.

We write a ε S to say that a is a member ofthe set S. In this section we shall diagram themembership relation as follows:

This is knot-set notation.In this notation, if b goes once under a, we

write a = {b}. If b goes twice under a, we writea = {b, b}. This means that the “sets” aremulti-sets, allowing more than one appear-ance of a member. For a deeper analysis of theknot-set structure see (Kauffman 1995).

This knot-set notation allows us to havesets that are members of themselves,

and sets can be members of each other.

Here a mutual relationship of a and b isdiagrammed as a topological linking.

ba

a

a ε b

Ω

Ω = {Ω}Ω ε Ω

b

a

a = {b}b = {a}

a = {b, b}b = {c, c}c = {a, a}b

a

c



Here are the Borromean Rings. The Ringshave the property that if you remove any oneof them, then the other two are topologicallyunlinked. They form a topological tripartiterelation. Their knot-set is described by thethree equations in the diagram.

Thus we see that this representative knot-set is a “scissors-paper-stone” pattern. Eachcomponent of the Rings lies over one othercomponent, in a cyclic pattern.

Remark. The connection between this for-malism and epistemic logic (Hintikka 1962)should be further explored. In epistemic logicthe basic expressions are of the form KaKbp(“a knows that b knows that p”).

One specific thing to explore is the prob-lem of common knowledge, which can onlybe reduced to an infinite number of K’s as in

E = KaKbKaKbKaKb…

denoting that “a knows that b knows that aknows that…”

We can write this as

E = KaF

F = KbE

indicating that

E = “a knows F”

F = “b knows E.”

Together these statements indicate com-mon knowledge or mutuality for a and b.Conversely, we can take the linked sets A = {B}and B = {A} as a statement of common knowl-edge.

Another avenue that should be explored isthe relationship between knot set theory andAczel’s theory of self-referential and non-wellfounded sets (Aczel 1988) and the relatedtreatment by Barwise and Moss (1996).

Quandles and colorings of knot diagramsThere is an approach to studying knots andlinks that is very close to our knot sets, butstarts from a rather different premise.

In this approach each arc of the diagramreceives a label or “color.” An arc of the dia-gram is a continuous curve in the diagramthat starts at one undercrossing and ends atanother undercrossing. For example, the tre-foil diagram in the following illustartion hasthree arcs.

Each arc corresponds to an element of a“Trefoil Color Algebra” IQ(T), where Tdenotes the trefoil knot. The algebra is gener-ated by colors a, b and c with the relations

a * a = a, b * b = b, c * c = c,

a * b = b * a = c, b * c = c * b = a, a * c = c * a = b.

Each of these relations in the diagramabove is a description of one of the crossingsin T. The full set of relations describes the col-oring rules for an algebra that contains theserelations and allows any two elements to becombined to a third element. This three-ele-ment algebra is particularly simple. If two col-ors are different, they combine to form theremaining third color. If two colors are thesame, they combine to form the same color.

When we take an algebra of this sort, wewant its coloring structure to be invariantunder the Reidemeister moves (illustratedbelow).

This means that when you make a new dia-gram from the old diagram by a topologicalmove, the resulting new diagram inherits aunique coloring from the old diagram. Thenone can see from this that the trefoil must beknotted since all diagrams topologicallyequivalent to it will carry three colors, while anunknotted diagram can carry only one color.

As the next diagram shows, invariance ofthe coloring rules under the Reidemeistermoves implies the following global relationson the algebra:

x * x = x(x * y) * y = x

(x * y) * z = (x * z) * (y * z)

for any x, y and z in the algebra (set of colors)IQ(T).

An algebra that satisfies these rules is calledan Involutory Quandle (Kauffman 1995),hence the initials IQ. Perhaps the mostremarkable property of the quandle is itsright-distributive law corresponding to thethird Reidemeister move, as illustrated below.The reader will be interested to observe that ina multiplicative group G, the following oper-ation satisfies all the axioms for the quandle:g * h = hg–1h.

