## Temporal Numbers (expected in April, 2021)

For the first time in history, imaginary numbers have a genuine meaning that unites between physics and mathematics!

The relation between “numbers” and “time” is extremely profound and fundamental, since “counting” necessarily implies some recurrence or repetition of the same primary “unit”, which is normally denoted as “one”, that then adds and multiplies in various ways to produce other quantities. For this reason, numbers are ideally represented as contiguous points on a straight line that has a starting point and direction, just like time. For this reason, also, when numbers whose square is negative were encountered in solving the quadratic equations, they were termed “imaginary”, as opposed to the “reals” that normally occupy the x axis, since it subsequently turned out that these numbers have a geometrical meaning that implies simultaneous counting in the x and y directions, which is not possible with our normal notion and perception of time - as a linear and one-directional flow, normally referred to as “the arrow of time”. Henceforth, the combination of imaginary and real numbers is denoted as “complex”, which were later extended to quaternions, octonions, sedenions and other hyper-complex systems.

Counting is the process of determining the number of elements of a set of objects in a group. The final count itself is expressed as one number, that can be thought of as a whole new unit. But, since it is composed of smaller units, it is not possible to determine its value without going through these sub-units, and hence counting them, which is essentially related to time.

This fundamental relation between time and numbers is usually neglected in mathematics, and that was the primary reason that made Euclidean geometry feasible. Mathematics, which is primarily based on the five axioms established by Euclid about two thousand years ago, worked out successfully and efficiently for all practical purposes and most abstract problems. However, after the advent of the Theory of Relativity in the early 19th century, many other non-Euclidean geometries were introduced. The main motivation was the newly discovered relation between space and time, together termed as “space-time”, which implied that physical space itself is not uniform anymore, as it was treated in Euclidean geometry which turned out to be an approximation that works only in normal conditions of relatively short distances and weak gravitational fields.

In normal Euclidean geometry, “space” is considered an abstract background that had to be absolutely uniform and infinitely extending in all (three) dimensions, and also idle throughout the flow of time. Accordingly, all mathematical operations, whether algebraic, arithmetic, and even infinitesimal calculus and other types of analysis, are efficiently performed in Euclidean space for all practical purposes in all fields of applied mathematics, including thermal (and statistical) physics and classical (Newtonian) mechanics. Non-Euclidean geometry is required only in some modern physics theories that deal with extreme conditions, such very large or exceedingly small dimensions as well as extremely high gravitational fields, all because “space” becomes fundamentally related to “time” and not uniform anymore.

However, the Theory of Relativity, whether in its Special or General versions, and in its classical or modern formulations; although it showed that space and time are connected, via the speed of light, it did not explain the nature of this connection and how exactly does it affect the ultimate structure of geometry. The reason for this essential deficit, as it turned out to be, is the lack of any apprehension of the ontological role of the speed of light c in the relation between space and time, apart from its cardinal presence in the equations. In fact, so far in modern physics, the constancy and invariance of the speed of light is considered an axiom, which has been experimentally verified but without any theoretical or philosophical explanation or interpretation.

As this book will clearly show, revealing the ontological reason behind this distinct and illogical behavior of light will have enormous consequences on the foundations of physics and mathematics, because it delineates the very structure of geometry, and it will be the gate for a new era in the history of science and philosophy.

According to the “Duality of Time Theory”, which is based on the “Single Monad Model”, the whole cosmos, including space and what it contains of matter particles, is being re-created internally in the inner “real” levels of time, before it kinetically evolves throughout the outer “imaginary” time that we normally encounter. Therefore, the points of this “complex-time geometry” are perpetually and sequentially fluctuating between existence and non-existence.

This chronological coming into being, one point at a time, by the only one real and unique single monad, makes the geometry intrinsically granular and self-contained, even without the need to any minimum time or length scales or quanta, because the unit of quantization is the frequency of re-creation, or the re-fresh rate, which is the ratio between the inner and outer levels of times. This ratio is nothing but the speed of light itself whose normalized value also corresponds to the fractal dimensions of geometry, which then becomes “genuinely hyper-complex”, since both its real and imaginary parts have the same nature (of time).

Actually, as we have shown in the first three volumes of the Single Monad Model series, this genuine complex-time geometry provided elegant solutions to many persisting problems in physics and cosmology, including causality, non-locality, homogeneity, the arrow of time, the mass-gap, the cosmological constant, super-symmetry and matter-antimatter asymmetry, in addition to reconciling the inherent contradictions between General Relativity and Quantum Mechanics, by providing the smooth link between “discreteness” and “continuity”.

Therefore, the profound implications of the Duality of Time propagate into Mathematics and Number Theory, because it delineates the original edifice of Geometry. In fact, this innovative theory removes all boundaries between Physics and Mathematics, not only because it explains all the fundamental interactions, and not only gravity, based on the new complex-time geometry, but even pure mathematics is now fundamentally relativistic, since its very abstract geometrical points are being dynamically created at the speed of light. Therefore, all kinds of objects in creation are nothing but “temporal (hyper-complex, or time-time) numbers”, and no truthful calculus can be performed without taking into account their relativistic nature!

Hence, the Duality of Time Theory made a substantial breakthrough in mathematics, physics, and cosmology, as well as natural philosophy, because it exposes a deeper level of time and original theory of creation that will cause a paradigm shift in our knowledge of the fundamental nature of the cosmos, with its corporeal and incorporeal structures.

