small Tao beauty Tao

 

SOMETIMES SMALL IS BEAUTIFUL

Perhaps no variable brings the problems of being alive so vividly and clearly before the analyst's eye as does size. The elephant is afflicted with the problems of bigness; the shrew, with those of smallness. But for each, there is an optimum size. The elephant would not be better off if he were much smaller, nor would the shrew be relieved by being much bigger. We may say that each is addicted to the size that is.
There are purely physical problems of bigness or smallness, problems that affect the solar system, the bridge, and the wristwatch. But in addition to these, there are problems special to aggregates of living matter, whether these be single creatures or whole cities.
Let us first look at the physical. Problems of mechanical instability arise because, for example, the forces of gravity do not follow the same quantitative regularities as those of cohesion. A large clod of earth is easier to break by dropping it on the ground than is a small one. The glacier grows and therefore, partly melting and partly breaking, must begin a changed existence in the form of avalanches, smaller units that must fall off the larger matrix. Conversely, even in the physical universe, the very small may become unstable because the relation between surface area and weight is nonlinear. We break up any material which we wish to dissolve because the smaller pieces have a greater ratio of surface to volume and will therefore give more access to the solvent. The larger lumps will be the last to disappear. And so on.

 

To carry these thoughts over into the more complex world of living things, a fable may be offered:

THE TALE OF THE POLYPLOID HORSE

They say the Nobel people are still embarrassed when anybody mentions polyploid horses. Anyhow, Dr. P. U. Posif, the great Erewhonian geneticist, got his prize in the late 1980s for jiggling with the DNA of the common cart horse (Equus caballus). It was said that he made a great contribution to the then new science of transportology. At any rate, he got his prize for creating - no other word would be good enough for a piece of applied science so nearly usurping the role of deity - creating, I say, a horse precisely twice the size of the ordinary Clydesdale. It was twice as long, twice as high, and twice as thick. It was a polyploid, with four times the usual number of chromosomes.
P.U. Posif always claimed that there was a time, when this wonderful animal was still a colt, when it was able to stand on its four legs. A wonderful it must have been! But anyhow, by the time the horse was shown to the public and recorded with all the communicational devices of modern civilization, the horse was not doing any standing. In a word, it was too heavy. It weighed, of course, eight times as much as a normal Clydesdale.
For a public showing and for the media, Dr. Posif always insisted on turning off the hoses that were continuously necessary to keep the beast at normal mammalian temperature. But we were always afraid that the innermost parts would begin to cook. After all, the poor beast's skin and dermal fat were twice as thick as normal, and it surface area was only four times that of a normal horse, so it didn't cool properly.
Every morning, the horse had to be raised to its feet with the aid of a small crane and hung in a sort of box on wheels, in which it was suspended on springs, adjusted to take half its weigh off its legs.
Dr. Posif used to claim that the animal was outstandingly intelligent. It had, of course, eight times as much brain (by weight) as any other horse, but I could never see that it was concerned with any questions more complex than those which interest other horses. I had very little free time, what with one thing and another - always panting, partly to keep cool and partly to oxygenate its eight-times body. Its windpipe, after all, had only four times the normal area of cross section.
And then there was eating. Somehow it had to eat, every day, eight times the amount that would satisfy a normal horse and had to push all that food down an esophagus only four times the caliber of the normal. The blood vessels, too, were reduced in relative size, and this made circulation more difficult and put extra strain on the heart.
A sad beast.

