Tao Types

The formal treatment of hierarchical logical levels was carried out by Bertand Russell and Alfred N. Whitehead in the first decade of this century, and appeared in final form in 1910 in the monumental work of mathematical logic with the title of "Principia Mathematica".
A major aim of this work was to preserve classical logic from paradoxes and antinomies. The simplest example is the kind of paradox:

"This statement is false"

If the statement is true then it is false, if it is false then it is true.
By examining the logical structure of this statement is known as having two characteristics simultaneously valid:

• The statement is self-reference, that is it refers to itself. For example the statement 'this apple is red"is not a paradox, since a statement is not an apple, while it may be true or false depending on whether the apple is red or not.
• The statement has the logical structure of the type "assertion of a negation" or -equally- "negation of an assertion", and being self-referential denies himself. For example the statement "this statement is true" is not paradoxical, if it is true is true and if it is false is false.

In the classical Aristotelian logic such a situation is devastating, and is determined by the principle of contradiction, which asserts the falsity of every proposition implying that a certain proposition A and its negation, the proposition that non-A, are both true at the same time and in the same way. In the words of Aristotle (Metaphysics):

« it is not possible to say truly at the same time that the same thing is and is not»

In order to preserve classical logic from paradoxical and self-referential problems Russell and Whitehead was assigned to the hierarchical logical levels, which they called "logical types" special rules, establishing a hierarchy of logical types that can not be broken, especially the rule that objects (elements) of a class (together) are at logical level lower than the class and, to prevent the formation of paradoxes, a class may not have as an element itself or anything that presupposes all the elements of a collection should not be an element of the collection itself.
With this split-level logical types on two different planes, one for the class and the other for its members , along with the rule to prevent logical links between the two planes, Russell and Whitehead intended to prevent the formation of the paradoxes in logic. With this subdivision and rule logic "self-swallowing" classes, in which an element of the class is the class itself,  become meaningless and without any logical validity.
A class of classes, ie, a metaclass, is not really a class, say for example that the set of all concepts is itself a concept is meaningless since it is a 'concept' of a higher logical type. The components of a Russellian hierarchy are among them as an element to a class, a class to a metaclass or one thing to their name.

The work of formal systematization and logical foundation of the Principia Mathematica posed the possibility of creating a unique logical-formal system and organize all of mathematics and then physics. The vision was that romantic due to the success of classical physics, namely that the universe was extremely complicated but fully describable if the logic and the math behind the physics were a complete formal system of description of the physical layer.
Thus arose the question of completeness and consistency of such system; it is complete if all true statements of mathematics are derivable "demonstrable" at its inside, it is coherent-consistent  if no internal contradictory statements can be derived, namely a proposition and its negation. A question of this sort comes in metamathematics, because it is a math survey on mathematics.For this purpose D.Hilbert launched the so-called "Hilbert's program" in the 20s: to demonstrate the completeness and consistency of the PM, or the attempt to axiomatization of mathematics. The main points were: 

• Formalization of all mathematics: all mathematical propositions should be written in a precisely formal language , and handled according to well-defined rules.
• Completeness: proof that all the statements that are true can be proved mathematically in the formalism.
• Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This test of consistency should preferably use only methods "finitistic" about finite mathematical objects.
• Conservation: proof that all the results of "real objects" obtained using the argument about "ideal objects" (as uncountable sets) can be tested without the use of ideal objects.
• Decidability: there should be an algorithm to decide the truth or falsity of any mathematical proposition.

The Russell and Whitehead's monumental effort to save the classical logic and the Hilbertian program that followed were literally swept away forever by the work of Kurt Gödel in1931, in particular the two incompleteness theorems, not by chance entitled "On formally undecidable propositions of Principia Mathematica and related systems", that demonstrate how a axiomatic theoretical building can not simultaneously satisfy the properties of consistency and completeness, and that no consistent system can be used to demonstrate its own consistency.
During those same years, the development of quantum physics and relativity poses a definitive end to the idea that complete formalization of the physical Universe, together with its logical foundations, was possible.

The work of Russell and Whitehead has today an historical significance, however paradoxes and paradoxical situations exist, as well as in logic, also in life. Its use was, unexpectedly, in the description of the interaction and communication with animals and humans by Gregory Bateson. In his words:

"What Russell and Whitehead had faced was a very abstract problem: logic, which they believed, had to be rescued from the tangles that arise when the "logical types", as Russell called them, are mistreated in their mathematical representation.