For the first time in history, and many centuries after their victorious introduction and successful applications on various physical systems, imaginary numbers have a genuine and fundamental meaning that unites between physics and mathematics! Because numbers are fundamentally connected with it, discovering the inner levels of time allowed simultaneous counting in various intersecting dimensions. Plainly-stated, the Duality of Time Theory explains how the dimensions of geometry are being structured in time, which is the ontological process that is creating both the abstract space-time and its material contents.

In fact, with the new innovative concept of the duality of time, (the hyperbolic) complex numbers (H) are more realistic than the “reals”, because they can objectively and efficiently describe all physical systems, while the reals are approached and justified only in the extreme and non-realistic conditions which implies motion at the speed of light; thus they can be adequately termed “imaginary”. However, this is also true for all Euclidean and Hilbert spaces, which can be expressed by normal complex numbers C, using the various extensions of hyper-complex systems such as quaternions, octonions and sedenions.

In fact, as we shall see in this book, the Euclidean geometry can efficiently describe existence in (three-dimensional) space at any instance of time, but without motion or change, i.e. without time! To describe the dynamic geometry, hyperbolic space is required. The reason why Euclidean or Hilbert spaces, even with Riemannian geometry, are not sufficient to the describe the real world is because it is being re-created or perpetually fluctuating between existence and non-existence, or vacuum and void. In other words, the Euclidean space is the geometry of existence at every instance of time, which makes the vacuum that is space without time. Because it is being re-created, (everything in) the world becomes void in time, i.e. between any two consecutive instances of time. The world is therefore constantly fluctuating between vacuum and void, which is the same ancient atomistic philosophy, reinterpreted out of the most original theological and philosophical conceptions, combined together with the principles and conclusions of modern physics and cosmology.

However, in this perpetual process of creation and re-creation, the state of vacuum, which is the spatial existence that can be expressed by normal complex numbers C, is splitting into two temporal and super-symmetrical hyperbolic spaces H that correspond to the corporeal and incorporeal worlds, which then instantly annihilate back into vacuum. The reason why this creation and annihilation process is not causing any devastation or destruction, as we would normally expect from such extremely energetic fissile reaction, even when it happens on the slightest atomic scales, is because it is happening sequentially in the inner levels of time before the physical mass is generated. Rather, this process itself is the fundamental origin of the instantaneous generation of matter particles and antiparticles, which then simultaneously repair and vanish into the past, just to be re-created again, perpetually. As we described in the “Ultimate Symmetry”, this perpetual and sequential dynamic formation causes the illusion of motion and continuity of spatial dimensions, while in fact all this multiplicity is being created from one metaphysical point, which is the Single Monad that is the only entity described by real continuous existence.

Moreover, this splitting of the three-dimensional Euclidean geometry into two hyperbolic planes on two arrows of time, which then describe the entangled corporeal and incorporeal worlds, means that the psychical entities have exactly the same structure as the physical objects, with the same kinds of elementary particles, atoms, crystals, and even rocks and mountains; but they are denoted as incorporeal because they exist in an orthogonal dimension of time. Therefore, the fermions in our physical world behave as bosons in the psychical world, and vice versa.

Therefore, in addition to solving all the previous puzzles, reconciling between the two contradictory theories of relativity and quantum mechanics, and uniting between physics and mathematics, the Single Monad Model and Duality of Time Theory also explains how the physical and psychical creations are developing on two orthogonal arrows of time, thus bringing the incorporeal worlds under the same umbrella of the laws that govern the physical cosmos.

These are only some glimpses of the many doors opened as a result of understanding the connection between the constancy and invariance of the speed of light and the fractal dimensions of geometry, made possible by the Duality of Time Theory. We have discussed some of these advances in the previous three volumes of the Single Monad Model series. In this book we will explore the consequences of this innovative view on various fields on the foundations of mathematics and its intersection with physics, including geometry, infinitesimal calculus and number theory. Many fundamental issues will be discussed, including the structures of number sets and their inter-relations, in addition to various fundamental problems such as the Yang-Mills conjecture and the Riemann hypothesis.

The Duality of Time Theory, that results from the Single Monad Model, explains how multiplicity is dynamically emerging from absolute Oneness, and vice versa, at every instance of time! This theory leads to the Ultimate Symmetry of Space and its instantaneous breaking into the physical and psychical creations, with two hyperbolically orthogonal arrows of time.

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• ... time! This theory leads to the Ultimate Symmetry of Space and its instantaneous breaking into the physical and psychical creations, with two hyperbolically orthogonal arrows of time.   ...

• ## ... Innovative View =>:

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• ... ly expect from such extremely energetic fissile reaction, even when it happens on the slightest atomic scales, is because it is happening sequentially in the inner levels of time before the PHYSICAL MASS is generated. Rather, this process itself is the fundamental origin of the instantaneo ...

• ## ... Number Sets =>:

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• ## ... Atomistic Philosophy =>:

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• ## ... Including Causal =>:

• ... e have shown in the first three volumes of the Single Monad Model series, this genuine complex-time geometry provided elegant solutions to many persisting problems in physics and cosmology, INCLUDING CAUSAL ity, non-locality, homogeneity, the arrow of time, the mass-gap, the cosmological co ...

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• ## ... Extreme Conditions =>:

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The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
Ibn al-Arabi [The Meccan Revelations: Volume I, page 156. - Trns. Mohamed Haj Yousef]

### The Sun from the West:

Welcome to the Single Monad Model of the Cosmos and Duality of Time Theory

### Message from the Author:

I have no doubt that this is the most significant discovery in the history of mathematics, physics and philosophy, ever!

By revealing the mystery of the connection between discreteness and contintuity, this novel understanding of the complex (time-time) geometry, will cause a paradigm shift in our knowledge of the fundamental nature of the cosmos and its corporeal and incorporeal structures.