The fable shows what inevitably happens when two or more variables, whose curves are discrepant, interact. That is what produces the interaction between change and tolerance. For instance, gradual growth in a population, whether of automobiles or of people, has no perceptible effect upon a transportation system until suddenly the threshold of tolerance is passed and the traffic jams. The changing of one variable exposes a critical value of the other.
Of all such cases, the best known today is the behavior of fissionable material in the atom bomb. The uranium occurs in nature and is continually undergoing fission, but no explosion occurs because no chain reaction is established. Each atom, as it breaks, gives off neutrons that, that if they hit another uranium atom, may cause fission, but many neutrons are merely lost. Unless the lump of uranium is of critical size, an average of less than one neutron from each fission will break another atom, and the chain will dwindle. If the lump is made bigger, a larger fraction of the neutrons will hit uranium atoms to cause fission. The process will then achieve positive exponential gain and become an explosion.
In the case of the imaginary horse, length, surface area, and volume (or mass) become discrepant because their curves of increase have mutually nonlinear characteristics. Surface varies as the square of length, volume varies as the cube of length, and surface varies as the 2/3 power of volume.
For the horse (and for all real creatures), the matter becomes more serious because to remain alive, many internal motions must be maintained. There is an internal logistics of blood, food, oxygen, and excretory products and a logistics of information in the form of neural and hormonal messages.
The harbor porpoise, which is about three feet long, with a jacket of blubber about one inch thick and a surface area of about six square feet, has a known heat budget that balances comfortably in Arctic waters. The heat budget of a big whale, which is about ten times the length of the porpoise (i.e. 1,000 times the volume and 100 times the surface), with a blubber jacket nearly twelve inches thick, is totally mysterious. Presumably, they have a superior logistic system moving blood through the dorsal fins and tail flukes, where all cetaceans get rid of heat.
The fact of growth adds another order of complexity to the problems of bigness in living things. Will growth alter the proportions of the organism? These problems of the limitation of growth are met in very different ways by different creatures.
A simple case is that of the palms, which do not adjust their girth to compensate for their height. An oak tree with growing tissue (cambium) between its wood, and its bark grows in length and width throughout its life. But a coconut palm, whose only growing tissue is the apex of the trunk (the so-called millionaire's salad, which can only be got at the price of killing the palm), simply gets taller and taller, with some slow increase of the bole at its base. For this organism, the limitation of height is simply a normal part of its adaptation of a niche. The sheer mechanical instability of excessive height without compensation in girth provides its normal way of death.
Many plants avoid (or solve?) these problems of the limitation of growth by linking their life-span to the calendar or to their own reproductive cycle. Annuals start a new generation each year, and plants like the so-called century plant (yucca) may live many years but, like the salmon, inevitably die when they reproduce. Except for multiple branching within the flowering head, the yucca makes no branches. The branching influorescence itself is its terminal stem; when that has completed its function, the plant dies. Its death is normal to its way of life.
Among some higher animals, growth is controlled. The creature reaches a size or age or stage at which growth simply stops (i.e., is stopped by chemical or other messages within the organization of the creature). The cells, under control, cease to grow and divide. When controls no longer operate (by failure to generate the message or failure to receive it), the result is cancer. Where do such messages originate, what triggers their sending, and in what presumably chemical code are these messages immanent? What controls the nearly perfect external bilateral symmetry of the mammalian body? We have remarkably
little knowledge of the message system that controls growth. There must be a whole interlocking system as yet scarcely studied.

the Places of Tao

The Dhaulagiri (8167 m., western Himalaya, Nepal) South Face represents perhaps the greatest unsolved climbing problem, and may be impossible. From the base of the wall to the summit there is a difference of about 4000 m. with a slope ranging from 50° to 90° on ice, with difficulty from M5 to M7+. In particolar the band of rocks which runs across the whole wall at 7200 m. is considered virtually impossible.

The best attempt on this wall has been done by Tomaz Humar in 1999:

Climbing till the band of rocks Humar was forced to traverse to reach the southest crest.

Tribute to Tao: Stanley Kubrick

Childwickbury Green Manor

Hertfordshire, England

Tao: second quantum leap

first quantum leap

the Paths of Complex Tao

To reach to the point that you do not know,

you need to take the road you do not know.

S. Giovanni della Croce

In the absence of paradigms for complexity, resulting in the impossibility to define methodologies for description and calculating/computing in replacement/integration of the previous used by classic science, Edgar Morin has proposed somepaths to complexity, or better to complexities, as - naturally - the same complexity is complex.

Morin states that the complexity shows as difficulty and uncertainty and not as clarity and answer typical of the paradigms of classical science. The problem is therefore to understand if it is possible to challenge of uncertainty and complexity.

Today, the biological and physical sciences are characterized by a crisis of the simple explanation, and then what appeared to be the residues of the human sciences such as uncertainty and disorder are part of the problem of scientific knowledge.

