Information and Communication

Mathematical Models

Vasil Penchev

• Bulgarian Academy of Sciences: Institute for the Study of Societies of Knowledge

• vasildinev@gmail.com

Thursday, October 8th, 18:15 – 19:00 Vilnius university: Faculty of Philosophy Models of Communication: Theoretical and Philosophical Approaches Vilnius, 8-10 October 2015

“A mathematical theory of communication”

• The concept and quantity of information are introduced by Shannon (1948) as a way and mathematical method for communication to be formalized as transmission and relation of messages

• The communication is formalized as transmitting messages in channels

• A general quantity of information measured in bits is introduced to describe mathematical both message and communication

About Shannon’s information

• It is inverse to thermodynamic entropy

• It depends both on the content and length of message as both are represented as mathematical functions

• This implies that the content of communication is indirectly postulated as a stage or result of calculation to be representable as any mathematical function

• Correspondingly, the length of message is the number of digits necessary for that function to be encoded

Noise and information in Shannon

Anything, which is not information, in Shannon’s mathematical theory of communication, is “noise”:

Thus “noise” means both:

- any change of information due to internal influence or the material realization of communication

- any additional component of communication, which is not (or cannot be) reduced to information

The philosophical formula of Shannon’s mathematical theory of communication

A comment: Information is the mathematical quantity, which can represented all stages of communication as an ideal process of calculation realizable in technical devices

Wiener’s “cybernetics”

• Wiener (1948), his colleague in MIT, used an analogical approach to define “cybernetics” as "the scientific study of control and communication in the animal and the machine“

• In fact, this is a rather philosophical generalization of Shannon’s approach for the mathematization of communication:

Shannon Wiener

Communication Control & Communication

Machine Animal & Machine

What information in Wiener is

So, information was understood as:

The “matter of control and communication”

Structured in units such as messages

Allowing of mathematical models general to the human being, the animal, the machine and even to any physical, chemical, and biological systems

Entropy as the quantity of disorder is opposite to that of information

The philosophical formula of Wiener’s cybernetics

A comment: Information is the mathematical quantity, which can represented all stages of communication and control as a process of calculation realizable in technical devices, animals and even any systems

Habermas’s “Theory of communicative action”

• Habermas (1981) introduced the model of communicative rationality based on communicative action

• The information approach to communication reduces it to transmitting rational messages and thus any message to some ordering or “logic” of what is communicated:

• The idea and corresponding ideal of communication is the collaboration for ordering nature, society, and the world in a rational way relevant to all communicators

The philosophical formula of Habermas’s theory of communicative action

A comment: Communicative action suggests another kind of rationality, rationality in society (or social rationality). It is different from that rationality directed to nature for achieving human aims and objectivities

Habermas’s philosophical background

• Though different, rationality in society and rationality to nature are similar anyway both being forms of rationality:

Rationality to nature is an infinite continuation of the finite human mind

Rationality in society is an discrete leap between two or more finite human minds (me ... others)

• Thus, infinite continuation to nature should be similar to discrete leap within the finite of society (intersubective transcendentalism)

Information vs communication in Habermas

• One and the same message can be and usually is interpreted rather differently by separate human beings

• Nevertheless, it has an invariant base shared by all or many enough: This is the information of the message

• It is not formal and yet less mathematical

• Information is not identical to communication

• Communication suggests discontinuity of information: Though information is one and the same, it is shared by different members of society

Communicative vs technical action I. Technical action

• Technical action embodies certain information being an intention or plan in a material of nature creating an artefact such as a technical device, etc

• Thus it continues the plan in a material object

• Action is what leaps from the ideal into the material and links them over the gap of infinity

Communicative vs technical action II. Communicative action

• Communicative action does not consider the Other as an object of nature, in whom to embody the message

• It is directed to mutual understanding, in which it will reveal its information depending on that mutual understanding but independent of different interpretations

• Its information is impossible without understanding and even does not exist before understanding

• Thus information exists only in a society

Communicative as technical action

• Though being so different, information in Shannon and Wiener and it in Habermas share a general and formal structure:

• This is the unit of information, a bit in Shannon and Wiener, which is the elementary choice between two equally probable alternatives

• This is the elementary communicative action sharing one and the same information in two individuals in the process of understanding

Technical as communicative action

• One can think of any bit of information as a hidden communicative action preliminary accomplished in a society

• Those bits of information are what are further embodied in the piece of nature for it to be modeled according to human intentions and plans

• Thus communicative action is embodied in information being social understanding and concordance and only then that information is embodied in nature by technical action in turn

A generalized formula of information and action

Comment: the formula elucidates that information and action incl. physical action pass from each to other Therefore they should share a common and thus general philosophical ground

