Fleeting thoughts: Secondary qualities, or the mappings between quality and quantity

The thesis is:

The relations between quality and quantity can be represented properly quantitatively, i.e. in terms of mathematics and more exactly, in terms of theory of set, which in turn serves as the common fundament of mathematics: The transition between quality and quantity is a mapping between an unordered set and a well-ordered set or between an ordered and a well-ordered state of a single set therefore needing the well-ordering theorem equivalent to the axiom of choice. Thus that transition is equivalent to a set of mappings between quality (or a set of qualities) and quantity (or a set of quantities) being measurable by the quantities of information or quantum information in units of elementary choices either bits or quantum bits (qubits). 

Two mappings, straight and reverse, should be investigated: The former maps any quality or a set of qualities into a single quantity by the following successive stages: (1) the set of quality or qualities is well-ordering for the axiom of choice is valid in general; (2) that set already well-ordered is easily metrizable, e.g. by a unit of distance between any two successive elements of the series. The latter, which is the reverse mapping from quantity to quality, can be defined by the exchange of the stages as follows: 

(2) the real continuum of any quantity is reduced to a well-ordered series of points (real numbers) of the continuum and thus, to sets of those points; (1) the well-ordering of the series is removed and thus the series is reduced to an unordered set of elements. This latter set can be interpreted for example as the set of different states of a single quality such as color or as the set of qualities of a single entity such as any object. The concept of choice and the quantity of information as the quantity of elementary choices is crucial for the distinction between the former and latter mapping therefore for that between quality and quantity.

The mathematical formalism of quantum mechanics based on Hilbert space can visualize both stages: 

– The straight mapping:

(1) The coherent state before measurement does not allow any well-ordering because of the theorems of absence of hidden variables in quantum mechanics. However it is necessarily well-ordered after measurement thus involving the well-ordering theorem and the axiom of choice. (2) The state is metrizable by that quantity, which is measured in the former stage. 

– The reverse mapping:

(2) The coherent state can be exhaustedly represented only by a single quantity. (1) This quantity is reduced to a certain set of qualities thus adequately reproducing the initial coherent state.

The conclusion is: The inseparable whole of a quantum system such as any physical object is transformed into quantity, which in turn is transformed into a set of secondary qualities serving to represent the inseparable whole of primary qualities of the object by itself. Quantity, and thus choice and information mediate necessarily those mappings between primary and secondary qualities.