The complexity is an obstacle, a challenge. It seems negative or regressive because it involves the reintegration of the uncertainty in a knowledge that was going towards the conquest of absolute certainty, absolute that is no longer possible.

Morin introduces some typical characteristics of complexity:

• the problem of contradiction

Morin indicates in Niels Bohr the author of the most logic braking of science. Formulating the Complementarity Principle in the Copenaghen interpretation of the quantum duality probability wave/particle for the first time in history of science with the classic aristotelian logic of or/or introducing with spectacular theoretical and experimental results the logic of and/and, moving from a single logic to a dialogic, that is accepting that two independent and dual logics coexist at the same time.

• the logic limitation

After the proof of the Gödel incompleteness theorems and the development of Tarski logic it became obvious that no system of explanation can explained itself completely by itself.

•   the meta-complexus

It is not possible to approach the complexity through a single and preliminary definition but it is necessary to follow different paths, so different that one might wonder if there are many and different complexities. All the various complexities such as wires are woven together to form the unity of the texture of complexity. So we arrive at complexus of complexus, namely that core of the complexity in which the various complexities encounter.

• the multi-dimensional thinking

The positive aspect that comes from complexity is the nedd of a multidimensional and dialogic thinking, where in the latter two logics, two natures are connected in a unity without thereby dissolve the duality in unity. The notion of dialogic is not a concept that avoids the logical and empirical constraints but is likely to face difficulties, to fight with the real.

La challenge of complexity makes us give up forever the myth of the total universe clarification, encouraging us to continue the adventure of knowledge which is a dialogue with the universe.

The aim of our knowledge is not to close but to open the dialogue with the universe. The Method of Complexity requires us to think without never closing the concepts. The complexity is just the conjunction of concepts that are fighting each other, cohabit with the complexity and conflict trying to keep from falling inside.

The complexity also leads to think in organizational form, that is to understand how the organization does not result in a few laws but, on the contrary, needs a highly developed thought.

The paths indicated by Morin that lead to the challenge of complexity are:

THE PATH OF IRREDUCIBILITY OF RANDOMNESS AND DISORDER

randomness and disorder are inevitably present in the universe and play an active role in its evolution, but we are not able to resolve the uncertainty caused by the notions of disorder and randomness. The same randomness is not sure to be an accident, or an accidental event of which that the causes are unknown.

THE OVERCOMING OF THE LIMITS WHICH ELIMINATED THE SINGULARITY, THE LOCALITY AND THE TEMPORALITY

it is not possible to delete the singular and the local using the universal. Indeed, it is necessary to connect these concepts, for example in contemporary biology the species are no longer considered as a framework within which the individual is a singular case. In contrast, every living species is considered as a singularity that produces singularity within the most diverse physical-chemical organizations that exist. We must connect the singular, the local and the universal

THE PATH OF COMPLICATION

this problem arised when it was realized that the biological and social phenomena had a infinite number  of interactions and di interazioni e inter-feedbacks.

COMPLEMENTARITY BETWEEN ORDER, DISORDER ANDORGANIZATION

comes into play the concept deployed by Von Foerster “Order fron Noise”: from a disordered motion may arise from organized phenomena.

THE PATH OF ORGANIZATION

the organization determines a system from different elements. Is a unit and at the same time a multiplicity unitas multiplex: it should not dissolve the multiple in the one, nor the one in the multiple. A system is something more and something less of the sum of its parts. Something more because brings out more of the qualities that would not exist without the organization, something less because this organization imposes some constraints that limit some of the potential found in the individual parts. The qualities that emerge, exercis feedback on the individual parts and may stimulate and express their full potential. For example, culture, language or education are properties that can only be a matter of social totality and, in turn, feedbacking on different parts of the society, allow the development of mind and intelligence of individuals.

HOLOGRAMATIC PRINCIPLE AND RECURSIVE ORGANIZATION

in the field of complexity emerges the hologramatic principle: not only the part is in everything, but everything is in the part. In trying to understand the phenomenon one must go from the parts to the whole and from the whole to the parts by adopting a non-linear and circular explanation. The hologram is a physical image that has the quality that each point contains almost all the information of the whole, such as criminal law, the fact that every cell of an organism contains the genetic information of the whole organism.

