In what way is the sandgrain the dune? In what way am I a part of the whole?

It seems like I have been pondering that question forever, and over the past few days, this inquiry has been looping almost incessantly through my mind.

But, today, I have a kind of answer.

Is it a correct answer? 'Oui, c'est correcte!', ... as one says here in Quebec, ... and by that one does NOT mean 'exempt de fautes' (faultless), but instead 'dont la qualite est convenable, acceptable', ... or roughly, ... 'it works for me.'

The answer has been implicit for some time, ... at least since I was five years old and was fascinated, as most kids are at that age, with the process of 'self-organization', ... how kids playing together could just 'make up stuff' and thereby induce harmony and order into the group, ... playing 'cowboys and indians' or some such thing, within which there were always possibilities for everyone, and each little natural-born thespian allowed his or her imagination to open up a new little character, which 'worked for them', into which they stepped for an hour or two.

Sure, the nominal content of that particular example may not have been 'politically correct', but the message at the time was in the medium, ... it was about whole-and-part harmony, ... cooperation, ... taking care of one another, ... opening up possibilities for everyone in the group, ... it was, strangely enough, about 'being who you are' by being someone else. And there was no way the cowboys were favoritized, ... everyone wanted to be indian, ... great chief White Cloud, ... strong and straight, who knew no fear and who could see right through 'white man who speak with forked tongue.'

I remember clearly my first day at school, ... I was shocked by this new form of organization. Sure, I already had tasted of it at home and elsewhere, .... but this time it wasn't a momentary imposition, ... it clashed directly with the commons of my childhood, occupying them like a panzer division occupying a pastoral commons in Europe, in the war which was just over. Some kids eagerly accepted this new discipline which was clearly symbolic of moving towards adulthood. Not me, ... while the teachers were smiling, ... I sensed something 'not right' about it, ... their adult behaviors were different from those of visitors to your house. Where were they 'taking us', and themselves?

Tomorrow, here in Montreal, there will be a two-day youth-government conference, 'Le Sommet du Québec et de la jeunesse'. In Saturday's 'La Presse', ... the 'summit' was in small print, beneath the banner headline 'Les jeunes n'y croient pas' ('Youth doesn't believe in it.').

... What don't they believe in?

"Plus des trois quarts des jeunes, 77% des 15 a 29 ans, ne font pas confiance aux partis politiques!" ('More than three quarters of youth, 77% of those from 15 to 29 years don't trust the political parties!')

The article goes on to say that 81% of the young females are 'totally sceptical' of politicians, compared to 75% of the young males, and they even doubt the good intent of the politicians.

But survey's aside, what are the youths themselves saying? ... The following quotes are from a youth spokesman from ROCAJQ ('Coalition of independent community organisations of Quebec')

"Les priorités d'action du gouvernement sont les mêmes que celles de son Plan d'action jeunesse 1998-2001 : la compétitivité des entreprises et la compétition entre les jeunes qui mettent hors-jeu les moins performants. " ... "... les jeunes qui vivent des difficulté ou qui ne s'identifient pas aux valeurs de compétition ne peuvent pas se reconnaître dans ce que l'on pourrait qualifier de mascarade autour du Sommet du Québec et de la jeunesse."

("The government's priorities are the same as in its 1998 - 2001 Action Plan for youth: competition in business and competition amongst youth which puts the less performant out of play." . . . "Young people who experience difficulty or who do not identify with the values of competition do not relate to this, what might be termed 'mascarade' which surrounds the Summit Meeting of Quebec and Youth')

"L'entrepreneurship est considéré comme la voie prévilégiée pour la création d'emplois, selon le communiqué de presse du gouvernement, et le chantier sur la formation réaffirme l'objectif étroitement économique de mettre l'éducation au service des entreprises. Rien sur la nécessité de parler d'accessibilité à l'éducation, de diminuer les frais de scolarité et l'endettement étudiant, d'aborder le problème de l'appauvrissement étudiant, de lier l'éducation à l'épanouissement individuel et social. Difficile d'ailleurs de ne pas remarquer que plusieurs "juniors" coprésidant les chantiers sont activement impliqués dans les associations des jeunes du monde des affaires

("Entrepreneurship is considered to be the privileged path for job creation, according to the government press release, and its development base reaffirms the narrowly economic objective of putting education in the service of the corporation. There is nothing [in the plan] about the need to speak of accessibility to education, of lowering the cost of scholarship and student indebtedness, to broach the problem of student impoverishment, to tie education to individual and social actualization. Moreover, it is hard not to notice [that] the several "juniors" co-presiding over these [shambles?] are actively involved in the youth associations of the business world... ")

"L'équité intergénérationnelle : diviser pour que règne l'inéquité sociale"

("The intergenerational equation : to divide, so that social inequality rules.")

