Borislav Dimitrov

This study is about the Quality. Here I have dealt with the quality that differs significantly from the common understanding of quality /as determined quality/ that arise from the law of dialectics. This new quality is the quality of the quantity /quality of the quantitative changes/, noticed in philosophy by Plato as “quality of numbers”, and later developed by Hegel as “qualitative quantity. The difference between the known determined quality and qualitative quantity is evident in the exhibit form of these two qualities. The exhibit form of the known determined quality from the law of dialectics /or it transformation/ is related with discreteness and abrupt changes. The exhibit form of the qualitative quantity /and it transformation/ is related with the continuity and gradual transition from one condition, to a different condition, without any abrupt changes. In my paper “Quality of the quantity”, I have argued that one of the most ancient implementation of quality of the number can be found in the dimensional mathematical model of point – line – surface – figure - introduced by Plato. The most whole presentation of the idea of quality of number in Plato is embeded in his teaching about the "eidical number". The quality of the quantity emerges as criteria for recognizing the difference between the eidical numbers and natural arithmetical number. The thesis concerning Plato is based on the The Unwritten Doctrine of Plato and one of the most original works in the history of philosophy written in the 20th century - “Arete bei Platon und Aristoteles” – “Arete in Plato and Aristotle” /Heidelberg 1959/ written by Hans Joachim Krämer. The new quality as the quality of the quantity /quality of the quantitative changes/, first noticed in philosophy by Plato as “quality of numbers” was developed in Hegel as “qualitative quantity”. Hegel proclaimed the Qualitative quantity, or Measure in the both of his Logics -The Science of Logic / the Greater Logic/ and The Lesser Logic/ Part One of the Encyclopedia of Philosophical Sciences: The Logic. In my paper I have offered the arguments that the concept of quality of the quantity should be enhanced with the adopted methodological approach of analogy with an implementation in the field of the Topology - Analysis Situs, developed by the Jules Henri Poincare. In the topology we could see homeomorphism as exhibit form of Quality on the Quantity. The explicit form of the quality of the quantity transformation is the continuous deformation – typically known in topology as homeomorphism. The concept of qualitative quantity is linked with the concept “structural stability” and nonequilibrum phase transition. The concept of structural stability is related with the topological homeomorphism. In his book “Synergetics: Introduction and Advanced Topics” /Springer, ISBN 3-540-40824/, in the Chapter 1.13. Qualitative Changes: General approach, p. 434-435, Hermann Haken explores and illustrate the structural stability with an example /figure 1.13, p.434/ given by of D'Arcy W. Thompson, the Scottish biologist, mathematician and classics scholar and pioneering mathematical biologist, Nobel Laureate in Medicine /1960/, the author of the book, On Growth and Form, /1917/. The quality of the quantity could be seen in the Herman Haken’s citation on the D'Arcy W. Thompson. My thesis is that the topological homeomorphism is the explicit form of the quality of the quantity transformation. The qualitative quantity change which becomes phenomenon, according to Émile Boutroux, is the subject of study in Cultural Phenomenology of Qualitative quantity. Our approach to this subject is Poincaré Model of the Subconscious Mind in Mathematics, which is the most suitable tool to unfold the arhetype of qualitative quantity.