The hologramatic principle should be connected to the recursive organization principle: a recursive process is a process in which the products and effects are at the same time causes and producers of what that produce them. The idea of recursion is therefore a breaking idea with the of linear idea of cause/effect, product/producer, structure/superstructure;for example the reproduction produces individuals that produce the reproductive cycle.

THE CRISIS OF CLARITY AND OF THE SEPARATION IN THE EXPLANATION

there is a break with the idea that truth is given by the clarity of ideas. The truth is also evident in the ambiguity and in the apparent confusion. It is no longer possible to make a boundary between science and non science, between subject and object, between organism and environment as occured previously for experimental science: it took a subject, extracted it from its context and placed it in an artificial environment, then modified it and checked its modifications.

In addition not to isolate a self-organizing system from its environment, it is necessary to connect this system to its environment, or to obtain a self-eco-organization. The concept of autonomy implies that a system is both open and closed (the result of an operation of the system still falls within the boundaries of the system). This type of system has to maintain its individuality and originality.

THE RETURN OF THE OBSERVER

it is not possible to eliminate the observer from the observations that are made. Always keeping in mind the hologramatic principle, the observer is in the society, but the society is also in the observer. Therefore the observer has to integrate himself into its observation and in its conception and should try to understand its socioculturalhic et nunc. Principio di integrazione dell’osservatore: regardless of the theory and whatever its content, must account that the observer is part of it: 

"... whatever the theory, and whatever its content, it must account of what makes it possible to produce the theory itself. If in any case is unable to account for this, it must also know that the problem is posed."

Immortal Dialogues of Tao: Have you ever considered ... Willard ... any real freedoms?

WILLARD "It smelled like slow death in there, malaria, nightmares. This was the end of the river allright."

COLONEL KURTZ "Where are you from Willard ?
WILLARD "I'm from Ohio, sir."
KURTZ "Were you born there ?"
WILLARD "Yes, sir."
KURTZ "Whereabouts ?"
WILLARD "Toledo, sir."
KURTZ "How far were you from the river ?"
WILLARD "The Ohio river, sir?"
KURTZ "Uh Uh..."
WILLARD "About 200 miles."
KURTZ "I went down that river when I was a kid. There's a place in the river...I can't remember... Must have been a gardenia plantation at one time. All wild and overgrown now, but about five miles you'd think that heaven just fell on the earth in the form of gardenias...
Have you ever considered any real freedoms? Freedoms - from the opinions of others... Even the opinions of yourself.
They say why..., Willard, why they wanted to terminate my command ?"
WILLARD "I was sent on a classified mission, sir."
KURTZ "Ain't no longer classified, is it? What did they tell you ?"
WILLARD "They told me that you had gone totally insane and that your methods were unsound."
KURTZ "Are my methods unsound?"
WILLARD "I don't see any method at all, sir."
KURTZ "I expected someone like you. What did you expect?"
Willard only shakes his head
KURTZ "Are you an assassin?"
WILLARD "I'm a soldier."
KURTZ "You're neither. You're an errand boy, sent by grocery clerks to collect a bill."

quantity and pattern of Tao

 

QUANTITY DOES NOT DETERMINE PATTERN

It is impossible, in principle, to explain any pattern by invoking a single quantity. But note that a ratio between two quantities is already the beginning of pattern. In other words, quantity and pattern are of different logical type and do not readily fit together in the same thinking.
What appears to be a genesis of pattern by quantity arises where the pattern was latent before the quantity had impact on the system. The familiar case is that of tension which will break a chain at the weakest link. Under change of a quantity, tension, a latent difference is made manifest or, as the photographers would say, developed. The development of a photographic negative is precisely the making manifest of latent differences laid down in the photographic emulsion by previous differential exposure to light.
Imagine an island with two mountains on it. A quantitative change, a rise, in the level of the ocean may convert this single island into two islands. This will happen at the point where the level of the ocean rises higher than the saddle between the two mountains. Again, the qualitative pattern was latent before the quantity had impact on it; and when the pattern changed, the change was sudden and discontinuous.
There is a strong tendency in explanatory prose to invoke quantities of tension, energy, and whatnot to explain the genesis of pattern. I believe that all such explanations are inappropriate or wrong. From the point of view of any agent who imposes a quantitative change, any change of pattern which may occur will be unpredictable or divergent.