I do not know if, while working within the "Principia," Whitehead and Russell had any idea that the object of their interest is essential for the life of humans and other organisms.

Whitehead certainly knew that playing with the types you can have fun and you can make the humor emerge. But I doubt that he ever passed the stage of fun and has come to realize that the game was not insignificant and that would throw light on the entire biology.
While not having to contemplate the nature of the human dilemmas that would have been revealed they avoided - perhaps unconsciously - to arrive at a more general understanding."

fractalised Tao

«Never-ending wonders pop out from simple rules, if these are repeated ad infinitum.»

Benoît Mandelbrot

the (not) certainty of Tao

Hieronymus Bosch
The Crowning with Thorns 
about 1485, oil on oak, National Gallery - London

In medieval iconography the four characters who surround Jesus symbolize the four human representations considered in the Middle Ages. In particular, the character in the bottom right, is still holding Jesus for the mantle and the ground freezes, represents the slavery of certainty in relation to the transcendental: "If I know, I already know"

Ascolta

Trascrizione fonetica

from H. Maturana, F. Varela, "Tree of Knowledge", 1984

Tao types

A possible cataloging , not exhaustive, of the various types of systems based on the structure, function, internal properties or of input/output can be:

• Absurd-Meaningless - Impossible, by constrast: Ingenious

These are systems for which the selection and composition of elements and/or the types of relationships/connections that are chosen do not have any sense or are inconsistent with fundamental laws.
It's worth noting that many fundamental discoveries were born to find meaning or links in natural or conceptual systems, where before it was thought there were not.

• Trivial

These are systems whose description/analysis/knowledge does not add any information.

• Simple

These are systems whose structure/functionality is easy to analyze/describe/implement.

• Elementary

These are systems that have basic features but whose structure/description is not necessarily simple.

• Deterministic / Random-Stochastic

This terms generally refer to the internal state of the system (defined by the set of internal state observable variables or to the characteristic of input/output. If a defined state of the system is a precise and unique state of the process or if a set value input produces always a defined output value then the system is deterministic - once determined these values are valid forever. If the values are descrivable by a probabilistic random variable the system is random, also known as dynamic random or stochastic, governed by a certain probability distribution on the values. Typically this means that some parts of the system/subsystem/processes are of random/probabilistic nature.

• with/ without Memory

In systems with memory the system status and/or the value of the input/output function depend on the state or the input of the past.

• linear / nonlinear

This term refers to the in/out function of the system, whether linear or not.
Typically a system is only linear in the range of values over a certain dynamic of values, the input / output becomes non-linear, such as saturation. A typical example of linear system are the amplifiers.

Linear systems are the only ones that can have a complete formal description of their features. A technique for dealing with nonlinear systems is to linearize them for a certain range of values, or simulate them by one or more linear systems.

• stationary / non stationary

Systems are stationary (or time invariant) when the internal parameters do not depend on time but are constant.

• static / dynamical

Among all the linear systems are of particular interest those lwhich are linear and stationary, for which the output signal depends only on the instantaneous value of the input signal. Conversely, there are linear dynamic systems, which are those systems for which the response depends - as well as input from the actual value - even from its past history.

•  open / closed / isolated / adiabatic

Open systems, first defined by von Bertalanffy, are systems that can thermodynamically interact with the external environment exchanging both energy (work or heat) and matter.

A closed system is a system that does not exchange mass with the external environment, while it may carry exchange of energy in all its forms (including heat) or work.An isolated system is a system that does not interact in any way with the environment, or which does not exchange neither mass, nor work and heat.

An adiabatic sistem is a closed system that can not exchange heat or matter with the external environment, but can exchange work.

• concentrated / distributed constants

It is a terminology normally applied to systems circuit. If the minimum wavelength of the signals passing through a circuit is large compared to the components/elements of the circuit is said to be with constant-concentrated, conversely if it is comparable is called with distributed parameters. A typical example are microwave circuits, where the wavelength of the signals is the order of cm. or mm.

• discrete time /continuous time

It is an alternative and more genera definition for the values/variables/processes of digital/analog types. If the variables or processes of the system are discrete in time, or only may take certain discrete values, the system is discrete-time, while if they have continuous values over time are continuous time.

