Metaphysics & Geometry

Metaphysics is the science of the universals; it is the science of essences, i.e. realities that are universal and unchanging,i.e. do not depend on time or any particular degree of manifestation. Metaphysics is not the same as speculative philosophy or theories of creation. Metaphysics is and is true whether or not Big Bang Theory of the Hypothesis of Evolution are true. The subject matter of metaphysics proper is the Truth as perceived by the so-called Divine Mind. Metaphysics is not a human knowledge but is the content of the Divine Intellect or Pure Intellect. Metaphysics is not invented or learned; it cannot really be taught; it is only discovered, i.e. remembered, by a Seeing that is altogether other-worldly and absolute. This science, contrary to all other human sciences, is not done by speculation or experimentation but through intellection which is a transcendental, and by no means mental, activity of the spirit.

Metaphysics is about Truth Itself and not what men and women, whether saints or philosophers, have thought or said about the truth. Metaphysics is about the architectonics of reality as a whole, i.e. totality, as opposed to empirical science which consider reality only insofar as it is sensible and measurable by minds and men. We cannot say, for example, that metaphysics is other than physics though it is also not identical with it. Metaphysics encompasses all other sciences and is their very ground, but the nature of its relation to human sciences is always misunderstood; this relation is best understood only within the framework of an analogy, of course until the attainment of metaphysical realization which is the same as the Liberation. So, we offer the following analogy that we hope clarifies the nature of the relationship between metaphysics and empirical reality or empirical sciences:

Consider a video game in which your avatar, i.e. your embodied character inside the game-world, is put in a world similar to ours and its mission is to explore this world and find its natural laws. Basically, in this game you are a scientist that has to use the tools of the game to find the most general regularities of that world, things such as  gravity, electricity, etc. Now, if we want to transpose the idea of metaphysical knowledge into the context of this game, what element of the game should we choose? Let us begin by saying that metaphysics is not anything of the nature of a fact or a piece of knowledge within the game-world and can be sought and found in the same manner that one seeks and finds the facts of a world. It is not something that our scientist can find by means of exploring inside the world game.

Within the context of our game, metaphysical knowledge can be compared not to any natural law or any fact of the game-world but to the programming code that is behind the very phenomenon called the game-world and has made that world possible. Our scientist cannot find this underlying code by exploring inside the game-world and experimenting on its phenomena; he himself and all his explorations, etc. are nothing but the manifestations of the programming code underlying the game. To access metaphysical knowledge, i.e. the code in this analogy, the avatar does not even have to look outside himself or to move around, for this knowledge is, as it were, inside him, i.e. known by realizing that “there is a game” and then trying to transcend it. Now, how a game can be programmed so to make such our scenario possible, and whether it is possible or not, is entirely irrelevant here. The point of analogy was only to show the radical nature of this science of metaphysics and its relation to all other empirical sciences which can know only what is inside the game and not what is behind it. Even the methods of inquiry are radically different: In empirical science we explore and experiment, but metaphysical truths, i.e. that programming code, is not susceptible to such methods of investigations; the only way of accessing such knowledge is to somehow exit or transcend the game, or more precisely, to wake up rather than to look out and explore.

It is natural that the methods of inquiry inside the game, methods designed to find empirical facts and natural laws, can in no way help one even become aware of the possibility of a programming code behind the game, let alone finding that code. All such methods, all empirical sciences, are sciences of phenomena, but metaphysics, being the science of essences, i.e. the science noumena as opposed to phenomena, has nothing to do with phenomena but the meaning behind them. In other words, metaphysics studies the metaphysical roots of phenomena. Therefore, the failure of empirical methods in becoming conscious of the possibility of metaphysical knowledge, let alone discovering it, is only a failure and a shortcoming and doesn’t mean that there is no metaphysical foundation, or code, behind the game, a code that is different from the facts and natural laws inside the game simply because these laws are themselves manifestations, or projections, of that very code. One cannot use these methods to prove or deny the existence of a program behind the game, and this indicates the futility of all attempts at denying that there is a metaphysical foundation behind all things. The ultimate proof of metaphysical truths and realities is metaphysical realization, and this realization has been known and done and is not something up for discussion or refutation. As we have said many times, those trying to deny metaphysical knowledge, which by the way has nothing to do with god or religion but encompasses them, are like blind men trying to dispute the reality of a rainbow. A true metaphysician, i.e. a Jivan Mukta, never bothers trying to prove to anyone the possibility and reality of metaphysical truth; he can at best show the way for the seeker of truth to see and realize for himself that which is never a matter of opinion or speculation of any kind: The only proof of the Face is the face Itself.  