Imperial Tao

the Teh of Tao

- 13 -

Success is as dangerous as failure.
Hope is as hollow as fear.

What does it mean that success is a dangerous as failure?
Whether you go up the ladder or down it,
you position is shaky.
When you stand with your two feet on the ground,
you will always keep your balance.

What does it mean that hope is as hollow as fear?
Hope and fear are both phantoms
that arise from thinking of the self.
When we don't see the self as self,
what do we have to fear?

See the world as your self.
Have faith in the way things are.
Love the world as your self;
then you can care for all things.  

stick no bills on Tao

black boxed Tao method

D: Daddy, what’s a black box?

F: A “black box” is a conventional agreement between scientists to stop trying to explain things at a certain point. I guess it’s usually a temporary agreement.

D: But that doesn’t sound like a black box.

F: No—but that’s what it’s called. Things often don’t sound like their names.

D: No.

F: It’s a word that comes from the engineers. When they draw a diagram of a complicated machine, they use a sort of shorthand. Instead of drawing all the details, they put a box to stand for a whole bunch of parts and label the box with what that bunch of parts is supposed to do.

D: So a “black box” is a label for what a bunch of things are supposed to do….

F: That’s right. But it’s not an explanation of how the bunch works.

D: And gravity?

F: Is a label for what gravity is supposed to do. It’s not an explanation of how it does it.

D: Oh.

Metalogue: "What is an instinct?"

Two are the methods that classical science has used with great success for simple and complicated systems. The question is wheter these methods can also be used for complex systems.

The first, reductionism and its inverse, constructionism, was rated inadequate by the work of Anderson.

The second is the black box method, historically derived from the theory of electronic systems, but widely studied and generalized in the first cybernetics, in particolare da Ross Ashby.

In the approach of the black box we know nothing about the interior of the system, we can only access it through the entrances where the introduction of in-put tand observe the outputs where to gather the out-put. This is a very common situation in physics, electronics, systems theory but also in biology, medicine and psychology, where the system undergoes a cause and effects are observed, reactions to a stimulus etc.. For example, an entire approach to psychology, the behaviorism, is entirely based on the black box, as well as medical drug testing procedures.

x	f	y
input	operation	output
indipendent variable	function	dipendent variable
cause	natural law	effect
minor premise	major premise	conclusion
stimulus	organism	reaction
reason	character	action
goal	system	behavior

A black box system is characterized by a input-output relation through an operator f which to any input x provides the output y: y=f(x).

trivial determined machine

If the operator f is known, by form or function or of the output/input relation for all inputs of interest, then the system is completely determined. Conversely, if f is unknown, and this applies in almost all cases, the way to determine it is to introduce input values to the inputs and observe the corresponding output values on output.

VonFoerster calls the black box "machine" and divides the machines into two categories, those without internal states and those with internal states, defining the first trivial and the second non-trivial machines.
In trivial machines one introduces a certain number of input values x1 x2 x3 ... and will collect the corresponding output values y1 y2 y3 ... If at an input value xn is always and only associated an output value yn, then the machine is history independent (it is a system without memory), is determined in a synthetic way, that is that machine has been designed to operate in that specific way and only in that way, it is analytically determinable, that is given a certain number of input values the machine is fully known and describable, and finally is predictable; that's why Von Foerster defines it trivial. In the example of figure if we give an input value A we get an output value B, if we give B we get C and so on, if we supply again on the first input A we'll get again B etc. In general, the machine is described by tables which associate with each xi of interest the corresponding yi.

