Dear Sir,
As we have already explained real numbers are discrete units. Fractions are also discrete units of the subdivision of the earlier unit by the denominator. But when you refer to "the count of real numbers between two different real numbers", do you mean to say that the real numbers are analog? If you have two apples, the interval between them is also full of apples? It is beyond our understanding. We assign the discreteness to dimension. In solids, it is fixed, in fluids, it is loose and is affected by the container and in plasma, it is unbound.
Mathematics is related also to the measurement of area or curves on a graph - the so-called mathematical structures, which are two dimensional structures. Thus, the basic assumptions of all topologies, including symplectic topology, linear and vector algebra and the tensor calculus, all representations of vector spaces, whether they are abstract or physical, real or complex, composed of whatever combination of scalars, vectors, quaternions, or tensors, and the current definition of the point, line, and derivative are necessarily at least one dimension less from physical space.
The graph may represent space, but it is not space itself. The drawings of a circle, a square, a vector or any other physical representation, are similar abstractions. The circle represents only a two dimensional cross section of a three dimensional sphere. The square represents a surface of a cube. Without the cube or similar structure (including the paper), it has no physical existence. An ellipse may represent an orbit, but it is not the dynamical orbit itself. The vector is a fixed representation of velocity; it is not the dynamical velocity itself, and so on. The so-called simplification or scaling up or down of the drawing does not make it abstract. The basic abstraction is due to the fact that the mathematics that is applied to solve physical problems actually applies to the two dimensional diagram, and not to the three dimensional space.
The numbers are assigned to points on the piece of paper or in the Cartesian graph, and not to points in space. If one assigns a number to a point in space, what one really means is that it is at a certain distance from an arbitrarily chosen origin. Thus, by assigning a number to a point in space, what one really does is assign an origin, which is another point in space leading to a contradiction. The point in space can exist by itself as the equilibrium position of various forces. But a point on a paper exists only with reference to the arbitrarily assigned origin. If additional force is applied, the locus of the point in space resolves into two equal but oppositely directed field lines. But the locus of a point on a graph is always unidirectional and depicts distance - linear or non-linear, but not force. Thus, a physical structure is different from its mathematical representation. Hence you cannot apply these so-called mathematical concepts to physics or information.
All information is in a relation between an object and a concept defining that object. Interconnectedness and interdependence are the laws of Nature. But then there is a concept of eigenfunctions and eigenvalues. You cannot make any combinations. There are rabbits and there are horns, but you cannot describe a rabbit with horns except in dreams.Kindly ponder over it.
Regards,
basudeba