In the paper “Quality of the quantity” I proposed a critical approach to the dialectical materialism’s cliché in the law of the transformation of quantity into quality determined by Engels in his “Dialectics of Nature”, derived from his reading of Hegel’s Science of Logic as the law of the passage of quantitative changes into qualitative changes. The subject of my criticism was the second law of dialectics, established by Marx in his Capital, with the statement that “development is a process whereby insignificant and imperceptible quantitative changes lead to fundamental, qualitative changes”, and his statements that “qualitative changes occur not gradually, but rapidly and abruptly, as leaps from one state to another” or "merely quantitative differences, beyond a certain point, pass into qualitative changes". The reason to mark the dialectical law of transformation of quantity into quality as a cliché is that these postulates of Engels and Marx are relevant only to the linear systems. The interplay of quality and quality is associated with the development and growth. Both the classical and non-classical approaches to the quality in the law of dialectics, are addressing the dialectical nature of change. The determined quality implies discontinuous change, a leap, and transformation is discursive. The exhibit form of determined quality is abrupt displacements in the equilibrium - revolution. The quality of the quantity implies continuous changes, and transformation is non-discursive. The exhibit form of qualitative quantity is transformation without leap or abrupt displacements in the equilibrium - evolution. I have argued that one of the most ancient implementation of quality of the number can be found in the dimensional mathematical model of point – line – surface – figure - introduced by Plato. The most whole presentation of the idea of quality of number in Plato is embeded in his teaching about the "eidical number". The quality of the quantity emerges as criteria for recognizing the difference between the eidical numbers and natural arithmetical number. The thesis concerning Plato is based on the The Unwritten Doctrine of Plato and one of the most original works in the history of philosophy written in the 20th century - “Arete bei Platon und Aristoteles” – “Arete in Plato and Aristotle” /Heidelberg 1959/ written by Hans Joachim Krämer. The title of Krämer’s first book reflects an earlier stage of the dissertation project - starting from an analysis of the arete concepts in both thinkers. Krämer recognized the Platonic origin of the Aristotelean doctrine of arete as mesotes and the ontological foundations of this doctrine that Aristotle ascribes to Plato, but that are not found explicitly in the dialogues. This new quality as the quality of the quantity /quality of the quantitative changes/, first noticed in philosophy by Plato as “quality of numbers” was developed in Hegel as “qualitative quantity”. Hegel proclaimed the Qualitative quantity, or Measure in the both of his Logics -The Science of Logic / the Greater Logic/ and The Lesser Logic/ Part One of the Encyclopedia of Philosophical Sciences: The Logic. The Quality of the quantity appears in § 106 of Hegel’s The Lesser Logic, part of the Encyclopedia of the Philosophical Sciences, where Hegel claims “the Qualitative quantity, or Measure” – where “the result of the dialectic however is not a mere return to quality, as if that were the true and quantity the false notion, but an advance to the unity and truth of both, to qualitative quantity, or Measure.” The quality of the quantity is derived from the § 105 The Lesser Logic, part of the Encyclopedia of the Philosophical Sciences, where Hegel states “That the Quantum in its independent character is external to itself, is what constitutes its quality.”, and ”In that externality it is itself and referred connectively to itself. There is a union in it of externality, i.e. the quantitative, and of independency (Being-for-self)-the qualitative.”, and “the Quantum when explicitly put thus in its own self is the Quantitative Ratio, a mode of being which, while, in its Exponent, it is an immediate quantum, is also mediation, viz. the reference of some one quantum to another, forming the two sides of the ratio. But the two quanta are not reckoned at their immediate value: their value is only in this relation.”, or “This contradiction is what forms the dialectic of quantity. The result of the dialectic however is not a mere return to quality, as if that were the true and quantity the false notion, but an advance to the unity and truth of both, to qualitative quantity, or Measure. /§ 106/ Around 1900, Poincare formulated a measure of an object’s topology, called homotopy. In particular, two mathematical objects are said to be homeotopic if one can be continuously deformed into the other. There are two possible definitions for homeomorphism with implication to the qualitative quantity. Quality of the quantity /or Qualitative quantity is Homeomorphous quality. Homeomorphous quality is quality of quantity that possesses similarity of form. Homeomorphous quality is quality of quantity that is continuous, one-to-one, and having continuous inverse. I have offered the arguments that the concept of quality of the quantity should be enhanced with the adopted methodological approach of analogy with an implementation in the field of the Topology - Analysis Situs, developed by the Jules Henri Poincare. In the topology we could see homeomorphism as exhibit form of Quality on the Quantity. The explicit form of the quality of the quantity transformation is the continuous deformation – typically known in topology as homeomorphism. There are colaborative evidences