If the machine has only one input and one output, then given an input value x1 with the corresponding value y1=f(x1) the machine is fully described for that value. If we have two inputs and two input values x1 and x2 to completely describe the machine is necessary to observe the outputs with the permutations x1x1, x1x2, x2x1, x2x2. In general, if N is the number of inputs (assumed equal to that of outputs) the number of input configuration to be calculated to completely describe the machine is N^N. For a machine with 4 inputs  and 4 outputs there are 4⁴=256 different configurations to compute, for a one with 10 input s and 10 outputs there are 10¹⁰=10 billions configurations; assuming that to compute one it is necessary one microsecond, one millionth of second, namely that any configuration is computable at any MIPS (million instructions per second), a 70s computer at 1 MIPS took 10000 seconds, namely little less than three hours to compute all the possible configurations, one of the 80s at 10 MIPS about 16 minutes, a Pentium III  less than  two minutes and a Pentium 4 about 1 second. The trivial machine seems to confirm the statement of Atlan that for a complicated system, having sufficient resources of time and money, you can come to have complete knowledge.

As  Von Foerster tells when we buy a any machine - a car, a blender, a washing machine - we expect, and the seller is ready to confirm, it's a trivial machine; then one day you turn the key and nothing happens, or press a button and floods the house, then go to a mechanic or technician and ask him to make the machine again trivial.

The black box model with trivial machine is and has been a successful tool in the analysis of simple and complicated systems, and its natural extrapolation is to add internal states in the machine to try to use it for complex systems. The presence of internal states within the system (with only one internal state we have the trivial machine) is one of the features of complex systems.

non-trivial internal states machine

In this case the black box system has no more one but two input/output operators: ithe first y=f(x,z) gives the output value as a function of the x and of the internal state z, the second one z=g(x,z1) gives the new internal state of the machine as a function of input x and of the previous internal state z1.

Proceeding as before with the trivial machine, providing an input x1 does not mean that the output is always y1, but can be any output y1z depending on the internal state z which at that time has the machine, state that can be modified by any of the input.

The number of configurations that must be calculated to provide a complete knowledge of the internal state machine is not great as in the trivial machines, neither astronomical, it is meta-astronomical. In the case of 2 inputs, 2 outputs and 2 internal states the possible configurations are 2¹⁶=65536, a number which is easily computable, but at the next step with 4 inputs, 4 outputs and 2 internal states the configurations  become 2⁸¹⁹²=10²⁴⁶⁶; to try to evaluate this number considering that the age of universe (around 13 billions years) expressed in seconds is about 10¹⁷ seconds and that to 2010 the most powerful supercomputer with parallel multi-processors (Tianhe-1A with 186368 core processors Intel EM64T Xeon X56xx at 2930 MHz with a 229376 GB memory) had a computing speed of about petaFLOPS, namely about 10¹⁸ floating point operations per second and assuming that an input configuration can be computed with 10 FLOPS we have that to compute the total number of 10²⁴⁶⁶ configurations is required a time 10²⁴³² times the age of the universe.

As Von Foerster says: "... I highly recommend you not to undertake such an enterprise ... you'll loose time, money ... everything".
The non-trivial machine is determined in a synthetic way as the trivial ones, it is dependent from history due to its internal states, it is analytically indeterminable, as clear from the example, and finally it is umpredictable, like all complex systems.

TRIVIAL MACHINE	NON TRIVIAL MACHINE
input-output stimulus-response	It has internal states that influence what it will do. When acting on it, it can change its internal state. With the same previous imput, not necessarily it will act in the same way.
It is determined in a synthetic way: when it was built, it was determined the way in which it was working.	Determined in a synthetic way: we can build it as we want
Analytically determinable: if analyzed, it produces a determined result.	Analythically indeterminable
Independent from history: whatever is the applied input, it does not remember it following the previous laws.	Dependent from history
Predictable: when given a certain input, we know wht the machine will do.	Unpredictable

As Von Foerster tells Ross Ashby had built a machine of this type and used it to assess students who applied to work with him. It was a machine with two inputs, two outputs and four internal states and was presented as an innocuous small metal box with two switches (inputs), two light bulbs (the output) and was powered by batteries.

During the interview, Ashby casually suggested to the student: "Why not see to understand how this box works?". The student was not very impressed - two switches, two lamps - and began to work and take down input /output tables. Let's say this happened at three in the afternoon. Ashby came home about six o'clock and returned to the office the next morning at seven. If the student was not there, was then asked "What happened with that problem?" and if the student told him he worked until midnight did not understand anything and was gone bored Ashby thought "It's not my man". If the student was there exhausted, sleepless, with a greenish color, surrounded by hundreds of tables he would say, 'You're tough, interested, but let me give you a hint: you'll never solve this problem ".