Now, what does all this have to do with geometry? Well, as we said metaphysics is the science of universal and immutable truths, realities that are unchanging, and hence do not depend on time or any becoming whatsoever. Thus, all theories of creation, modern or traditional, religious or secular, are only human theories and speculations and at best have symbolic value, for they all presuppose a becoming of some sort. In metaphysics, there is no becoming. Creation is a human fact; from a metaphysical, i.e. transcendental, point of view, creation and becoming are absurd and unreal concepts. But since we find ourselves as humans in the midst of a world of becoming, our thinking is conditioned by it and can understand things mostly in terms of being and becoming. At the same time, because of the transcendental nature of metaphysics it can only be understood through analogies until the point of metaphysical realization in and through which all metaphysical truths are grasped immediately and intuitively and with absolute certainty.

A proper and successful analogy is that which is based on a fundamental but common feature present in both sides of the analogy. Since immutability and the absence of time and becoming is the most fundamental and definite characteristic of metaphysics, the most appropriate analogical means for its expression should be another science that is devoid of the concepts of time and becoming. We know that all empirical sciences such as physics, biology, etc. are in fact about change and becoming, so they cannot possibly provide enough ground for a successful analogical method of communicating metaphysical truths. Of all the sciences accessible to man, the only one that is devoid of the concept of change and time is geometry, and in general pure mathematics. There is a direct correspondence between most propositions of geometry and abstract algebra, and this to the extent that we can even say geometry and abstract algebra are only two different ways of expressing same mathematical truths: In abstract algebra we are dealing with abstract symbols and their relationships but in geometry we are dealing with spatial relationships, so it is like abstract algebra only with the advantage of visualization. For this reason, geometry is a little more suitable for communicating metaphysical truths because it allows some space for visualization which is a great tool in all learning.

Thus, we believe and we can show that metaphysical truths, which are entirely independent of the concepts of god, creation, divine, etc. but are at the same time the very source and ground of all these concepts and realities, is best conveyed through geometry. And it is no surprise that geometry has always been considered a sacred science and present almost in all esoteric schools such as Pythagoreanism, Platonism, Neoplatonism, Eastern and Western religious esoterisms, and also in Freemasonry. For example, the letter G at the center of the symbol for Freemasonry consisting of square and a compass, traditional tools of architecture, stands both for God and Geometry, in which God is viewed as the Great Architect of the Universe who creates by means of immutable and intelligible relationships, i.e. geometrical forms. We only add that geometry is not to be seen as identical with metaphysics, and we should not think that in reality there are points and circles; but geometry and its concepts are used as the best symbolic means of expressing intelligible realities that are beyond any spatial relationships. In other words, geometry is the reflection of intelligible truths in the plane of human mind which can understand things only in terms of the conditions of time and space. The reflection of the sun in the pond is not the sun but can tell much about the sun, at least about its form. Metaphysics too exposes the universal forms by which truth can manifest Itself, the Truth Itself and in Itself remaining always unmanifest and only known by being it, a mode of knowledge accessed upon metaphysical realization, i.e. Supreme Identity.

It is outside the scope of this article to delve into geometry and show the metaphysical truths that are reflected in it. Our goal was only to show, firstly, the radical nature of the science of metaphysics and its relationship to all other spheres of human knowledge, and secondly, to suggest the best way of approaching this science, which is done through geometry. We hope that in the future we can devote a few articles to some aspects of metaphysics using geometrical truths. The main problem that makes metaphysics difficult to understand for the average mind is that metaphysics is not a science for everyone and requires a very refined and objective intelligence not deformed by the assaults made against the mind by the universal compulsory education. One needs to know some abstract algebra and be very well familiar with geometry, particularly the theory of functions, power series, Taylor Expansions, etc. in order to perceive the subtle metaphysical truth reflected in these forms. However, metaphysical truths, which is a way toward liberation and Supreme Identity, is not the only way. Religion and spiritual disciplines have the same end in view. While metaphysics is the way to Truth for the born philosopher, religion is a way for everyone, even the last man. The spiritual man if he succeeds may come to see the Face of Truth, but the metaphysician if he succeeds comes to see the Whole of Truth.

We end this piece by mentioning the simplest of all metaphysical truths whose reflection is found in the geometry of a circle:

Where there is a circle there is also a center, as no circle can exist without having and belonging to a center. A center, a singular point, however, can be without belonging to a circle. But what is a circle really? A circle is nothing but the expression of its center in terms of spatial relations. All the points on the periphery are the outward projections of the central point. In other words, a circle is nothing but a dimensionless point in expansive mode: Circle is the expression of a point. Or, when a point utters itself, the circle is that utterance.

Note that in expressing itself the center has not become, or turned into, a circle; the center doesn’t become the circle or cease to be the center; the center just is, and it is all that there is, for the circle is nothing but the center in its expansive mode: Circle is only a point’s dream. In reality, there is no expansion or contraction; there is only expression, the sound, The Word.