In an additive and commutative version ofthis axiom we can write a * b = 2b – a. Herethe models that are most useful to the knottheorist are to take a and b to be elements ofthe integers Z or elements of the modularnumber system Z / dZ = Zd for some appro-priate modulus d. The knot being analyzedrestricts the modular possibilities. In the caseof the trefoil knot the only possibility is d = 3,and in the case of the Figure Eight knot(shown after the Reidemeister moves below)the only possibility is d = 5.

This analysis then shows that there cannotbe any sequence of Reidemeister moves con-necting the Trefoil and the Figure Eight. Theyare distinct knot types.

y

z

x

z = x * y

b

a

c

b = a * cc = b * aa = c * b

T

x * x x

xx

x

x * x = x

I.

x

xy

(x * y) * y

x * y x

(x * y) * y = x

II.

y z

x (x * z) * (y * z)

x * zy * z

x * yx (x * y) * z

y * z

(x * y) * z = (x * z) * (y * z)

III. y z



Here is the example for the Figure EightKnot.

We have shown how an attempt to label thearcs of the knot according to the quandle rule

a * b = 2b – a leads to a labelling of the FigureEight knot in Z/5Z. In our illustration we haveshown that there is a compatible coloringusing four out of the five elements of Z/5Z. Ifyou apply Reidemeister moves to the diagramfor the Figure Eight knot you will see thatother versions of the knot require all five col-ors. It is interesting to prove that there is nodiagram of the Figure Eight knot that can becolored in less than four colors.

It should be noted that the knot diagramsgive a remarkable picture of non-associativealgebra structure and that each arc-label a ina diagram is both an element of the algebraand a transformation of the algebra to itselfvia the mapping Oa(x) = x * a.

Note that the right distributivity of thisoperation has the equation

Oa(x * y) = (x * y) * a = (x * a) * (y * a) = Oa(x) * Oa(y)

That is, we have

Oa(x * y) = Oa(x) * Oa(y).

The right distributive law tells us thateach quandle operation is a quandle homo-morphism. That is, each quandle operation isa structure-preserving mapping of the quan-dle to itself. This is an underlying algebraicmeaning of the third Reidemeister move.Since the mappings Oa are invertible, we see

that any quandle Q is in 1–1 correspondencewith a certain collection of automorphismsof itself. In this sense a quandle is a reflexivedomain with a limitation on the allowablecollection of self-mappings. In fact we have avery simple fixed point theorem for quandlessince

Oa(a) = a * a = a.

Every element of the quandle is fixed by itsown automorphism.

Since we take [Q, Q] to be the set of map-pings of Q to itself of the form Oa(x) = x * a,we see that any quandle is a reflexive domainof a restricted sort. (Not every set theoreticmapping of Q to Q is realized in the abovemanner.)

How far is the quandle from being a reflex-ive space in the full sense of the word? Let uslook at the fixed point construction. Wedefine

G(x) = (x * x) * F

for a given element F of the quandle. Is it then the case that (x * x) * F = x * g for

some g in the quandle? The answer is, yes, but for a very simple

reason: We have x * x = x so that (x * x) * F = x * F

and consequently (F * F) * F = F * F. In fact,F * F = F, so F is already its own fixed point.We therefore see that in a quandle the fixedpoint theorem is satisfied automatically dueto the axiom x * x = x for all x.

On the other hand, if F:Q → Q is an arbi-trary mapping from Q to Q, then we cannotexpect that F will have a fixed point. Forexample, in the trefoil quandle TRI, supposewe define F(a) = b, F(b) = c and F(c) = a. ThenF has no fixed point. Note that F is a structure-preserving mapping. (In this case the compo-sition of F with itself three times fixes every-thing. If we make transformations that arepermutations of finite sets, then they may befixed-point free, but some powers of them willcertainly have fixed points.)

We have F(x * y) = F(x) * F(y) for all x andy in TRI. For example, F(a * b) = F(c) = a =b * c = F(a) * F(b).

In order to extend TRI to a reflexive (right-distributive) magma we would have to add anelement f to the algebra such that x * f = F(x)for each x in TRI, take the consequences ofthat and continue. We leave the exploration ofthis extension to the reader.