offered in “Quality of the Quantity” linking the exhibition form of qualitative quantity with the concept “structural stability” and the concept of nonequilibrum phase transition. The concept of structural stability is related with the topological homeomorphism. In his book “Synergetics: Introduction and Advanced Topics” /Springer, ISBN 3-540-40824/, in the Chapter 1.13. Qualitative Changes: General approach, p. 434-435, Hermann Haken explores and illustrate the structural stability with an example /figure 1.13, p.434/ given by of D'Arcy W. Thompson, the Scottish biologist, mathematician and classics scholar and pioneering mathematical biologist, Nobel Laureate in Medicine /1960/, the author of the book, On Growth and Form, /1917/. The quality of the quantity could be seen in the Herman Haken’s citation on the D'Arcy W. Thompson. Exploring the invariance in deformation and transformation of the forms against spatial or temporal deformation, Haken wrote: “Figure 1.13, p.434 /“Synergetics: Introduction and Advanced Topics”/ shows two different kind of fish, namely, porcupine fish and sun fish. According to the studies by D'Arcy W. Thompson of the beginning of the twentieth century, the two kinds of fish can be transformed into each other by a simple grid transformation. While from the biological point of view such a grid transformation is a highly interesting phenomenon, from the mathematical point of view, we are dealing here with an example of structural stability. In a mathematician’s interpretation the two kinds of fish are the same. They are just deformed copies of each other. A fin is transformed into a fin, an eye into an eye and etc. In other words, no new qualitative features such a new fin, occur. In the following we shall have structural changes /in the widest sense of word/ in the mind.” Under the illustration set in Figure 1.13, p.434 /“Synergetics: Introduction and Advanced Topics”/, Haken wrote – “the porcupine fish and the sun fish can be transformed into each other by a simple grid transformation. /After ”D'Arcy W. Thompson: On Growth and the Form, ed. By J.T. Bonner, University Press, Cambridge, 1981/.” Hermann Haken’s example are illustrated here with the original Thompson’s illustration of the transformation of the fish Argyropelecus olfersi into the fish Sternoptyx diaphana by applying a 70° shear mapping. The reverse transformation is possible simply with manipulating the grid and shear mapping. The non discursive transition of quality or the phase transformation and nonequilibrum phase transition linked the qualitative quantity with evolution. There is a great example of the quality of the quantity find in the evolution of biological spieces illustrating the non discoursive qualitative change. This example is the fish Latimeria chalumnae or Coelacanth. The Coelacanth/Latimeria is a dual being consisted by two determined qualitis – one of the fish and other of the lung-fishes and tetrapods, but interesting fact is that no one of these qualities is not dominant. The Coelacanth/Latimeria embeds the two qualities in one. The Coelacanth/Latimeria is like the vesica pisces. These two qualities are not distinctive blending in one – the quality of the quantity. Not a fish and not an animal, The Coelacanth/Latimeria is something else – the third one. The Coelacanth/Latimeria is just great evolutionary illustration of the phase transition, such a synergetic being. Illustrating the Growth and Form after the D'Arcy's famous work, On Growth and Form and proving his concept of allometry. The qualitative quantity could be illustrated with the concept of allometry, with the conclusion given by D'Arcy Thompson: "An organism is so complex thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly." In the conclusion of my paper “Quality of the Quantity” I have established some links between the nature of qualitative quantity and the plasticity and elasticity as exhibit forms of quality of the quantity, arguing that the dialectic of the qualitative quantity can be found in the nature of organic life. A good example for the dual nature of the quality of the quantity is the nature of the fluid crystals. A liquid crystal is a fluid like a liquid but is anisotropic in its optical and electro-magnetic characteristics like a solid. The concept of qualitative quantity is extended the theory of neuroplasticity - the plasticity of the brain.

In “Quality and Time” /1990/, I have continued to improve and develop my concept of qualitative quantity with some implication concerning ontology dilemma and cognitive problems, in particular related with Henry Poincare's theory for the Time and Simultaneousness seen as an implementation of Qualitative quantity. The common understanding in dialectics represents the quantity as related with time, and the quality as related with space. If the Time is defined as the qualitative motion of the being, Space is defined as the quantitative motion of the being. The change of paradigm established by the physical theory of relativity in particular special theory of relativity proposed the reconceptualization of interdependence between quality and quantity, space and time. I have enhanced my concept of the qualitative quantity with some implication illustrating the concepts with the findings of the Russian astronomer and astrophysicist Nikolai Kozyrev, the concept of Thermodinamical Time of Albert Veinik, and the concept of Thermodynamics of Irreversible Processes of Ilya Romanovich Prigogine. The principal distinction between the qualitative and quantitative aspects of Time given by the Russian astronomer and astrophysicist Nikolai Aleksandrovich Kozyrev /1908-1983/ is experimentaly proven with the so called Kozyrev