Abstract

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.