Dear Sir,
Your essay is as much a pleasure to read, as it is conceptually very close to our essay published on May 31, 2013, though both approach the subject from totally different perspectives. Thus, though there are some differences in our analysis, the final outcome is strikingly the same.
"Analyticity implies the directional aspect of causality and vice versa" may not be a correct statement. Consider an example: A B → C D.
Here a force makes A interact with B to produce C and D. The same force doesn't act on C and D as they don't exist at that stage. If we change the direction of the force, B acts on A. Here only the direction of force and not the interval between the states before and after application of force (time) will change and the equation will be:
B A → C D and not B A ← C D.
Hence it does not affect causality. There can be no negative direction for time or cause and effect. However, "Time scale can be shifted without any restriction", as we chose it.
The concept of minus and plus infinity is mathematically void. Mathematics is related to numbers, which is a property of particles by which we differentiate between similars. If there are no similars, it is one. If there are similars, it is many. Depending upon the times of perception, many can be 2,3,4....n. Zero is the temporal absence at "here-now". We must have prior knowledge of the object labeled as zero to perceive its absence. Hence neither positive nor negative signs could be assigned to zero. Infinity is like 1 - without similars. But while the dimensions of one are fully perceptible, the dimensions of infinity are not perceptible. Hence it is not a number. If it passes through zero, then we can perceive at least one end of it. But zero is absence at "here-now". Thus, it produces a contradiction. In infinity cannot pass through zero. Complex numbers are not physical. You have correctly pointed out that "They vanish with correct transformation back into the domain of reality, i.e., positive real values". Hence they cannot be used in programming.
We have shown in various threads here and elsewhere that division by zero is not infinity, but leaves the number unchanged. This would solve many problems of current physics. Space and time are examples of infinities that coexist. We use arbitrary segments of these infinities. Whatever can be measured as effect is past, which continuously gets converted to cause whose effect will be the future. Hence we cannot control past and cannot know future. Present is only the fleeting interface between the two. Yet, it is most important to the observer because everything "happens" - perceived - only at here-now, because it is the only parameter that is ever changing and when measurement can be taken, whereas past is frozen and future is non-existent. Consciousness goes beyond time and space in a self-recreating world. We have described its mechanism in many threads.
We agree that mathematical space is different from physical space. Mathematics measures only the numbers of steps in an interaction, the distances involved including amplitude, wave length, etc and the quanta of energy applied etc. 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.
The left hand side of every equation depicts free-will, as we are free to change the parameters as we want. The right hand side of every equation depicts determinism, as the outcome will follow a fixed pattern subject to some special conditions signified by the equality sign.
To know or understand or solve something is to predict its behavior in a given situation, when such prediction matches observed behavior. Something makes meaning only if the description remains invariant under multiple perceptions or measurements under similar conditions through a proper measurement system. In communication, as in perception, it is the class or form that remains invariant as a concept. The sequence of sound in a word or signal ceases to exist, but the meaning remains as a concept. In Nature, same atoms (or numbers signifying objects) may combine differently to produce different objects. The concept arising out of each combination acquires a name (word, message) that remains invariant through all material changes and even when they cease to exist. Since the past is frozen and future is non-existent, only the past could be measured - hence the result of measurement known. And since past through present is the cause for the future, it can be influenced.
You are absolutely correct in analyzing Wheeler's view according to which "restriction to yes/no questions ascribes only rational numbers to what he considered reality. This means dichotomy. On the other hand, he needs the trichotomy (smaller, equal to, or larger than) of real numbers as to arrive at complete sets of vector spaces and in particular singularities. Given there are only yes/no questions in reality; doesn't this preclude real singularities?" More information in this regard can be seen in our essay.
Regards,
basudeba