• discrete states-events / continuous states-events

Similar to the precedent for the system internal states.

• synchronous / asynchronous / syncronicity

In the first two types we refer in general to the comparison between the internal processes of the system or of some internal processes and other external to the system boundary. If the two processes have a correlation, usually in time, of the one-to-one among them, such as the type of cause and effect, these processes are synchronized with each other,  while if they are temporally independent are called asynchronous.

The term syncronicity was introduced by Carl G. Jung in 1950 to describe a connection between events, psychological or objective, which occur synchronously, ie at the same time, but between which there is not a relation of cause and effect but a clear commonality of meaning. By extension, a system is synchronic when has relations between internal and/or external processes of synchronic type.

• Paradoxical or Oscillators

The oscillatory systems are those where the internal states, processes or the system's output are of oscillation type , with a period, usually in time. The oscillation of the state or output can be generated by a system internal or internal/external paradox , for example of the type "if not then yes - if yes then not"; in this case the state or the system output becomes a continuous oscillation of yes and no.

• Strange Loop

The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter to describe a situation where by moving up or down through a hierarchical system one finds oneself back where one started. Strange loops may involve self-reference and paradox. By extension strange loop systems are those where internal processes are of this type.

• Chaotic

A dynamical sistem is of  chaotic type if it has the following three main characteristics:

1. Sensitivity to initial conditions, that is to infinitesimal variations of the boundary conditions (or, generally, of the inputs) match with finite variations in the output. As a trivial example: the smoke of burning matches under the most very similar  conditions (pressure, temperature, air currents) follows trajectories from time to time very different.
2. Unpredictability, ie it is not possible to predict in advance the performance of the system on long times compared to the characteristic time of the system starting from assigned boundary conditions.
3. The evolution the system is described, in the phase space, by many stochastic trajectories as seen by an external observer, which all remain confined within a defined space: the system that is not evolving toward infinity, for any variable; in this case we speak of 'attractors' or even 'deterministic-chaos'.

A chaotic system is deterministic in general, that is regulated by a very precise law that requires  to assume a certain state (given its previous history and the law). The special feature of chaotic systems (a result of which are often confused with the random systems) is the fact that the upgrading law strongly depends on the initial conditions: a tiny change in initial conditions lead the system in a state far from what it could have achieved without this small variation.

• Fractals

The term fractal was coined in 1975 by Benoît Mandelbrot, and derives from the Latin fractus fraction; in fact fractal images are considered by the mathematical objects with fractionary dimension. A fractal is a geometric object that is repeated in its structure the same on different scale, or that its aspect does not change at any magnification. This feature is often called self-similarity.  The distinctive feature of fractals is that while the generation rules are relatively simple, their result produces meta-infinites.
By extension, fractal systems are those where processes or even the elements are of a fractal type.

• Synergistic

The term synergetics was introduced by Hermann Haken within theframe of laser physics in the 70-80 years .Is commonly defined as a combined action of two or more elements, resulting in enhanced efficacy compared to their simple summation. It follows that a synergy is a simultaneous action of two phenomena, forces, or other entity, which strengthens the individual effects. A system is synergistic if one of its internal or external/internal processes have a synergy.

refer to: (Center for Synergetics, Stuttgart University)

• Undecidable

It is a characteristic that occurs in a particular set of dynamic systems called finite automata, in particular for the Turing machine (TM). A TM (similar in all effects to a computer) is a formal system which can be described as an ideal mechanism, but in principle feasible, which can be in well-defined states (state machine), operating on strings according to strict rules and is a computing model. In a system of this type one pose the halting problem, or if it always possible in a TM, which has unlimited evolution, described a program and a given finite input, decide whether this program will end or will recur indefinitely. It was shown that there can be no general algorithm that can solve this problem for all possible inputs.

• Complicated/Complex

The term complicated derives from complicatus, ie "with folds", and can be explained by the socalled classic science, while complex (from complexus, ovvero "weaving") cannot be explained by classical science.