That was it! That was my feeling on my first day of school, .. that we were being forcibly divided up by the adults and 'the better ones' were going to be put 'over here', and the poorer ones', 'over there', ... no more ecological relating, which seemed to emanate naturally from within us, to open up a niche for every individual 'community' constituent, so that they might cultivate their 'thing', whatever it was.

Half a century later, .. I'm still trying to 'parameterize that equation', ... and like I say, I have a kind of answer.

It's been implicitly with me for quite some time, in fact, and has been slowly taking on form and substance. In my 1997 essay in *Complexity*, 'Complexity and the 'Learning Organization', I spoke of managing 'creative interference latencies' in 'interorganizational phase space.'. Of course, some readers were irritated, ... and one of them wrote in asking the editor of *Complexity* how he ever could have published such 'techno-babble.'. Strangely enough, every time I get a response like that, when reductionist circuitry starts to smoke and fritz, I know I am closing in on an answer, ... not 'the answer', ... but 'an answer', ... oui, c'est correcte.

This morning's version of it comes after abandoning my mind to cycle through anything and everything associated with this question of why certain people cannot see what to others is obvious, ... i.e. 'creative interference latencies in interorganizational phase space'. All I can think of is that they've never been to Omak, ... never drunk longnecks while playing eightball, ... looking up across the Okanagan river to the cliff where they have the suicide run every year, ... twenty whooping and hollering indians on horseback, galloping off into space over the cliff edge, ... then falling, tumbling, sliding, down into the rapid-running waters of the Okanagan, ... some of them actually making it across and into the stadium where they are greeted by the cheers of the stampede crowd.

Now without getting into the tao of CILIIPS, there's no way these dudes could ever survive.

Ok, we've been through this a few times before, so we can cut to the quick.

In relativistic curved space, as in pool, we consider not only the explicit 'actualities' but also the implicit 'possibilities'. Every time we move a ball we change the possibilities 'seen' by every ball. This 'possibility space' is termed 'reciprocal disposition' and it is a mathematically defined thing, even though it is 'implicit'. It is implicit because it is 'unbounded'. Draw two small disjoint (non-overlapping) circles on the surface of an orange and then try to explicitly specify the shape of the 'reciprocal disposition' seen by either of those little circles. You can't do it because that shape you're looking at is 'unbounded'.

If you want to get explicit, all you can do is to specify the shape of the perimeter of the two small circles (or whatever shapes you put on there) and 'talk' about the 'containing space'. Now a computer could do a numerical simulation of that reciprocal space for you, ... but there would be something missing, ... and that would be the unique 'shape' seen looking out from the center of each of those circles. That's what the skilled pool player is interested in, ... 'what reciprocal disposition shape does each ball see before I shoot?', ... and 'what reciprocal disposition shape does each ball see after I shoot?' and these are questions about single events but questions which concern the topographic evolution of the possibility space.

I'll leave it to you to figure out whether a computer can hold these questions in its mind and come up with the optimum parameters for taking the shot, ... not forgetting, of course, that the perturbation of balls is subject to deterministic chaos and that the individual balls can change their objective depending on the shape of possibility space, and that the optimization of that unbounded shape bleeds into past and future in an implicit rather than explicit manner.

What I'm interested in is the accounting principles. For those constituents of the system who attend to the harmonic cultivation of possibility space, so that their 'brother constituents' will have greater rather than lesser possibilities after the move, ... credits are nevertheless commonly confined to 'actuality space' results. In the west, this is as it must be; i.e. according to the dictates of the material-causal precepts of the Newtonian paradigm. There can be no credits for boosting the 'creative interference latencies in interorganizational phase space', ... because these latencies cannot be explicitly measured and as the business dictum goes, ... 'you can't management what you can't measure'.

But some teams achieve better results than others and some constituents achieve better results than others, ... how is this handled?

