This essay looks at Jakościowa teoria informacji (‘A Qualitative Theory of Information’, 1970) as a single construction: what material it is built from, by what distinction Mazur separates code from information, and why the whole classification of transmissions follows from a single theorem. Concept by concept, in the order of the definitions, the domain is covered by the guide to qualitative information theory, and in twelve steps by the reading trail ‘Information: from message to transinforming’. What remains here is the structure itself. A separate page, Mazur and Western cybernetics, sets Mazur's concepts alongside those of Shannon, Wiener and Ashby; below, Shannon is mentioned only where the difference follows directly from the definitions.
The question with which the book begins
In the preface Mazur distinguishes three possible theories of information. The quantitative theory, by far the most developed, counts information. The evaluative theory, then only just emerging, assesses its usefulness for decisions. The qualitative theory is meant to answer what information is, what kinds of it there are and what informing is; the name is Mazur's own proposal (p. 11). He published a first outline in the article “Matematyczna definicja informacji” (‘A Mathematical Definition of Information’, 1965), developed it in a separate chapter of his 1966 book, and presented the concept of informing in a 1968 conference paper (pp. 11–12).
Mazur finds a theory that measures well but does not say what it measures. Shannon defined the amount of information by a formula, that is, by an explicit convention, and Mazur regards such a procedure as a model of scientific rigour (p. 16). The trouble lies in the name: since it is known what the amount of information is, it is easy to believe that it is also known what information itself is. Yet no one has proposed such a convention for information itself (p. 19).
Mazur points to situations in which a quantitative approach is needed, yet a measure based on probability is of no use. A history syllabus covers more or fewer facts, and every one of them has already happened. A map shows Stockholm to the north of Budapest, although neither the cities nor the points on the map occur with any probability. Geometry states that one angle is half of another, and angles do not happen as events at all (p. 17). The break with content, too, turns out to be only partial: a telecommunications office may send a telegram written in lower-case letters and deliver it in capitals, because it assumes that the recipient will treat ‘a’ and ‘A’ as members of the same class of signs (p. 22).
The material and two directions
The object of study is control, and the material is the message: any physical state in the control path, provided it can be distinguished from another. Messages at one point of the path form a transverse set of messages; messages arising one from another along the path form a longitudinal set of messages. At the output of the source are the originals, and at the input of the receiver are the images.
The chapter on informing opens with an observation that brings order to the whole book: as conversions of some messages into others, codes and items of information do not differ in any way. In one and the same set of messages one could see the passage of one chain into another in both arrangements; the duality disappears only once it is taken into account that the messages lie in a definite path (p. 81). From then on a code is a transformation along the path, and information is a transformation across it. Mazur announces this distinction as early as the preface, as the axis of the theory (p. 12).
So no substance called information flows along the path. What travels is a link: the same transformation appears first between the originals, then between successive intermessages, and finally between the images. Mazur compares this to a wave on water, in which the particles of water move up and down while the shape of the surface shifts horizontally (pp. 85–86).
The same measuring instrument can therefore be described in two ways. Looking at forces, one sees the work that deflects the pointer; looking at proportions, one sees informing, because the information on the side of the originals is the proportion in which the measured quantity has changed, and on the side of the images — the proportion in which the pointer has moved; no single value is information. Only the second description matters for the theory: in a heating appliance what counts is the work done, in a control path — what becomes of the information, and the energy used up is a cost that would ideally be brought down to zero (p. 85).
One classification, two applications
Since code and information are transformations, everything that Chapter 4 establishes about transformations carries over to both concepts. Hence the structure of the middle of the book: the chapter on transformation has 23 definitions, and the chapters on code and on information 14 each, arranged in parallel. The theorems are parallel as well. Composing the successive conversions in a chain gives a resultant transformation, and composing a conversion with its inverse gives a trivial one; the theorems for codes and for information state exactly the same results (4.2 and 4.3, 5.1 and 5.2, 6.1 and 6.2). Mazur points out that the proofs can be followed without mathematical logic: a transformation is a special kind of relation, and he has gathered the rules for using it in a separate chapter (p. 12). The full list of numbered definitions is shown in the definitions in order view.
One theorem, every variety of transmission
Informing requires at least four messages: two originals and two images (p. 82). Theorem 7.1 states what the information between the images is made up of: the inverse code that leads from the first image back to the first original, the information between the originals, and the code that leads from the second original to the second image (pp. 82–84).
Two things follow from this. The controlling system reacts to the information between the images, because it does not see the information between the originals at all; the two may agree or not, and whether they agree depends on the choice of codes (p. 84). Transinforming is the case in which the formula reduces to the equality of the two items of information (p. 87), and Chapter 8 examines, in seven theorems, which codes ensure this equality.
The formula also assumes that the coding chains are separate and complete. The following chapters drop these assumptions one at a time (p. 84). Non-separate chains give pseudoinforming, incomplete ones give disinforming; their varieties are traced by the trail ‘Pseudoinformation and disinformation’. A message that belongs to the transverse set but to no coding chain gives parainforming, and comparing information from different paths gives metainforming. The classification in the tables at the end of the book (p. 208) is thus the result of reasoning, not a catalogue. Mazur says so explicitly in the preface: examining the general formula of informing reveals every possible variety of it (p. 12). The terms that arise from this fall into a table of prefixes and bases — the families of terms view.
Where quantity returns
Probability does not appear in the definitions. For Mazur a message occurs only once, and what information theory calls a repeated event is a class of messages that are treated as identical in some respect: n tosses of a coin are n different outcomes, and ‘heads’ and ‘tails’ are merely the names of two classes (p. 204).
What can be counted, on the other hand, is transformations. To describe every message of a chain made up of n different messages, n items of describing information are needed (Theorem 13.1, p. 190). To single out one distinguished message, fewer items of identifying information suffice: as many as the binary logarithm of n (Theorem 13.5, p. 198). When the sizes of the classes are divided by the total number of messages, the formula for the average number of items of identifying information takes the form of Shannon's formula (p. 203). Mazur explains the difference between the two numbers as follows: to describe one message of a set is at the same time to describe them all, whereas to identify it is merely to establish that none of the others is the one sought, without saying what the others are (p. 202). In this theory the amount of information becomes a special case: it answers the question of identification, and only there does it make sense.
Mazur shows this with examples in which identification is out of the question. The number of independent equations should equal the number of unknowns: with too few, the chain remains undetermined, and surplus equations either follow from the others, like redundant information, or contradict them, like parasitic information (pp. 205–206). A textbook is sensibly measured by the number of items of describing information, because teaching is not about singling out one distinguished message. An examination is different: a pupil who asks classmates which questions have been coming up is estimating probability, that is, seeking items of identifying information (p. 206).
What the definitions do not contain
The very structure of the definitions also shows what the theory does not assume. None of them speaks of meaning or content, of the sender's intentions or of probability; the kind of transmission is decided solely by the links between messages: what the codes are and how their chains are built. That is why the same concepts describe the pointer of a measuring instrument, a map of terrain, a bilingual dictionary and communication between people, and each of them can be checked on its entry page, where the definition is given with its number and the page of the book.