Left distributivityWe have written the quandle as a right-dis-tributive structure with invertible elements. Itis mathematically equivalent to use the for-malism of a left distributive operation. In leftdistributive formalism we have A * (b * c) =(A * b) * (A * c). This corresponds exactly tothe interpretation that each element A in Q isa mapping of Q to Q where the mapping A[x]= A * x is a structure-preserving mappingfrom Q to Q.

A[b * c] = A[b] * A[c].

We can ask of a domain that every elementof the domain is itself a structure-preservingmapping of that domain. This is very similar tothe requirement of reflexivity and, as we haveseen in the case of quandles, can often be real-ized for small structures such as the Trefoilquandle.

As said before, a magma is a domain Mwith an operation * that is left distributive.Magmas are more general than the link dia-grammatic quandles. We take only the analogof the third Reidemeister move and do notassume any other axioms A magma with noother relations than left-distributivity iscalled a free magma.

The search for structure-preserving map-pings can occur in rarefied contexts. See, forexample, the work of Laver and Dehornoy(2000; Kauffman 1995), who studied map-pings of set theory to itself that would pre-serve all definable structure in the theory.Dehornoy realized that many of the problemshe studied in relation to set theory were acces-sible in more concrete ways via the use ofknots and braids. Thus the knots and braidsbecome a language for understanding the for-mal properties of self-embedded structures.

Structure-preserving mappings of set the-ory must begin as the identity mapping sincethe relations of sets are quite rigid at thebeginning. (You would not be able to map anempty set to a set that was not empty forexample, and so the empty set would have togo to itself.) The existence of non-trivialstructure-preserving mappings of set theoryquestions the boundaries of definability andinvolves the postulation of sets of very largesize. See Piechocinska (2005) for a goodexposition of the philosophical issues aboutsuch embeddings and for an approach towholeness in physics that is based on theseideas.

0

1 2

3

5

2 × 1 – 0 = 12 × 2 – 1 = 32 × 3 – 1 = 5→ 0 = 5

Z / 5Z = {0, 1, 2, 3, 4 } with 0 = 5

b

c = 2b – a = a * b

a



It is worth making a remark here aboutsets. Consider the collection Aleph of all setswhose members are themselves sets and suchthat any investigation into membership willjust reveal more sets as members. Typical ele-ments of Aleph are the empty set { }, the setwhose member is the empty set { { } } and ofcourse various curious constructs that haveinfinitely many members such as { { }, {{ }},{{{ }}}, {{{{ }}}}, … } and we may even con-sider sets that are members of themselves(eigen-sets!) such as { { { { { … } } } } }.

The key thing to understand about Alephas a class of sets is that any member of Alephis, by definition, a subset of Aleph. And anysubset of Aleph is, by definition, a member ofAleph. This is a beautiful property of theclass Aleph, and it is a paradoxical propertyif we imagine that Aleph is a set! For if Alephis a set, then we have just shown that Aleph isin 1–1 correspondence with the set of subsetsP(Aleph) of Aleph. If X is any set then wedenote the set of subsets of X by P(X). Can-tor’s Theorem (proved here in Section 8 andrelated in that section to the fixed point the-ory of reflexive domains) tells us that for anyset X, P(X) is larger than X.

This means that there cannot be a 1–1 cor-respondence between Aleph and P(Aleph) ifAleph is a set.

We can only conclude that Aleph is not aset. It is a class, to give it a name. It is anunbroken wholeness whose particularitieswe can always consider, but whose totalitywill always elude us. The way that the totalityof Aleph eludes us is right before our eyes.Any particular element of Aleph is a set andis a collection of sets as well. But we cannotcomplete Aleph. Any attempt to approxi-mate Aleph as a set will always have somesubsets that have not been tallied inside itselfand so the set of subsets of the approxima-tion will grow beyond that approximation toa new and larger domain of sets. Philosoph-ically, this observation of the unreachabilityof Aleph, the set of all sets, as a set itself isvery interesting and important. We see herehow a perfectly clear mathematical conceptmay always remain outside the bounds of theformalities to which it refers and yet thatconcept is indeed composed of these formal-ities. It is the leading presence of the ulti-mately huge and unattainable Aleph that

leads us to consider exceeding large sets inthe pursuit of a flexibility in the self-embed-dings of set theory. At the end of Section 8,we take an alternative view of Aleph and con-sider what would have to change if Alephwere admitted to be a set.