Generally, what is not accounted for in terms of causal ingredients is put down to probabilities. The probabilities of random chance or the probabilities of nature or nurture or whatever, which are generally only invoked on the downside while the upside results are explained in terms of skills; i.e, ... 'this job is turning out to be tougher than we thought', ... 'we didn't get the support we needed'. The skilled pool player who rarely 'snookers' himself and always leaves the balls with 'good possibilities' in hand, ... may never have to make a really tough shot. Whereas the player who gets himself in a mess all the time, may develop a knack for getting out of some of his difficult situations, and what management and the gallery crowd rises out of the seats for are the superb 'shots'. The 'shape-making', is implicit, and tends to be invisible to the unskilled eye, because it is spread out over space-time and not concentrated in 'causal events'. 'Shot-meisters' are Gods and 'shape-meisters' are shite. But the fact is that the skilled player can improve the possibilities for multiple constituents when he makes his move and this nonlinear effect never shows up on the actualities accounting books.

More than this. In the real life realm of teams, when one constituent foregoes the culturally encouraged full frontal competition and instead leverages his moves to improve the possibilities of multiple brother constituents, he is improving the community possibilities.

More than this, still, this leveraging of possibility space opportunities can be on a 'life' basis, ... e.g. the team constituent who 'fields' troublesome things for a sick colleague who's off work so that they won't be grown into monsters by the time he returns.

If you think about it, all 'moves' of the constituent of an ensemble lead to two entries on the curved space-time books, ... his 'actuality space credits' which are the only credits he'll receive in the euclidian-newtonian accounting, and his 'possibility space credits' which are implicit AND UNBOUNDED.

What does this mean, 'unbounded'?, ... what it means is that what he contributes here, he contributes to latent order in a space-time reservoir which goes on, and on, and on, ... without bound. As in the old Frank Capra movie 'It's a Wonderful World', ... if you improve someone's possibility space (e.g. save their life), the effect goes on forever.

So this possibility space cultivation skill which we can see in the synthetic example of the game of pool, ... is something like 'negentropy', ... something which pertains to the 'whole' of space-time.

What then, are the Quebec students complaining about?, ... They are in effect complaining that the socio-political establishment is transforming people into agents of entropy, and communities into entropy factories, because the orientation to zero-sum competition is in head-to-head conflict with leveraged cultivation of possibility space.

Those people who have or are working 'independently' towards a 'birds nest on the ground', so to speak, tend not to want to mess with the 'accounting system', and one can slide into this mainstream economic flow profile more easily than we'd like to think. As the 'La Presse' article also said, ... "Il est toutefois interessant de noter qu'une brisure marquee survient a 25 ans. Avant ca, on ne veut pas de baisses d'impots. L'option recolte 5% d'appuis chez les 15 a 19 ans et 10% chez les 20 a 24. Mais soudainement, chez les 25 a 29, c'est 27% qui trouvent que les baisses d'impots ne sont pas une mauvaise idee."

("It is meanwhile interesting to note that a marked discontinuity appears at age 25. Before that, there is a dispreference for lowering taxes. This option was supported by 5% of 15 to 19 year olds and 10% of 20 to 24 year olds. But suddenly, with the 25 to 29 year olds, 27% of those surveyed find that the lowering of taxes is not a bad idea.")

As it turns out, there is no place for the measurement of 'possibility space' effects if one is using the euclidian space convention because 'reciprocal disposition' does not even exist in euclidian space. We see it, ... we feel the effects of it, ... and we may respond to it, ... but there is no place to put 'negentropy' or 'entropy' on the books. And because it cannot be measured in euclidian space, ... our space convention of choice in the west, ... we say that our entropy production is zero, ... we say, ... that 'you can't manage what you can't measure', ... even as we are bringing in bumper crops of it.

I beg to differ, ... I think the answer to our entropy production is childs play.

* * *

Date: Mon, 21 Feb 2000 01:09:19

To: bart kosko, john casti, ian stewart

From: ted lumley <emiliano@sympatico.ca>

Subject: Where Fuzzy Logic Meets Complexity

... for your possible interest, ...

