IV.4.1 Hierarchy of Number Sets

The natural numbers , which are the positive integers , were used by people in the Stone Age for counting things like livestocks and enemies. The zero was used as a number representing the state of not having something, while the negatives came along much later to allow debt collectors. The number line was filled with later by a continuum of rational fractions, which are the ratio of two whole numbers, that allow solving division problems, then the irrational numbers, such as  and  were also interwoven into the number sets, to make the real number line . Nevertheless, many other classes of problems have been found that have no answer using these normal numbers, so new number groups are invented to solve these problems, such as imaginary numbers and hyper-reals, all this in trying to achieve completeness that should allow solving every possible problem in mathematics. However, G del s Incompleteness Theorem ended this ambition by proving that there is no system that can solve every problem, though the principle holds correct for most problems.

The imaginary, or complex, numbers  first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by mathematicians, thus complex arithmetic became an important part of mathematics, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. In the 18th century, Euler Equation is used to define the natural logarithm of negative numbers, while Cantor introduced his diagonal argument to define the real numbers absolutely and rigorously, which also established the countable and uncountable infinities. These various number sets are depicted in Figure IV.9, which also shows the hyper-complex numbers .

Figure IV.9: The hierarchy of the main number sets, which are: the naturals N, the integers Z, the reals R, the split-complex H (hyperbolic), the complex C and infinity. All these number sets are generated from the unity number: one, and they are all included within infinity, that is also included within each set, when observed from a lower dimension, even in the unity itself when observed from inside, while also infinity is equal to one when observed from outside.

The use of complex numbers extended the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number  can be then identified with the point  in the complex plane, which makes the connection between numbers and geometry. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone. The term imaginary was coined by Descartes in 1637, although he was at pains to stress their imaginary nature. Complex numbers now have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. With the advent of modern scientific theories, as more problems arise, new types of numbers are introduced, including the split-complex, or hyperbolic, numbers , which are applied in Clifford quaternion algebra.

However, although it is the ornament for precision and accuracy, mathematics is primordially based on a deadly erratic approximation that permeates all its fields of the most fundamental levels, and that s the abstraction of complicated physical entities into simple numbers. While this is extremely helpful and necessary to accomplish the various daily activities, it may ultimately fail as soon as the system becomes complicated, and this is certainly the case when we want to describe the Universe itself. For this reason, mathematics and theoretical physics is often confronted with profound conceptual problems, such as infinities and singularities, when they encounter microscopic or macroscopic systems. These problems cannot be resolved without redefining the most elementary conceptions of abstract numbers and mathematical operations.

As we have already introduced above, with the Duality of Time postulate, complex numbers become genuine concepts that underlie the creation and dynamic structure and formation of space-time dimensions. Most current mathematics is mainly based on spatial geometry, that is normally represented in terms of Hilbert space, as a generalization of Euclidean geometry to any number of dimensions. However, even a simple Euclidean plane is an abstract concept that does not exist in nature, or even the line of real numbers, except as a limit when we keep counting for infinity; itself is not any practical concept!

Figure IV.10: The hierarchical structure of the hyperbolic number dimensions. The abstract three-dimensional space R³ is only realized instantaneously as a result of the pairing between the two orthogonal psychical and physical arrows of time, otherwise we are normally evolving in in H³that is R²+T.

Therefore, instead of relying on purely abstract spaces that do not actually exist in nature, fundamental mathematics should be first established on actual physical entities, by taking into account their natural structure and behavior that cannot be initially abstracted into simple numbers. Only after establishing these bases, some systems can be then idealized and extended into other extreme situations, after setting some well-defined level of approximation that is always necessary to apply any mathematical operation. Even the simplest fundamental operations, such as , for example, is only correct under some ideal assumptions, such that the two physical entities being added are absolutely exactly identical. Otherwise, if we suppose mathematics is originally ideal, we may not be able to estimate all the inevitable errors in describing real physical systems. Therefore, the concept of errors must be incorporated to the fundamental equations of mathematics as they are derived, and not just when we apply them to any specific situation. This is because we are always starting from practical systems that have no ideal predefined mathematical structure.

A circle, for example, is supposed to be an ideal object with a perfect continuous symmetry around its center. Under this assumption we can then calculate its area according to the equation: . However, this is correct only when we consider that the circumference is infinitely continuous. But this is conceptually wrong when we consider any actual circular object that is necessarily made of some discrete atoms. The same can be said even on any simple line segment. Of course, practically, we don t work on the level of atoms in our daily life, and that s why mathematics is working very well for normal daily counting and arithmetic. The problem only becomes manifest on the subatomic levels, or similar extremes, for example when we want to describe the atoms themselves, in which case we have to consider the fine structure of their background geometry. This is similar to the famous coast-line problem, as it will be discussed further when we talk about fractals in section I.2.

Nonetheless, the main physics theories are based on some idealized Euclidean geometry, that do not take into account the granular structure and behavior. Even the theory of General Relativity is primarily based on space-time continuum that utilizes Riemannian manifolds, which are necessarily supposed to be differentiable, and that s why they fail at extreme gravity or short distances. Therefore, the main problem is that all physics theories are primarily dependent on some predefined background geometry. In contrast, with the Duality of Time Theory, the starting point is the absolute metaphysical oneness from which the geometrical dimensions are dynamically formed in the inner levels of time, while in itself it does not need any background geometry.

Therefore, in order to start dealing with real objects, we need to establish new mathematics that is primarily based on actual dynamic space-time geometry, and not just a simple abstract space that is realized only at the speed of light. In reality, therefore, since we are captured by time, we are now living inside this ideal three-dimensional space, which means that our actual dimension is less than three, but also more than two, as we shall see further below when we discuss fractal geometry. Accordingly, the hierarchy of number sets should be according to the groups shown in Figure IV.9, which is based on the fact that the dimensions of Euclidean space are obtained only at he speed of light, which case the outer level of time is null. Equivalently, we can also say that the physical world, which is described by hyperbolic geometry , is a result of splitting the Euclidean vacuum  into two levels of time as we introduced above.