Enough said about the abstract reaches ofthe magma.

We should not expect that any givenstructure is a reflexive space. But it is possibleto create languages that can expand indefi-nitely and thus partake of the ideal of reflex-ivity.

8. Church and Curry

In this section we point out how the notionof a reflexive domain first appeared in thework of Alonzo Church and Haskell Curry(Barendregt 1984) in the 1930s. This methodis commonly called the “lambda calculus.”The key to lambda calculus is the construc-tion of a self-reflexive language, a languagethat can refer and operate upon itself. In thisway eigenforms can be woven into the con-text of languages that are their own metalan-guages, hence into the context of natural lan-guage and observing systems.

In the Church-Curry language (thelambda calculus), there are two basic rules:

1. Naming. If you have an expression in thesymbols in lambda calculus then there isalways a single word in the language thatencodes this expression. The application ofthis word has the same effect as the applica-tion of the expression itself.

2. Reflexivity. Given any two words, A and B,in the lambda calculus, there is permission toform their concatenation AB, with the inter-pretation that A operates upon or qualifies B.In this way, every word in the lambda calcu-lus is both an operator and an operand. Thecalculus is inherently self-reflexive.

Here is an example. Let GA denote theprocess that creates two copies of A and putsthem in a box.

In lambda calculus we are allowed to applyG to itself. The result is two copies of G next toone another, inside the box.

This equation about GG exhibits GGdirectly as a solution to the eigenform equa-tion

thus producing the eigenform without an infi-nite limiting process.

More generally, we wish to find the eigen-form for a process F. We want to find a J so thatF(J) = J. We create an operator G with theproperty that

GX = F(XX)

for any X. When G operates on X, G makesa duplicate of X and allows X to act on itsduplicate. Now comes the kicker.

Let G act on itself and look!

GG = F(GG)

So GG is a fixed point for F. We have solved the eigenform problem

without the excursion to infinity. If you reflecton this magic trick of Church and Curry youwill see that it has come directly from the pos-tulates of Naming and Reflexivity that we havediscussed above. These notions, that thereshould be a name for everything, and that wordscan be applied to the description and productionof other words, allow language to refer to itselfand to produce itself from itself. The Church-Curry construction was devised for mathe-matical logic, but it is fundamental to the logicof logic, the linguistics of linguistics and thecybernetics of cybernetics.

An eigenform must be placed in a contextin order for it to have human meaning. Thestruggle on the mathematical side is to controlrecursions, bending them to desired ends. Thestruggle on the human side is to cognize aworld sensibly and to communicate well andeffectively with others. For each of us, there isa continual manufacture of eigenforms(tokens for eigenbehavior). Such tokens willnot pass as the currency of communicationunless we achieve mutuality as well. Mutualityitself is a higher eigenform. As with all eigen-forms, the abstract version exists. Realizationhappens over the course of time.

GA = AA

GG = GG

X = X



9. Cantor’s diagonal argument and Russell’s paradoxLet AB mean that B is a member of A.

Cantor’s Theorem. Let S be any set (S can befinite or infinite).

Let P(S) be the set of subsets of S. ThenP(S) is bigger than S in the sense that for anymapping F: S → P(S) there will be subsets Cof S (hence elements of F(S)) that are not ofthe form F(a) for any a in S. In short, thepower set P(S) of any set S is larger than S.

Proof. Suppose that you were given a way toassociate to each element x of a set S a subsetF(x) of S. Then we can ask whether x is amember of F(x). Either it is or it is not. So letus form the set of all x such that x is not amember of F(x). Call this new set C. We havethe defining equation for C:

Cx = ~F(x)x.