The definition of complexity is itself complex - many authors in different fields have proposed their own definition of complexity. Moreover, the distinction between complicated and complex is not clear, both systems have in general the following features:

1. structure with many elements already in themselves complex
2. non-linear interactions among the elements
3. open system type
4. structure very often of net-type
5. necessity for the description of hierarchical and/or logical levels

Generally the characteristics of a complex system are that its elements are undergoing continuous changes individually predictable, but which is not possible or is very difficult to predict a future state.Moreover, complex systems can present a emergence behaviour, or a situation in which a system exhibits inexplicable properties on the basis of the laws governing its components. This fact arises from non-linear interactions among the system's comoponenets. Furthermore the system has adaptive characteristics, or modifies its parameters, elements and processes  according to different inside or outside situations.
In general all living systems are complex while artificial systems can be very complicated but not necessarily complex, or with a degree of complexity much lower than the living ones.

The best example of a complicated system is Internet, a set of transmission networks based on the same protocol, which at present interconnects some hundreds of millions of machines among clients, hosts, servers and networks equipments such as switchs and routers. Though Internet, as well as being very complicated, has indeed a certain degree of complexity - due to its network structure, the number of elements, the interactions between them and the hierarchy levels of the communication protocol used (IP) - still has a complexity much less than the smallest living organisms, such as those unicellular like  protozoa, or single cells.

Other examples of complex systems are cellular automaton, the earth's crust, considered as a dynamic system in the plate tectonics, all the ecosystems (even the simplest), the economical systems, the social systems, the nervous system , the climate systems,  local or global.

• Hologramatic

by extension of the physical case, obtained with laser interference, they are complex systems in which any partial subset arbitrarily divided contains the structure of the entire system.

exercises of Tao

In the S bus, in the rush hour. A chap of about 26, felt hat with a cord instead of a ribbon, neck too long, as if someone's been having a tug-of-war with it. People getting off. The chap in question gets annoyed with one of the men standing next to him. He accuses him of jostling him every time anyone goes past. A snivelling tone which is meant to be aggressive. When he sees a vacant seat he throws himself on to it. Two hours later, I meet him in the Cour de Rome, in front of the gare Saint-Lazare. He's with a friend who's saying: "You ought to get an extra button put on your overcoat." He shows him where (at the lapels) and why.

meta-cyberTao

The description of systems with feedback developed by cybernetics in the 40s and 50s is not enough when one considers systems that have different logical levels of description such , in general, complex systems, which contain many feedbacked subsystems and have several retroactive stabilization processes at many, such, for example, as metabolism:

The notion of complexity also marks the transition from first to second cybernetics: according to the first cybernetics an objective and external reality exists independent of the observer. In the system approaches according to the second-cybernetic the system, that is the set of elements that are interacting in the observable reality, it is comprehensive, dynamic and consists of two subsystems individual/environment in which the role of the observing subject is fundamental and should be taken into account in describing the global system. In complex systems, such as an animal, person, family or organization, we observe the system under observation and describe it exactly as the system observes and describes us.

Each of the two subsystems evolve according to its own logic and implement their own changes, but it is the other subsystem that determines the conditions that define the transformation that each subsystem implements.

In the second cybernetics subject and object, knower and the known are part of the same system. The individual and the environment are no longer places of property whose identification and description is under the responsibility of the outside observer, but location of mechanisms governed by their own rules, which determine the footage of their reciprocal interactions. The reality, no longer independent of the observer that actively builds it, becomes the personal reality of each individual, and any personal vision of reality comes from a specific relationship between knower and known, which reflects self-specific constraints. The individual thus becomes a cognitive system with its own internal coherence that can filter the reality, and structured through a set of constructs and beliefs that allow them to organize their behavior in response to the environment, according to its own self-constraints. The second cybernetics, seen as meta-cybernetics of the first cybernetics or simple, cybernetics, is characterized by subsequent orders of feedback:

and from the recursion between the orders of feedback and those of calibration which follow the feedback process:

taken from:

A simple feedback brings the system to calibrate itself to stability. A feedback of feedback leads him into a state of meta-calibration. A further level of feedback brings the system in what is called a stand-alone system, where the relationships and interactions that define it as a whole are determined only by the system itself, or autopoietic system, with a meta-meta-meta-calibration of closed organization.

triplofonic Tao

 to have haD
the idEa
  his Music
would nEver
     sTop
  the Range
  of hIs
     vOice
would have
no limitS
   nexT
    foR him
 to leArn was
   in Tibet
after that Out
into vocal Space

— John Cage, Mesostic for Demetrio Stratos (1991)