Mirrors. In physics time is identified with the set of real numbers. Their mathematical architecture is very rich: their construction involves closely interwoven structures of order and topology and several algebraic structures. Evidently the mathematical properties of a straight line should conform to the real properties of the physical time. The structure of order creates the succession of time instant. The additive group of number addition forms the metric to measure physical time intervals. The number multiplication group enables us to choose arbitrary units for measuring time. The real-number line topology induces the physical time continuity. However, physics does not contain a necessity and sufficiency analysis of the real line axiomatics (containing one and a half or two tens of postulates) for describing the actual properties of time. A reason for that is the usual absence of an explicit non-mathematical concept of time in physics. N. A. Kozyrev made a physical interpretation of the concept of time. A substantial interpretation of N.A. Kozyrev’s concept of Time enriched the dynamic picture of the World by introducing a new entity, possessing “active properties” and coinciding with neither matter, nor field, nor space-time in its usual understanding. This entity is difficult not only for intuitive or logical perception but also for a verbal description since a proper complex of concepts and images for dealing with the new ideas is yet to be developed. /A. P. Levich/. N.A.Kozyrev’s causal mechanics contains a substantial time construction. There is a material carrier /a certain medium/, directly “converting” cause into effect”. N.A.Kozyrev imagined time as “a mighty flow embracing all the material processes in the universe, and all the processes taking place in these systems are sources feeding that flow” /Kozyrev 1963, p.96/. According to N.A.Kozyrev, “time flows into a system through a cause to an effect” /Kozyrev 1971, p.118/. “There is an impression that time is pulled inside by a cause and gets denser at the location of an effect” /Kozyrev 1971, p.129/. “... In every process of Nature time can be formed or spent” /Kozyrev 1971, p.129/. Kozyrev pointed out the sharp contradiction between the second law of thermodynamics which brings nearer the thermal death of the Universe, and the absence of any signs of equilibrium in the observed diversity of the Universe. Back in 1970, Albert I. Veinik introduced the expression Chronal field comprehending time as elementary particles, chronons, and thereby able to vary in speed. These chronons create a chronal force field, especially by mass being set to spin. A.I. Veinik experimented with gyroscopes and their variation in gravity. The term chronal comes from the Greek word kronos, meaning time. Assuming a connection between the detected fields and the rate of the course of time, A.I.Veinik coined the term "chronal detector". A.I. Veinik experimentally discovered that it is possible to alter the rate of any process /including the process of a radioactive decay/ by subjecting that process to the influence of torsion radiation. This fact is stipulated by the ability of torsion fields to affect the inertial forces in any circulating mechanical system and demonstrated rigorously by G.I.Shipov. In accordance with Albert Veinik’s concept, the qualitative quantity in the Torsion field has explored. Torsion fields are able to change the rate of any physical process, for instance, they significantly alter the oscillation frequency of quartz crystals. Thus this property can be employed in torsion field detectors. The possibile affect upon the oscillation frequency of a quartz plate by torsion radiation was experimentally discovered by N.A.Kozyrev, and later was employed in various torsion detectors developed by A.I.Veinik. Since the superposition of a torsion field and a gravitational field in a local area of space may result in the reduction of gravity in this area /the so called "torsion compensation of gravity"/, then the influence of torsion radiation upon any physical object may result in a reduction in weight of that object. This significant property of torsion fields was discovered in the 1950s by N.A.Kozyrev, and later, it was confirmed in the investigations conducted by A.I.Veinik, M.M.Lavrentiev, and others. The implementation of the qualitative quantity was explored in Prigogine’s definition of dissipative structures and their role in the thermodynamic systems away from equilibrium. The quest for a creative and irreversible role of time in the natural sciences and social sciances is the quest of the quality of the quantity. Prigogin’s work is seen by many as a bridge between natural sciences and social sciences. In his Thermodynamic Stability Theory Prigogine points that there is a basic distinction between “reversible processes” and “irreverible processes”. At the very core of the second law of thermodynamics we find the basic distinction between “reversible” and “irreversible processes” and only irreversible processes contribute to entropy production. The second law of termodynamics expresses the fact that irreversible processes lead to one-sidedness of time. The positive time direction is associated with the increase of entropy. The qualitative quantity is related with the fundamentsl question set by Prigogine - Can we extrapolate this stability property further away from equilibrium? The inclusion of thermodynamic irreversibility through a non-unitary transformation theory leads to a deep alteration of the structure of dynamics. The Quality of the quantity is the task of philosophical inquiries of the dissipative structure theory and the pioneering research in self-organizing systems.