Where Fuzzy Logic Meets Complexity:

~~~consider the curved space of 'relativity' as being characterized by a spherical outer) surface upon which vehicles are moving (similar to the earth).

If there is more than one vehicle, ... a 'reciprocal disposition' exists mathematically and the geometric shape of the reciprocal disposition is visualizable (reciprocal disposition is undefined in euclidian space)

since the shape of reciprocal disposition is 'unbounded', it cannot be explicitly specified. i.e. it can only be implied 'relationally'; i.e. relative to the specifiable bounded areas constituted by the 'footprint' or space occupied by the vehicles. nevertheless it can be visualized and satellite monitoring could provide the 'reciprocal disposition' information to the vehicle drivers, ... a view of constituent-container relationships.

activity within this spherical space, corresponding to activities upon the surface of the earth, given constituent accessibility to reciprocal disposition information, is governed not only by the physical properties and behaviors of the vehicles, but also by the 'shape' of the reciprocal disposition which changes simultaneously with the movement of any vehicle and can influence constituent behaviors.

the additional 'possibility space' information (opportunity-purpose 'attractor' topography) innate in the implicit shape of reciprocal disposition is beyond the 'sum of the information concerning the properties and behaviors of the constituents' for this spherical space.

since the overall behavior of this 'system' cannot be deduced by the behaviors of its constituents, it qualifies as a complex system. in the context of complex systems, the innate ambiguity between material entities and their containing environment has been described in terms of 'initial conditions'. As Ilya Prigogine says in 'The End of Certainty';

"No physical concept is sufficiently defined without the knowledge of its domain of validity." [i.e. the relativistic space-time containing environment]... "We need a "divine" point of view to retain the idea of determinism. But no human measurements, no theoretical predictions, can give us initial conditions with infinite precision."

from a fuzzy logic point of view, in spherical space, the constituent is simultaneously 'itself' and 'its containing environment', since each constituent has an associated, uniquely definable 'reciprocal disposition' which is a qualitative (relativistic and implicit) function of its containing space and which, as einstein says; "... is a medium which is itself devoid of 'all' mechanical and kinematical qualities, but helps to determine mechanical (and electromagnetic) events."

for example, the behavior of a sanddune cannot be deduced from the behaviors of its sandgrains, since the sandgrain is itself the topography within which it is moving (it is its own reciprocal containing space). dune and atmosphere are likewise reciprocally related, which gives rise to the aesthetic dune curves.

material entities, being labels we apply to relatively stable space-time features we choose to abstract from space-time therefore suffer from the same fuzziness as 'fact', in the manner pointed out by professor kosko in 'The Fuzzy Future';

"You cannot prove a fact.

A proof assumes that some statements or premises are true. Then it derives a conclusion from them. You would have to assume at least one fact of the world to derive a fact of the world. But to assume that a fact is true is to reason in a circle. The point is to prove a fact is true. Suppose you assume that the sky is blue. Then you can derive the fact that the sky is either blue or red. So what? The truth of the conclusion still depends on the truth of the assumption." ... "no-one has found a binary fact of the world. No one has gotten the science right to more than a few decimal places. Maybe no one ever will. Maybe even God could not produce a binary fact. Or maybe a God is just the thing or force that can produce binary facts."

in summary, relativity theory and spherically curved space give us a mathematically (topologically) definable 'reciprocal' informational view which insists that all 'things' are simultaneously and inseparably their own containing environment. that is, all matter is 'fuzzy'. this 'fuzziness' emerges in both an ontological context (fuzzy existence and fuzzy logic), and an ontogenetical context (fuzzy evolution and fuzzy initial conditions). the additional information available to the observer using the curved space convention, being implicit (purely relational), cannot be explicitly captured and specified and is therefore inaccessible to explicit logical treatment, as in binary computing.. that is, curved space information has both explicit and implicit components, ... as in the complex information formulations of Denis Gabor.

this 'possibility space' information, which goes beyond informational representation available to the bivalent, euclidian space convention, provides exploitable insight on the coevolution of possibility space and actuality space which are obscured in the reduction of experiential observations into bivalent, euclidian 'probability' formulations. the additional information is 'invisible' when one is 'standing on the ladder of bivalence' (using the euclidian space convention) since 'reciprocal disposition' is undefined.

studies of 'community as complex system', as in the case of exceptional and dysfunctional teams, bring out the importance of including observer access to possibility space - actuality space coevolutional information, since this information transcends that furnished by simple 'probabilities', allowing the 'curved space observer' to cultivate a whole-and-part harmonic system behavior akin to that in a natural ecology, and to avoid the infusion of dysfunction and 'snookering' which accompany management approaches which reconcile possibility and actuality by means of one dimensional probability.