As every circle has a center and is nothing but a homogeneous expression of that center, every world, i.e. degree of manifestation, has an origin and is nothing but the homogeneous expression of that origin. This origin is exactly what is known as God in all spiritual traditions of the world.

In more abstract terms, a circle is the Taylor Expansion of a dimensionless point. Here, there are no two realities but only one reality with different names. To see and understand the exact logical and ontological relationship between the center and the points on the periphery of the circle consider this analogy: In life you are one person but with different roles or relationships. Say you are a man; to your wife you are a husband; to your son you are a father; to your father you are a son; to your employer you are the worker, and to your employee you are the boss; you can also be an uncle, a grandfather, a nephew, etc. As a father your relationship to your son is different from your relationship to your wife as a husband, etc. All these names, i.e. son, father, boss, husband, cousin, etc. which are different roles and relationships pertain to you, one and the same person. All these names designate different relationships that can be potentially established, or activated, within one and the same person; some of them may be only potential, for example you are only potentially a father when you have no kids, but once you have a child you have activated that potentiality which was within you all along and was nothing apart from you. Now, the logical and ontological relationship between you and all these names, i.e. relationships, is the same as the logical and ontological relationship between a central point and all the other points on the periphery of a circle spanned by that central point. Therefore, there are no many points; there is no becoming. There is one reality and multiple expressions, the expressions being only potentialities within one and the same point which is the basis of all Reality.   

 

 

 

Yin-Yang: The Geometric Equivalent of The Golden Ratio

I have discovered that Yin-Yang and Golden Ratio are different representations of one and the same metaphysical principle. The metaphysical principle of our interest is Unity Within Duality which states the complementary character of conflicting tendencies. This is of course similar to the Hegelian trinity in his interpretation of history: Thesis-Antithesis-Synthesis.

After giving a short account of Yin-Yang and Golden Ratio I proceed to prove their equivalence. Since after a year of research I have found no one else who has discovered this equivalence I stick a copyright notice at the end of the article just for the publication purposes in case the finding turns out to be of significance.

Yin-Yang and Golden Ratio are the two representations of the principle that maximally opposing tendencies, or aspects, complement one another to form a unity: The former represents it geometrically and the latter represents it numerically.

Yin-Yang in Chinese philosophy is the geometrical expression of the following concept: The apparently opposing and contradictory tendencies complement one another; being interconnected they add up to form a unity. In the Yin-Yang symbol a whole circle is divided into two maximally opposing aspects. The two aspects enjoy maximal opposition both in color and orientation: The color of one cut is the negation of the color of the other; and the orientation of one is the inverse of the orientation of the other.

YinYang

Notice that the two opposing sides have the same shape; they are the opposites of one another only in the attributes color and orientation. In other words, one side is the negation and inversion of the other side. But this cannot be done with just any shape; not every shape can be inverted and then added to itself to form a perfect circle which represents unity.

Golden Ratio is a number in mathematics; it has the numerical value of 1.61803398875…. When the ratio of two quantities is this number we say that they are in golden ratio. This number has huge significance in mathematics because it is the corner stone of spiral geometry which is observed everywhere in nature. It is also the limit of the Fibonacci Sequence. In fact when the ratio of two lengths is the golden ratio they have an inherent tendency to make up harmonious and beautiful structures.

golden-ratio

It is time to see how Yin-Yang and Golden Ratio are equivalent to one another, both being the representations of one principle.

We saw that Yin-Yang represents the idea that maximally opposing aspects add up to form a unity; each aspect is both the negation and the inversion of the other aspect. Numerically speaking, one being the inverse of the negative of the other. Let us try to represent this idea numerically, using number instead of shape: We need a number that when maximally opposed to itself, namely first negated and then inverted, can be added to itself to form a unity which is represented by number 1.

We don’t yet know if such number exists, but it will soon be proved that it actually does. First remember that by negation of a number of course we mean its negative, -2 being the negative of 2. By inversion we mean the inverse of the number, 1/2 being the inverse of 2; thus, the inverse of the negative of 2 is -1/2; but notice that when we add 2 and -1/2 we get 3/2 and not 1. We are after a number that gives us 1.

