## Paths in the Intuition of Structures

Units of data can be set to whatever physical, a bit, a byte, a word, a sentence, a book, a symphony, but they have to be set to some kind of thing before we can start to think about the information they carry. This goal however is still out of reach unless we predicate about each data input in order to distinguish them. This is due to the fact that information is not data but structure. Once some unit of data has been given a type, we have a pair of qualified data, also known as a symbol, but still it is not, properly said, information. This pair of data may bring information in itself (because it is compound) but doesn't constitute an information, except when associated to something else. Real information beggins when we consider this symbol as a property of something more compound, a property of a structure. A bit is obviously the simplest symbol when - once typed (at least by a common name) - it is seen as a discriminating value (true, false, 1, 0). Nevertheless, the common perspective we have on information involves data units that already are compound objects and thus some kind of (implicit) coarse graining is included in information exchanges. Setting the data unit to a size unit happens mostly in computers, otherwise data units are more likely to be met as structural. In language data units are grammatical.

A set of one or more symbols constitutes a structure (independently of its possible reference) and any such structure can show unlimited complexity by variation of the involved symbols and data units. The reason of this state of affairs lies in the fact that symbols can be made out of anything that can be communicated - except other symbols. Without deeper explanations, let's say that in the world of data structures it has been experienced that the interfacing objects (symbols) can't be composed of objects of the same kind (symbols) but only of the 'terms' they bind as types or properties in or of something else we define a structure.

A symbol can link some data taken as type to some other data, maybe in another data unit, considered as property. Sometimes the same input can be set as type and property at the same time: 'word' can be a property of type 'word'. On the other hand, data units can vary and this case is made obvious in binary properties, for a sole bit is required to set them as property, while their type requires more than one bit. Similarly a symbol can link some pure data taken as type to some existing structure for property, or the type can be a whole complex structure (like the definition of prime numbers) when the property is a small number. When both the type and the property are constituted of nested structures, the resulting symbol shows deeper complexity while still not being informative as long as we don't at least associate it to some named object.

Though important concepts like type and property, meaning and reference, participate in such a preamble, we are not going to discuss them here. It doesn't matter too much - once accepted the tripartite feature exposed above - to define precisely what is a property or what is reference for our purpose. We only need to evoke these notions here, in order to emphasize some of their relations. Structure is a complicated fact that we want to reduce to a simple expression and there are surely philosophical and scientific theories that match this conception of information.

With such standard means we gain a filter to layer between any system and its input, including our mind and the world. The three sets (data,symbols,structures) can be instantiated graphically.

In this figure, data are printed points on the y axis, structures and symbols (in cyan) along the x axis. Each point includes one fixed value, so that these three sets look like three panels. For structures and symbols the fixed value is y and it is set to 0 and 1400 respectively. Data are constrained to the axis by an x set to 0. The z axis is reserved to represent the calibrated size of the objects while the forward (for data) and right directions represent the time of input. An other property of information could certainly replace size on such a schema, but the time dimension (here calibrated) is required .

Several things are observable on this graph. One structure-point is circled to signal the target of interest. The size of the structure is rather big and it has been introduced early into the system. This is shown by a high left position above the x axis. If we had the knowledge of the content of our target structure in terms of nested structures we could of course say more. This is what happens in the following graph where this content is indicated by smaller circles around the involved points of the graph.

5 nested structures here. 2 of them (on the right) appear at a later stage in the development of the system, when their host (the studied structure) already exists. So we know it's a dynamic structure. It is a characteristic of something dynamic to get updates or additions in time just like shown on the graph. What this structure means in terms of symbols is available through some new highlight of its content.

In these groupings we guess the content of each nested structure by time proximity, more precision can be reached drawing the appropriate lines between points.

Further we may want to bring out the involved data so as to get a complete image of the analyzed compound.Suppose we asked different mathematicians to write a complete proof of a common theorem, and arrange their output (needed axioms, definitions, theorems as structures, subproofs, corollaries etc.) according to the distinctions presented here and the same data unit (propositions). Then we would obtain objectively different paths on such a graph even in the case of an identical proving scheme. Provided the proofs are valid, none of the resulting information paths would be worse than another - except considered according to some criterion, like size - because they all would lead to the proved theorem. Thus is distinguished mechanically what we may now call - somewhat quickly - intuition paths. The same can be done more generally for emission and reception of cognitive and even artistic structures, because these paths unhide important moments in the process of receiving input and building information.

If this type of analysis doesn't put forth indications on how the carried information was meant at its source, then it may help us understand how it was meant to be taken (in our example we would call that pedagogy).

Note: The proposed system implies symbols and data are not duplicated but reused. Symbols and structures can be added at any time without new data input. Structures can be reused or duplicated (if ever required). So the graph shows immediately wether a newer structure is a mere arrangement of old symbols and data in the development of the system or if it also enlists newer inputs.

Emmanuel Rens, Genève, 2018. home