The simple Ashby machine is perhaps the most concrete physical object that can represent the abstract mathematical concept of infinity. In general, the definition of infinity is represented through the concept of limit, but in the physical reality, even considering astronomical numbers such, for example, the estimated (uncertain) total number of atoms in the visible universe, that is  10⁷⁹  - 10⁸⁵ , or the estimation of the computational capacity of the entire universe calculated by Lloyd in 10¹²⁰ elementary logical operations with an accumulation of information as to 10⁹⁰ bits, one does not get the meta-stronomical numbers of the simple Ashby non-trivial machine.

Computational capacity of the universe

Seth Lloyd

Phys.Rev.Lett. 88 (2002)

Merely by existing, all physical systems register information. And by evolving dynamically in time, they transform and process that information.The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The universe is a physical system. This paper quantifies the amount of information that the universe can register and the number of elementary operations that it can have performed over its history. The universe can have performed no more than 10^120 ops on 10^90 bits

The W. Ross Ashby Digital Archive

The Heinz von Foerster Page

i feel good, i feel bad, i feel Tao

is a matter of quality
or a formality
I do not remember well, a formality
  like to decide to shave the hair
to eliminate coffee, cigarettes
to do away with someone
or something, a formality formality
or a quality issue
I'm fine I'm sick
I do not know how to be
I do not know where to stay
do not study do not work do not watch TV
I do not go to the movies do not sport
I'm good I'm bad I do not know
what to do I do have nor art nor part
I have nothing to teach
is a matter of quality
or a formality
I do not remember well, a formality

