THE MATHEMATICS OF ARTISTS' STAMPS (An Abstract Algebraic Analysis of Effects)*
1.1. Although the memory images of the single stamps will be indistinguishable, we still find that our memory definitely preserves and separates certain groups of stamps. A sensation remains as to the differences between the stamps of two artists or of artists from two different countries.
1.2. It is as if we separated the stamps on the basis of their similarity!
What is the explanation to this contradiction known ever since Antiquity?
1.3. In order to find an explanation we have to inspect the kind of connection that is established between pairs of - seemingly indistinguishable - stamps. This connection bears the name of similarity in everyday language, while it is called a tolerance relation in mathematics.
2. THE MATHEMATICAL ANALYSIS
A tolerance (or similarity) space <M,t> is a tolerance relation on the set M of stamps and their details. The stamp designs of an exhibition or the totality of stamps seen at various times are - together with some suitable tolerance relation - instances of tolerance spaces.
our example set M will consist of five stamps, denoted by capital
letters, i. e.
L is a set of properties:
stamp has certain properties and some properties are common to several
stamps. The relation which holds between stamps and properties can be
interpreted as a one-to many function from M to L. By
definition the tolerance relation holds between two elements X
and Y of M if
i. e. the two stamps share some property. In our example, the possible intersections are the following:
The more common properties elements of M have, the more similar they are. The greatest similarity will obtain for the intersections [bcd] and [bdf]; this grade of similarity is marked 3 on the edge of the graph. Similarity is weaker in case of two common properties [bd], its grade being 2, while similarity is very weak if only one property, [c] is common, its grade being 1. There is one pair of stamps of 0 similarity grade - i. e. of dissimilarity -, namely D, E. While that has no bearing in the world outside consciousness, to our consciousness this fact appears as non-tolerance, i. e. we perceive a "quality", non tolerance namely, where there is actually no occurrence of tolerance.
The intersections can be regarded as extracts or condensations of the properties of two or more stamps. The recognition of the intersection [bcd] is then nothing else but a generalization from the properties common to the stamps A, B and C while the intersection [bdf] is a generalization of the stamps C and D.
The group of properties [bd] represents a generalization on a higher level and is, in fact, a further generalization of the generalization described by the intersections, or groups of properties, [bcd] and [bdf]. Generalization on the highest level is represented by the intersection [c] which is valid for all elements of the set M with the exception of D, i. e. for all its stamps.
The intersections can be regarded then as representing concepts of varying specificity for our consciousness. Such structures are generally known in case of certain relatively simple concepts (fig. 2.). With works of art, the difference stems from the fact that the concepts representing the images are much more complex and their conscious interpretation would necessitate lengthy discussions. They are concepts, all the same, and this is the crux of the matter. The spectator "experiences" these concepts without "knowing" them.
3. CONCLUSIONS CONCERNING THE NATURE OF THE CONTRADICTION DISCUSSED IN 1. 2.:
3.1. Although the difference between two stamp designs is marginal at the first glance, the accumulated differences in properties will register in our consciousness above a certain threshold, after skimming through a number of designs. This perception is, however, not due to a storing of elementary properties in the mind, since it is exactly this process which our mind is unable to perform. After having seen a certain number of images, differences will be perceived, apparently abruptly, from a certain stage on because there have emerged those, mostly subconscious, generalizations which are represented in our example by the intersections [bcd] and [bdf]. [bd] and [c] respectively. They could be constructed on the basis of similarities, although it is perversely the differences that they "ordered".
3.1.1. It is as if casting a glance on a stamp, our consciousness automatically combined the common, or seemingly common, properties of the stamps so far seen and then went on with that activity until the combinations evolved into a definite structure. As soon as the ordered state takes - subconsciously - the shape of that structure, items become distinguishable. The resulting structure is called a Lattice in abstract algebra. In our example which is very simple, this resulting structure is represented by a graph: its vertices are concepts of varying complexity, while its edges the connections between these concepts.
22.214.171.124. Now we are in a position to explicate qualitatively the threshold, above which the mind is theoretically able to distinguish items: it is where the mind becomes able to perform two different generalizations at the same time. That happens in our example if the spectator is exposed to the images D or E after having seen A, B and C. A, B, and C allow namely for one generalization only, and C with D in themselves do the same. The performance of two different generalizations (i. e. [bcd] and [bdf]) makes way for some kind of order among the more or less similar items which correspond to these generalizations. Distinguishable groups can thus be formed.
126.96.36.199. Consequently, the spectator's decision and act, to have a closer look at some image is not governed by chance (see 1.2.2.). He will instinctively select an image which can supplement the properties already stored in the depth of his consciousness, in order to make the images still more similar and undistinguishable, making the abstraction process, at the same time, easier.
4. METAPHYSICAL CONCLUSIONS:
The spectator experiences the properties of real or fictional occurrences, for instance, of stamps. These "dumb" properties form structures in his soul by way of the deterministic operations of generalization. Structures generate the meanings of words. These meanings are, however, as dumb in themselves as the spectacle itself. Structures namely do not "express" anything except forming classes. The conceptual spaces created by abstract structures in the depth of consciousness evoke the impression of "complete", "everlasting", of "ancient" knowledge in the individual who has begun to pay closer attention to the workings of his own mind. We could say that, having processed a certain number of occurrences, his mind will produce a conceptual space able to produce all the conceptual combinations necessary for "complete" knowledge.
4.0.1. This knowledge is in this direct form not only totally "dumb" but it may be dangerously much for the individual. It is not advisable for everybody to confront this knowledge. If one has been attempting to observe only this complete knowledge for a considerable time, it will - in Nietzsche's words - look back at him. Anyone who has ever tried, knows how unpleasant that can be. And failing to learn how to treat this "looking back" or how to retreat in time, may get a person into temporary or lasting trouble.
4.1. A work of art is nothing other but a quick glance cast into this "dumb" knowledge, short enough to save the looker from that glance, from that insight into knowledge. And then a work of art is the expression (e. g. painting) of the thing he has seen, with the help of the syntax (the "verbs" of painting), with the help of - human - forgery.
4.2.3. The identity of "events" is revealed in their similarities or tolerance relations. In case of stamps and stamp designs this similarity will become an outstanding feature; that is why it is easier to perceive here. In art forms like stamps, it is this looking in, this insight, that becomes massive. Everybody wants to report his/her participation in that great mystery. This experience is so overwhelming and has by now become so widespread that everybody has to tell about it only a few people and the number of informed and mutually informed people will be very great. It is the Act itself that is important: the excitement of the revelation. To demonstrate: All of us. Now. By now. The mystery has become an everyday event to be experienced by anybody. And from this point of view the "depth" of that insight is already irrelevant. Depth is after all but one of the many properties. It is dumb as all properties are. It is, like all properties, not a value in itself. Properties are equal, just as good or bad are. Their mortal danger stems from here: the truth with its whole weight can be born for no longer than a minute at most, for the seconds of creative inspiration, the recognition of the structure.
5. Here is a new property then, one of the countless number of properties: stamp art.
Ungváry is a mech. engineer and art critic living in Budapest.
He works on the modelling of thinking and conception using the theory
of classification and the general systems theory.