Is C = F(a) for some a in S?If C = F(a) then for all x we have F(a)x =

~F(x)x.Take x = a. Then F(a)a = ~F(a)a.This says that a is a member of F(a) if and

only if a is not a member of F(a). This showsthat indeed C cannot be of the form F(a), andwe have proved Cantor’s Theorem that the setof subsets of a set is always larger than the setitself.

Q.E.D.Note the problem that the assumption that

C = F(a) gave us.If C = F(a), then F(a)a = ~F(a)a. We would

have a fixed point for negation. But there is nofixed point for negation in classical logic!

If we had enlarged the truth set to

{T, F, I}

where ~I = I is an eigenform for negation,then F(a)a would have value I. What does thismean? It means that the index a of the corre-sponding set F(a) would have an oscillatingmembership value. The element a would belike Groucho Marx, who declared that hewould not join any club that would have himas a member. We would be propelled into setsthat vary in time.

Note that our proof of Cantor’s Theoremhas exactly the same form as our earlier proof

of the existence of fixed points for a reflexivespace. The mapping F: X → P(X) takes therole of the 1–1 correspondence between Dand [D, D]. The reader will enjoy thinkingabout this analogy. In the Cantor Theorem wehave used the non-existence of a fixed pointfor negation to deduce a difference betweenset X and its power set P(X). In the study of areflexive domain we have shown the existenceof fixed points, but we have seen that suchdomains must be open to new elements andnew transformations.

Note also how close Cantor’s Theorem isto Russell’s famous paradox.

Russell devised the set R defined by theequation

Rx = ~xx.

An element x is a member of the Russell setif and only if x is not a member of itself.

To see the contradiction, substitute R for xand get

RR = ~RR.

This appearance of an eigenform for nega-tion tells us that we either must concede tem-porality to Russell’s construction R, or elsebanish it from the world of sets.

10. The secret

What is the simplest language that is capableof self-reference?

We are all familiar with the abilities of nat-ural language to refer to itself. Why this verysentence is an example of self-referentiality.The American dollar bill declares, “This bill islegal tender.” The sentence that you are nowreading declares that you, the reader, are com-plicit in its own act of reference. But what isthe simplest language that can refer to itself?

The simplest language would have a sim-ple alphabet. Let us say it has only the letter R.The words in this language will be all stringsof R’s. Call the language LS. The words in LSare the following:

R, RR,

RRR, RRRR,

and so on.

Two words are equal if they have the samenumber of letter R’s.

Each word makes a meaningful statement ofreference via the rule:

If X is a word in LS, then RX refers to XX.

RX refers to XX, the repetition of X.

Thus RRR refers to RRRR (not to itself),and R refers to the empty word.

There is a word in LS that refers to itself.Can you find it?

Let us see.RX refers to XX.So we need XX = RX if RX were to refer to

RX.If XX = RX, then X = R.So we need X = R.And RR refers to itself.The little language LS looks like a pedantic

triviality, but it is actually at the root of reflex-ivity, Gödel’s incompleteness Theorem,recursion theory, Russell’s paradox and thenotion of self-observing and self-referringsystems. It seems paradoxical that what lookslike a trick of repeating a symbol can be soimportant. The trick is more than just a trick.

The Russell paradox (see the previous sec-tion) continues to act as a mystery at the cen-ter of our attempts to relate syntax andsemantics. In that center is a little trick of syn-tactical repetition.

I would like to think that when we eventu-ally discover the true secret of the universe itwill turn out to be this simple.

The snake bites its tail. The Universe isconstructed in such a way that it can refer toitself. In so doing, the Universe must divideitself into a part that refers and a part to whichit refers, a part that sees and a part that is seen.

Let us say that R is the part that refers andU is the referent. The divided universe is RXand RX = U and RX refers to U (itself). Oursolution suggests that the Universe dividesitself into two identical parts, each of whichrefers to the universe as a whole. This is

RR.

In other words, the universe can pretendthat it is two and then let itself refer to the two,and find that it has in the process referred onlyto the one, that is, itself.