the number of Tao is not the quantity of Tao

NUMBER IS DIFFERENT FROM QUANTITY

This difference is basic for any sort of theorizing behavioral science, for any sort of imagining of what goes on between organisms or inside organisms as part of their processes of thought.
Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three, there is a jump. In the case of quantity, there is no such jump; and because jump is missing in the world of quantity, it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.
Even when number and quantity are clearly discriminated, there is another concept that must be recognized and distinguished from both number and quantity. For this other concept, there is, I think, no English word, so we have to be content with remembering that there is a subset of patterns whose members are commonly called "numbers." Not all numbers are the products of counting. Indeed, it is the smaller, and therefore commoner, numbers that are often not counted but recognized as patterns at a single glance. Card-players do not stop to count the pips in the eight of spades and can even recognize the characteristic patterning of pips up to "ten."
In other words, number is of the world of pattern, gestalt, and digital computation; quantity is of the world of analogic and probabilistic computation.
Some birds can somehow distinguish number up to seven. But whether this is done by counting or by pattern recognition is not known. The experiment that came closest to testing this difference between the two methods was performed by Otto Koehler with a jackdaw. The bird was trained to the following routine: A number of small cups with lids are set out. In these cups, small pieces of meat are placed. Some cups have one piece of meat, some have two or three, and some cups have none. Separate from the cups, there is a plate on which there is a number of pieces of meat greater that the total number of pieces in the cups. The jackdaw learns to open each cup. Taking off the lid, and then eats any pieces of meat that are in the cup. Finally, when he has eaten all the meat in the cups, he may go to the plate and there eat the same number of pieces of meat that he got form the cups. The bird is punished if he eats more meat from the plate than was in the cups. This routine he is able to learn.
Now, the question is: is the jackdaw counting the pieces of meat, or is he using some alternative method of identifying the number of pieces? The experiment has been carefully designed to push the bird toward counting. His actions are interrupted by his
having to lift the lids, and the sequence has been further confused by having some cups contain more than one piece of meat and some contain none. By these devices, the experimenter has tried to make it impossible for the jackdaw to create some sort of pattern or rhythm by which to recognize the number of pieces of meat. The bird is thus forced, so far as the experimenter could force the matter, to count the pieces of meat.
It is still conceivable, of course, that the taking of the meat from the cups becomes some sort of rhythmic dance and that this rhythm is in some way repeated when the bird takes the meat from the plate. The matter is still conceivably in doubt, but on the whole, the experiment is rather convincing in favor of the hypothesis that the jackdaw is counting the pieces of meat rather than recognizing a pattern either of pieces or of his own actions.
It is interesting to look at the biological world in terms of this question: Should the various instances in which number is exhibited by regarded as instances of gestalt, of counted number, or of mere quantity? There is a rather conspicuous difference between, for example, the statement "This single rose has five petals, and it has five sepals, and indeed its symmetry is of a pentad pattern" and the statement "This rose has one hundred and twelve stamens, and that other has ninety-seven, and this has only sixty-four." The process which controls the number of stamens is surely different from the process that controls the number of petals or sepals. And, interestingly, in the double rose, what seems to have happened is that some of the stamens have been converted into petals, so that the process for determining how many petals to make has now become, not the normal process delimiting petals to a pattern of five, but more like the process determining the quantity of stamens. We may say that petals are normally "five" in the single rose but that stamens are "many" where "many" is a quantity that will vary from one rose to another.
With this difference in mind, we can look at the biological world and ask what is the largest number that the processes of growth can handle as a fixed pattern, beyond which the matter is handled as quantity. So far as I know, the "numbers" two, three, four, and five are the common ones in symmetry of plants and animals, particularly in radial symmetry.
The reader may find pleasure in collecting cases of rigidly controlled or patterned numbers in nature. For some reason, the larger numbers seem to be confined to linear series of segments, such as the vertebrae of mammals, the abdominal segments of insects, and the anterior segmentation of earthworms. (At the front end, the segmentation is rather rigidly controlled down to the segments bearing genital organs. The numbers vary with the species but may reach fifteen. After that, the tail has "many" segments.) An interesting addition to these observations is the common circumstance that an organism, having chosen a number for the radial symmetry of some set of parts, will repeat that number in other parts. A lily has three sepals and then three petals and then six stamens and a trilocular ovary.
It appears that what seemed to be a quirk or peculiarity of human operation - namely, that we occidental humans get numbers by counting or pattern recognition while we get quantities by measurement - turns out to be some sort of universal truth. Not only the
jackdaw but also the rose are constrained to show that for them, too - for the rose in its anatomy and for the jackdaw in its behavior (and, of course, in its vertebral segmentation) - there is this profound difference between numbers and quantity.
What does this mean? That question is very ancient and certainly goes back to Pythagoras, who is said to have encountered a similar regularity in the relation between harmonics.
The hexago-rectangle discussed in section 5 provides a means of posing these questions. We saw, in that case, that the components of description could be quite various. In that particular case, to attach more validity to one rather than to another way of organizing the description would be to indulge illusion. But in this matter of biological numbers and quantities, it seems that we encounter something more profound. Does this case differ from that of the hexago-rectangle? And if so, how?
I suggest that neither case is as trivial as the problems of the hexago-rectangle seemed to be at first sight. We go back to the eternal verities of Saint Augustine: "Listen to the thunder of that saint, in about A.D. 500: 7 and 3 are 10; 7 and 3 have always been 10; 7 and 3 at no time and in no way have ever been anything but 10; 7 and 3 will always be 10."

No doubt, in asserting the contrast between numbers and quantities, I am close to asserting an eternal verity, and Augustine would surely agree.
But we can replay to the saint, "Yes, very true. But is that really what you want and mean to say? It is also true, surely, that 3 and 7 are 10, and that 2 and 1 and 7 are 10, and that 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 are 10. In fact, the eternal verity that you are trying to assert is much more general and profound than the special case used by you to carry that profound message." But we can agree that the more abstract eternal verity will be difficult to state with unambiguous precision.
In other words, it is possible that many of the ways of describing my hexago-rectangle could be only different surfacings of the same more profound and more general tautology (where Euclidean geometry is viewed as a tautological system).
It is, I think, correct to say, not only that the various phrasings of the description of the hexago-rectangle ultimately agree about what the describers thought they saw but also that there is an agreement about a single more general and profound tautology in terms of which the various descriptions are organized.
In this sense, the distinction between numbers and quantities is, I believe, nontrivial and is shown to be so by the anatomy of the rose with its "5" petals and its "many" stamens, and I have put quotation marks into my description of the rose to suggest that the names of the numbers and of the quantities are the surfacing of formal ideas, immanent within the growing rose.

winter Tao