Computational spaces
Calculations are carried out in spaces that fit the different representations. Naturally, the starting point is the Euclidean space with generalisations extended to multiple dimensions. Note that this approach to space has also changed over time as we gained more experience and knowledge. This change affected our understanding of the third dimension, the horizon and perspective, and thus the representation of world views. The Cartesian orthogonal coordinate system was also a powerful driver of this development. The concepts of the relativity of spaces emerged only in the 19th century, marking the beginning of the history of modern mathematics and physics.
Having referred to the conceptual background, we can see that the spaces of changes, their dimensions and the measures and measure norms of changes as multiple routes originate from the abstraction circle of stability and change. The transformation operations which describe changes in these abstract spaces also have their roots in the abstraction circle of changes. At the same time, counting relies on algebra, that is on the system of rules which determine how and with what constraints instances of counting can be connected. The concepts of changes connected the separation and connection points of the past, the present and the future to spaces, and shaped all our ancient and modern notions and doubts relating to these event processes.
This thinking process and approach created all the approximation and modelling tools which shape modern systems theory and originate from the Euclidean space; the diverse distance norms and the evaluation methods calculated on the basis of such distance norms; the interlacing model relations of time and frequency; and the multi-dimensional number concepts of event connections.
The simplest model for summarising processes is motion; and it was the linearity of motion and one-dimensional calculation that generated the approximation methods for calculating further complexities, such as the methods of matrix technology.
The various types of norms were created by the generalisation of the concepts of distance and dimension. These also drew on the empirical models as terms such as scale suggest. Originally, the two-dimensional polar coordinate system was used; distance and direction were taken from the starting point and then from the given centre (e.g. Rome). The orthogonal coordinate system and world view and the aspect of the third dimension also emerged in relation to the construction of buildings. This opened the way for projections, perspectives and representation problems later on, raising the idea of the relativity of distances in relation to the desire to overcome distances.
This new world view liberated thinking from elemental starting points and experiences but for the sake of logical consistency all generalisation efforts were made so as to ensure that elemental definitions fit into the new system as unique cases.
Generalisations followed two main directions: one was to increase the number of dimensions, partly in order to reflect to the fact that many physical and chemical characteristics determine the states of systems apart from location. The other direction focused on the issue of adding up distance sections, considering that the process of getting from one place to another and the difficulty of routes are far from being balanced.
Mirroring and rotating methods, including the duality and relativity of certain coordinate solutions suggested that coordinates are relative and can be used for more general purposes. This led to the discovery of the complex variable space and the relations between infinite planes. The use and scaling of coordinates helped to clarify the concepts of norms and measures.
The various distance exponentiation and root extraction methods were developed specifically to summarise and average the above mentioned values, still bearing in mind that the elemental instance (i.e. the calculation of the distance between two points) must remain unchanged.
For this reason special significance was assigned to the square or second power forms of formulas as well as to a new system of concepts – in addition to the evaluation of route and process values on the way – which handled the increasingly practical problem of infinite dimensions. The practical problem is the evaluation of the route between starting and ending states, that is the aspect of the new steady (stationary) states, and the desire to reach and maintain a state of stability which is crucial for systems.
This is the point where the methods describing the complexly interrelated states of systems, the task to create theoretical/mathematical procedures for the various images of the representation spaces of states, and the issues of system usage and the robust, disturbance-tolerant and practicable stability of systems meet. This reflects the theoretical and practical relationship between finite and infinite, the relationship between philosophy and engineering. With the norms and measures, as well as with their above-mentioned relation to infinity, the Euclidean space was given a more general and more specific conceptual extension.
The distance degrees viewed from the aspect of changes form series which is particularly important as the careful monitoring of group properties enable us to extend the process of calculations. These groups constitute spaces in line with the original generalisations. These spaces are complete spaces due to the convergence of space elements running to infinity, so their borders also form part of the space which exhibit the properties of the whole space. Convergence running to infinity would give the new steady states identified in this space. The inclusion of space borders becomes especially important when transforming the complex variable infinite plane into a unit circle for interpretation and practical computing purposes.
We can also find the everyday examples of distance variants. Norm 1 can be considered as the straight path simplified to its purest form, the next one as the Manhattan distance (or city block distance) which assesses the dotted line of the route round the block of buildings, and the tendency to infinity as the opportunity of the king to move from one square to any adjacent squares on the chessboard.
For us these event and calculation spaces come to life when events are shaped as a consequence of certain effects. Newton’s laws of motion can be thought of as an analogy: in a Newtonian system the motion of bodies is induced and their path is influenced by external forces. Systems are also observed and controlled in line with this scheme. Group concepts determine the possibilities of calculation operations, and the real and the abstract world practice these operations on sets according to their generalised and specific properties.
Our computational and mathematical models and practical observation opportunities force us to implement the scheme with linear (or at least approximately linear) models, i.e. with models enabling additivity and homogeneity. Additivity is the ability to simply add the independent results of effects, and homogeneity is the proportional change in response to the effects.
This simplification allows the application of the matrix technique which can be easily handled with the tools of computer technology. First, we will deal with the matrix representation tools of the past, the present and the future. Events are represented by the matrices of the spaces of signals received about the states, effects are represented by matrices describing the processes, and time relations by Hankel and Toeplitz matrixes. The already mentioned application conditions and calculation simplifications are completed with the causality hypothesis, the role of the past in the present, and the relation and independence of the future. This is clearly represented by the structure of the two matrix performing the shift.
For us the most important space extension is the extension of Banach spaces. These are metric vector spaces with the norms interpreted on them, that is with the mathematical concepts with which the relations are calculated. The vector space interpretation includes the functions of complex variables as well. Finite normed Banach spaces are the Lebesgue spaces (l spaces) and Banach spaces where scalar products are defined are the Hilbert spaces.
In modern systems theory, similarly to modern physics, non-Euclidean spaces play an important role. At this point, however, we only deal with the hyperbolic spaces introduced by Joseph Fourier. Functions transformed into hyperbolic spaces usually transfer the shapes of the Euclidean space tending to infinity into circles or spheres through inversion, thus representing finite norms which can be well assessed. In our application, spaces with such properties are the Hardy spaces.
In these space transformations the group rules are valid to the chosen analytical functions.