The Universe plays hide and seek with her-self, pretending to divide itself into two whenit is really only one. And that is the secret ofthe Universe and that is the universal sourceof our trick of self-reference.



11. The world of recursive emergence and creativityWe have repeatedly insisted that a formalfixed point or eigenform is associated withany transformation T in any domain whereinfinite composition of transformations ispossible. Thus we make E = T(T(T(T(T(…))))) and find that E = T(E). This is thesymbolic fixed point that sometimes corre-sponds to a stability in the original domainof the recursion. We have also seen that onecan take a seed z for the recursion and repeat-edly form

z, T(z), T(T(z)), T(T(T(z))), …

in a temporal sequence or recursive process.Then the finite products of this process canexhibit similarity to the infinite eigenform,and they can also exhibit novelty and emer-gence structure in ways that are most sur-prising. It is this appearance of creativity andnovelty in recursive process that makesreflexivity more than abstract mathematicsand more than a philosophical idea.

The purpose of this last section is toexhibit an example involving cellular autom-ata that illustrates these ideas and gives us aplatform for thought. In this example, we areusing an algorithm that I call “7-Life.” It is avariant of the Life automaton of John H.Conway (Gardner 1970).

Conway’s automaton is governed by therule B3/S23 which means that a white squarein the grid is born (B) when it has 3 neigh-

bors and it survives (S) when it has exactly 2or 3 neighbors. Life has the property thatthere are many intriguing formations andprocesses, but statistically most configura-tions die out to a collection of isolated staticpatterns (still lifes) and oscillating patternsthat do not grow and do not interact outsidethemselves.

We should mention that there are a vastnumber of different cellular automata. Agood start in learning about these structuresis the book by Stephen Wolfram (2002).Wolfram’s book concentrates almost entirelyon one-dimensional cellular automata andachieves a qualitative classification of thebehaviors found in a comprehensive class ofthe simplest types of line automata. Wolframfinds that a number of these simplest autom-ata are computationally universal in thesense that they can simulate a universal Tur-ing machine on the one dimensional latticeof the automaton. The automata that we arelooking at in this section, and 7-Life in par-ticular, are two dimensional and hardly con-sidered by Wolfram in his treatise. Conway’sLife is also not analyzed in Wolfram. In fact,it was shown that Conway Life is Turing uni-versal by Conway and his collaborators priorto the onset of Wolfram’s work in the 1980s.We mention this background and the differ-ence in dimensionalities to give the readersome perspective so that he will not be sur-prised and wonder “Why is this phenome-non not discussed in Wolfram?” Indeed thephenomenon of the remarkable emergenceof complexity from simple algorithms is thetheme of Wolfram’s work. This theme plays

significantly in all algorithmic mathematicsand in all significant studies of cellularautomata.

6-Life, defined by the rule B36/S23 iswell-known, as a search on the Internet forLife automata will reveal. 6-Life does nothave the qualitative self-sustaining proper-ties that are evident in 7-Life, but there areemergent structures there as well. We con-centrate here on 7-Life because the long-term self-sustaining interactions of thisautomaton make it ideal for studies of longterm evolution and the emergence of forms.

7-Life has the rule B37/S23 and has manyof the properties of Life, plus the phenome-non that many starting configurations grow,self-interact and produce streams of gliders.The gliders are five-square formations(occurring in Life as well) that occur sponta-neously and regenerate themselves, appear-ing to move along diagonal directions in theprocess. The most striking property of 7-Lifeis the long term persistence of such self-interacting configurations, growing slowlyin complexity over time.

In Figures 1, 2, and 3 we indicate theresult of applying the 7-Life algorithm to asimple and not-quite symmetrical startingconfiguration, shown in Figure 1. In Figure 2we see the result of 33911 iterations of theprocess. We now have a galaxy of complexinteractions. The small entities radiatingaway from the galaxy are gliders, as describedabove, and if a reader were to watch the pro-cess using a computer program, he or shewould see a teeming, seemingly randommass of activity. Then in Figure 3 we see that

Figure 1: The starting configuration. Figure 2: After 33911 iterations. Figure 3: After 49281 iterations.



after 49281 iterations something new hasemerged. It seems that a highly patterneddragon is emerging from the chaos of thecomplex process. The tip of this dragonmoves forward relentlessly.