regards,

ted lumley

p.s. the following 'anecdote' illustrates how curved space reciprocal disposition information, by going beyond the properties and behaviors of material entities, can inform on issues of possibility-actuality codependencies (;-}

><><><>

>

>Sherlock Holmes and Doctor Watson went on a camping trip. After a good meal and a bottle of wine they lay down for the night, and went to sleep. Several hours later, Holmes awoke and nudged his faithful friend: "Watson, look up at the sky and tell me what you see."

>

>

>Watson replied: "I see millions and millions of stars."

>

>

>"What does that tell you?"

>

>

>Watson pondered for a moment. "Astronomically, it tells me that there are millions of galaxies and billions of planets. Astrologically, I observe that Saturn is in Leo. Horologically, I deduce that the time is approximately a quarter past three. Theologically, I can see that God is all powerful and that we are small and insignificant. Meteorologically, I suspect that tomorrow will be a beautiful day. What does it tell you?"

>

>

>Holmes paused for a moment. "Watson, you dickhead. Some bastard has stolen our tent."

>

><><><>

>Date: Tue, 22 Feb 2000 17:57:49

>To: bart kosko, john casti, ian stewart

>From: ted lumley <emiliano@sympatico.ca>

>Subject: p.s. to 'Where Fuzzy Logic Meets Complexity' fypi,...

>

>

>...for your possible interest;

>

>p.s. to 'Where Fuzzy Logic Meets Complexity'

>

>... mathematically, the issue appears to define itself by the following simple and practical problem;

>

>vehicles (motional agents) whose destinations are a function of their containing environment (ensemble in which they are constituents) move about on a common surface. according to what information (rules or whatever) shall each driver move so as to work towards all drivers having 'open road' in front of them?

>

>the drivers, or system designers, can choose between two space conventions, 'euclidian' and 'non-euclidian' (i.e. curved, finite and unbounded)

>

>observations:

>

>1. no matter the form of the euclidian surface, the problem may be defined in terms of the coordinates of the objects which move upon that surface, both the surface and the objects being referenced to a common euclidian coordinate system (reference frame). space is not a 'participant'.

>

>2. on the non-euclidian surface (an example would be the (featureless) surface of the earth seen as spherical space), the problem cannot be fully defined by means of the coordinates of the moving objects. this is because in non-euclidian space, the objects must be referenced to themselves. what this translates into is that the objects are referenced to the shape of inter-object space ('reciprocal disposition'). i.e. the reference frame is relativistic and 'floats' with the movement of the objects within it. (the drivers would always be moving into the openings which were forming.). space is therefore a 'participant'.

>

>3. the containing reference frame, in 2. is relative to the constituents of the space, thus the coordinates of the objects are in terms of its reciprocal disposition, ... with the latter changing simultaneous to the relative movement of any part of any member of the constituency of objects.

>

>4. in the non-euclidian case, the movement of any object induces a transformation of the reference frame and thus the dynamics of the individual constituents cannot be distinguished from the dynamics of the entire system, if the system behavior implies that the constituents are capable of sensing and responding to the shape of space (openings which they can move through) [1].

>

>5. in the general case we not only have 'individuals' but also 'groups' which behave as 'individuals', thus we must also consider local dynamics within more global dynamics, as for example, convoys of vehicles moving over the surface or tornados within a hurricane. (or Darwin's 'individual' and 'family').

>

>in this case, the referencing must be 'from the outside in', since the ultimate referencing source is the relational interference on the basis of the overall globe or overall finite and unbounded surface (or at least the next level out. [2]

>

>6. if we have more than one level of non-euclidian referencing, ... referencing which is dynamic and relative to the motional agent, ... then each individual agent must act simultaneously on the basis of his local reciprocal disposition and his group's global reciprocal disposition. in a convey of vehicles, he must consider his dynamic within the group simultaneously with group's dynamic within the global container. as he moves locally, then, he must imagine how he and his local team are moving with respect to the containing (more global) ensemble. he will thus have a 'real' informational signal, and an 'imaginary' informational signal, as shown in the 'theory of communications' of denis gabor.

>

> * * *

>

>while the above appears to characterize the mathematical problem, ... that if one chooses the non-euclidian assumption, one must manage one's movements on the basis of a 'floating datum', the continually transforming shape of space (reciprocal disposition), since it is a more general level of information than the movements of particular constituents of space (which cannot be isolated since they simultaneously change the 'shape' of space). and in the presence of three geometrical levels, ('e.g. metaphorically, 'organ, organism, environment'), which seems to establish the general case, the reference information becomes complex; i.e. the motional agent acts locally (real component) but subordinates his local movements to his imagined picture of his local systems motion within the more global containing space..