Production of information

in society

Utilization of information

in technics

Communicative action

Technical action

Understanding

Com. 1 Com. 2

A cell of memory

„0“ „1“

Choice

Alternative 1 Alternative 2

The ground of communication

• That ground might be choice and ordering

• Thus any communicator is representative and represented by its participation in common ordering

• Ordering is a result of choice

• Information elaborated by understanding and communicative action is the quantity of choices and thus ability of ordering

• The essence of technical action is the ordering in a certain pattern

Rational communication • However the model of rational

communication interchanging information whether reduced to a mathematical formula or a result of understanding in a society is unable to represent and even imagine many other sides of communication as an aim and objectivity by itself

• All other possible aspects of the messages and communication are either ignored or reduced to some obstacle or “noise” in the process of communication and ordering

Mathematics

• The same approach of rational communication and information is relevant even to mathematics and its foundation:

• Peano arithmetic, which is usually accepted as the ultimate element of mathematics is easily to be underlain by processes of information and calculation such as in a Turing machine

• Even the completeness of the so generalized arithmetic can be proved using two or more Turing machines (a quantum computer)

• That completeness corresponds to Gentzen’s or intuitionistic approaches for completeness

Choice as the base of rationality and mathematics

• Then the quantity of information and thus the corresponding ideal of rational communication underlie mathematics by means of the concept of choice

• Indeed the axiom of choice in mathematics allows of any even infinite or coherent class to be enumerated and thus reduced to a single Peano arithmetic

• Furthermore it allows a Peano arithmetic to be chosen between two independent ones

Information as the quantity of choices

• Information is interpreted formally as the quantity of elementary choices such as:

• Bits (in the case of finite messages, series or sets): a bit is an elementary choice between two equally probable alternatives; or

• Qubits, i.e. quantum bits (in the case of infinite ones): a qubit is that generalization of a bit referring to an elementary choice among an infinite set of alternatives

• A qubit (after Kolmogorov’s algorithmic definition of information) is equivalent to a transfinite bit, i.e. to an elementary choice between two independent Peano arithmetic

After choice, or utilizing information

• The ideal (and result) is well-ordering guaranteed by a relevant axiom or principle such as the axiom of choice or the well-ordering principle (theorem) equivalent to each other

• The essence of technical action is implementation of a certain pattern equivalent to a series of choices, i.e. information

• However that model for technical use is elaborated before that by communicative action in society, i.e. by the production of information in understanding

The inherent, but hidden link

• One can deduce an inherent link between the mathematical models of communication, the concept of and quantity of information and Habermas’s “communicative rationality”

• All those divide disjunctively the state before or after choice therefore postulating the choice itself

• Then rationality can be separated from irrationality entirely within the choice and its mechanism, and rationality accepted the choice as granted somehow beginning after it

Information

The “black box” of choice:

The ground of both rational

communication and information

Production of information

in society

Communicative action

Technical action

Utilization of information

in technics

Rational communication and information in European tradition

• All those are grounded on rational and empirical tradition in philosophy and especially in European philosophy

• There exists:

• The practice of separating rationality from irrationality disjunctively

• Representing irrationality as some “black box”, e.g. that of choice

• Postulating that “black box” as a an initial element of rationality

On that background

• It tends to all different from the rational and ordered to be generalized and ignored as the “irrational”

• Then, the irrational sides of communication can be studied only as obstacles and “informational noise” or as far they admit some rationalization by means of any more or less relevant model preferably mathematical

Conclusions: Shannon’s mathematical theory of

communication,

Wiener’s cybernetics, and even

Habermas’s theory of communicative action

share the European approach of rationality to communication

Its essence consists in dividing the rational from the irrational, and then “bracketing” the latter as a “black box” and initial element for rationality

Literature I. Habermas’s Theory of communicative action 1. Habermas, J. 1984-1987 The theory of communicative action. Vol. 1:Reason and the rationalization of society; Vol. 2: Lifeworld and system : a critique of functionalist reason, Boston: Beacon Press. 2. Habermas, J. 2001 On the pragmatics of social interaction: preliminary studies in the theory of communicative action, Cambridge, Mass.: MIT Press. 3. Honneth, A., H. Joas (eds.) 1991 Communicative action: essays on Jurgen Habermas's The theory of communicative action, Cambridge, Mass.: MIT Press.

Literature:

II. Information theory and cybernetics

1. Wiener, N. 1948 Cybernetics or control and communication in the animal and the machine, New York: J. Wiley: The Technology Press.

2. Shannon, C. 1948 “A mathematical theory of communication,” Bell System technical journal, 27(3): 379–423; 27(4): 623–656.

3. Kolmogorov, A. 1968 “Three approaches to the quantitative definition of information,” International Journal of Computer Mathematics, 2(1-4): 157-168.

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