The body of the dragon interacts with theglider radiation and begins to roil in the cha-otic process. So far, the growing tip of thedragon has not interacted with any gliders.

Figures 4 gives a close-up of the tip of thedragon and Figure 5 isolates the generator,GG, of the dragon itself. This configurationGG of 16 squares in mirror symmetry, whenplaced on an otherwise blank lattice, willgenerate the dragon in the 7-Life algorithm.

What has happened is that this 16-squaregenerator GG has appeared in the course ofthe complex interactions, and it has hadenough room to move forward in its ownpattern – forming the dragon behind it andperiodically regenerating itself. The genera-tor of the dragon, GG, is not our invention.GG is a natural consequence of the complexprocess of 7-Life. GG emerges, but withmuch lower probability than the gliders. The

result is an appearance of novelty and cre-ativity in the complex process as it happensover time. We can only speculate what morecomplex entities would eventually emerge in7-Life over many more iterations.

In the same way, DNA emerges from thecomplex process of the world of the earthand sun.

We see from this example that eigenformsthat are processes, such as the self-generatingGG, can and will emerge of their own accordfrom complex systems based on recursion.In this sense, such systems begin to generatetheir own reflexive spaces. The novel andself-reproducing forms that emerge fromthem can be seen in a similar light.

All these observations are made by an ob-server. The observer is clever only in the dis-tinctions that he or she makes, and that isenough to found an entire universe.

12. Discussion

In this paper we have covered a number ofmathematical structures related to the con-cept of reflexivity. We have defined the no-tion of a reflexive domain D as a domainwhere the elements of that domain and themappings of the domain to itself are in 1–1correspondence.

In such a context, every object is inher-ently a process, and the structure of the do-main as a whole comes from the relation-ships whose exploration constitutes thedomain. There is no place to hide in a reflex-ive domain, no fundamental particle, no ir-reducible object or building block. Any givenentity acquires its properties through its re-lationships with everything else. The sense ofsuch a domain is not at all like the set theo-retic notion of collections or unrelatedthings, or things related by an identifiableproperty. It is more like a conversation or animprovisation, held up and moving in itsown momentum, creating and lifting soundand meaning in the process of its own ex-change. Conversations create spaces andevents, and these events create further con-versations. The worlds appearing from re-flexivity are worlds nevertheless, with thoseproperties of partial longevity, emergence ofpatterns, and emergence of laws that we havecome to associate with seemingly objectivereality.

Note1. According to Webster’s New Collegiate

Dictionary (1956) a gremlin is “one of theimpish foot-high gnomes whimsicallyblamed by airmen for interfering with mo-tors, instruments, machine guns, etc.;hence any like disruptive elf.”

Figure 4: The growing tip.

Figure 5: The generating tip GG.

Louis H. Kauffman has a B.S. in mathematics from MIT and a PhD in mathematics from Princeton University. He is Professor of Mathematics at the University of Illinois at Chicago. Kauffman works on topology, knot theory, quantum topology, form and cyber-netics. He is the founding editor and editor-in-chief of the Journal of Knot Theory and Its Ramifications and the editor of the book series on Knots and Everything. He is the author of the column Virtual Logic for the journal Cybernetics and Human Knowing. Kauffman is the author of numerous books and papers on knot theory and is well-known for his discovery of the bracket state sum model of the Jones polynomial and for his discovery of a generalization of the Jones polynomial called the Kauffman polynomial, for his discovery of virtual knot theory and other topological structures. He is the recip-ient of the 1993 Warren McCulloch award of the American Society for Cybernetics and the 1996 award of the Alternative Natuaral Philosophy Association for his work on dis-crete non-commutative electromagnetism. He was president of the American Society for Cybernetics from 2005 to 2008. Kauff-man plays the clarinet in the Chicago-based ChickenFat Klezmer Orchestra.

THE AUTHOR



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Received: 15 May 2009Accepted: 7 July 2009

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