>

>as henri laborit has suggested, this geometry seems to characterize nature (ecology), from the atom to molecule to cell to organ to organism to community to culture to global society to environmental whole.

>

>from a practical point of view, then, the dynamical agents in this system (the drivers) can choose between the simpler euclidian convention which has them design their response upon the configuration of objects, or the more complex (relativistic) non-euclidian convention which has them design their response upon the (sphere within sphere) configuration of space. as Poincare says, ...the basis for choosing is 'whatever works best for the experiential issues one is dealing with'.

>

>experiential data sets differ in their demands on space conventions. if i am repairing machinery, the euclidian space convention may be adequate. with other datasets, the situation may be different. consider the following dataset discussed by a colleague in the systems sciences, ... 'human dynamics on earth'.

>

>>>interconnectedness via media, information technology, and world

>>>getting 'smaller' via transport and communication technology and

>>>burgeoning populations which pushe everything 'closer together' -

>>>before, people could imagine that neither events, nor populations,

>>>nor pressure resultant on exploding numbers of interactions - were

>>>so 'close' or came so thick and fast in time. They had time to

>>>'digest' an event, or situation, and respond in due course - now,

>>>that is changed - our apparatus for dealing with our world

>>>(perception, thinking, intuition) was 'built' for a slower pace - the

>>>'constructed' world and its pace of activity, are outstripping our

>>>capacity for dealing with it. (m. dodds-taljaard)

>

>under these circumstances, it would seem that the non-euclidian convention and the simultaneous dual-level referencing would be an advantage, ... and that is indeed what comes directly from my own investigations into exceptional versus dysfunctional teams. exceptional teams are characterized by their provision of a realtime view of the team in the context of its relationships with its containing environment, to each member of the team, so that the actions of individual teammembers SIMULTANEOUSLY account for local and global needs, possibilities and opportunities. dysfunctional teams are characterized by having members who focus on local activities out of the context of the global dynamics in which they are immersed.

>

>the implicit choice of non-euclidian space by the exceptional teams, also gives rise to the notion that one's actions may produce 'entropy' (discord of whole and part on a nested, multilevel basis) and/or 'syntropy' (harmony of whole and part on a nested, multilevel basis), and that these 'qualities', which though they cannot be explicitly measured since they are purely relativistic 'ordering' characteristics of unbounded space, .... can be 'intuited', as Einstein discusses in 'Geometry and Experience' [3]

>

>regards,

>

>ted lumley

>

>[1] "Only the genius of Riemann, solitary and uncomprehended, had already won its way to the middle of the last century to a new concept of space, in which space was deprived of its rigidity, ... in which its power to take part in physical events was recognized as possible." (Einstein). "This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived." (Max Born, speaking of 'spherical space')

>

>[2] According to Poincaré, all geometric systems deal with the same properties of space, although each of them employs its own language, whose syntax is defined by the set of axioms. In other words, geometries differ in their language, but they are concerned with the same reality, for a geometry can be translated into another geometry. There is only one criterion according to which we can select a geometry, namely a criterion of economy and simplicity. This is the very reason why we commonly use Euclidean geometry: it is the simplest. However, with respect to a specific problem, non-Euclidean geometry may give us the result with less effort. In 1915, Albert Einstein found it more convenient, the conventionalist would say, to develop his theory of general relativity using non-Euclidean rather than Euclidean geometry. Poincaré's realist opponent would disagree and say that Einstein discovered space to be non-Euclidean. (Internet Encyclopedia of Philosophy)

>

>[3]"Can we picture to ourselves a three-dimensional universe which is finite, yet unbounded?

>The usual answer to this question is ``No,'' but that is not the right answer. The purpose of the following remarks is to show that the answer should be ``Yes.'' I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental image to which, with some practice, we shall soon grow accustomed.

>

>First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To 'visualise' a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that relation of solid bodies with respect to their reciprocal disposition (contact) which corresponds to the theory of a finite universe. There is really nothing new in what I have to say about this; but innumerable questions addressed to me prove that the requirements of those who thirst for knowledge of these matters have not yet been completely satisfied. So, will the initiated please pardon me, if part of what I shall bring forward has long been known? " (Albert Einstein, from 'Geometry and Experience', an expanded form of an address to the Prussian Academy of Sciences in Berlin on January 27th, 1921. )

* * *

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