Let us call our desired number x and try to write down the concept behind Yin-Yang in the language of numbers and see if such number exists: We have x as our number; negate it to get -x; invert the negation to get -1/x. To express Yin-Yang is to add up these two numbers and have them equal 1. Thus, the algebraic equivalent of Yin-Yang would be the equation [x+(-1/x)]=1 which can be simplified as the following:

X – (1/X) = 1

This already looks like Yin-Yang :). For those with mathematics background this equation should blow their minds; it is none but the infamous equation that is the very definition of the Golden Ratio: Golden Ratio is the exact solution, and the only solution, of this quadratic equation. You will soon see that the second solution is just the negative of the inverse of the Golden Ratio, its maximally opposing complement which we call the Golden Complement. Let’s call this equation, the algebraic form of Yin-Yang, the Golden Equation since its solutions are the Golden Ratio and the Golden Complement; thus, this equation has two solutions in real numbers:

X – 1/X = 1               The Golden Equation (Equivalent of Yin-Yang in Algebra)

First Solution:        X1 = 1.61803398875…              The Golden Ratio       

Second Solution:   X2 = -0.61803398875…            The Golden Complement

X1 + X2 = 1             The Golden Equation in terms of its two complementary solutions

The first solution is the Golden Ratio and the second solution is interestingly the inverse of the negative of the first solution, namely the Golden Complement. See that the two solutions corresponding to the two complementary aspects of Yin-Yang add up to unity, hence completing the Yin-Yang equivalent in algebra. Notice that the Golden Complement has the exact same decimals as the Golden Ratio itself which is the reason why the decimals cancel out in the Golden Equation, hence leading to unity.

We saw that there actually exists a unique number whose inverse of its negation can be added to it to become unity, the number 1, hence expressing the Yin-Yang principle in the realm of numbers, and this unique number is none but the Golden Ratio, the solution of the Golden Equation. Thus one side of Yin-Yang corresponds to the Golden Ratio and the other side corresponds to its maximally opposing counterpart, the negative of its inverse which is the Golden Complement, and the two adding up to unity by definition. Only the Golden Ratio has this property, and one can see as we showed above that the principle behind Yin-Yang, leading to the Golden Equation in algebra, is the very definition of the Golden Ratio.


Digression:

{Notice that the two solutions of the Golden Equation, which correspond to the two maximally opposing aspects of Yin-Yang, have these two interesting properties:

 X1 + X2 = 1                     X1 × X2 = -1

If these two solutions, the Golden Ratio and the Golden Complement, are considered to be the diagonal elements, or eigenvalues, of a 2×2 matrix called the Golden Matrix G, then G is an orthogonal matrix with determinant -1 and trace 1, hence:

Det(G) + Tr(G) = 0

Which is only an interesting observation besides our main point.}

End of Digression


Therefore, the principle that the maximally opposing tendencies complement one another and form a unity is represented geometrically as Yin-Yang and algebraically as the Golden Equation X-1/X=1 whose two solutions, corresponding to the two aspects in Yin-Yang, are the Golden Ratio and the Golden Complement. We saw that of all numbers one and only one, the Golden Ratio, has this property. More precisely, the Golden Equation X-1/X=1 is the algebraic equivalent of Yin-Yang from which the Golden Ratio is extracted.

Yin-Yang is the Golden Equation in geometry. Golden Equation is Yin-Yang in algebra.

This equivalence between Yin-Yang and Golden Equation which I discovered in the fall of 2013 has blown my mind to this day. Who could guess that the Golden Ratio is Yin-Yang in numbers while Yin-Yang is the Golden Ratio in geometry! Fascinating.

*[Note for the mathematician: In constructing the Golden Equation, the algebraic equivalent of Yin-Yang, I assumed a bijection from plane shapes to real numbers, hence defining the equation over R, the field of real numbers. Naturally a bijection from the inverse and the negative of a shape would fall onto the inverse and negative of numbers. Thus, in constructing the bijection from Yin-Yang to real numbers the only proper way of translating maximal opposition, namely both negation and inversion, would be to use both the additive inverse and the multiplicative inverse of a number, the only possible ways of defining opposites among real numbers, hence the field properties of the Commutative Division Ring R. In other words, the only way to translate the relation between the two opposing aspects of Yin-Yang is to relate a real number to the multiplicative inverse of its additive inverse, hence mapping the two different operations present in Yin-Yang, namely the spatial inversion and color-negation. It is seen that the only number whose multiplicative inverse of its additive inverse can be added to itself to become unity is the Golden Ratio.]

Copyright © Tomaj Javidtash and NOEMAYA, 12/30/2014

Unauthorized use and/or duplication of this material without express and written permission from this blog’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Tomaj Javidtash and NOEMAYA with appropriate and specific direction to the original content.

The 4th Dimension

If we were exclusively 3 dimensional beings, then we would not be able to experience a 3 dimensional world. Why? Below is the reason:

Construction:

  1. Circle is a geometrical construct that needs two dimensions for its construction.
  2. The construction of a circle is possible only in a 2-D space.
  3. Two dimensional space is a necessary and sufficient condition for the construction of a circle.
  4. When we say that a certain geometrical or topological construct is N dimensional we mean that an N dimensional space is the necessary and sufficient condition for its construction.

Perception:

  1. Circle as a 2-D construct can be perceived fully only from above, from above the 2-D space in which it is constructed.
  2. To fully perceive a circle one needs to transcend the circle’s space.
  3. One has to be transcendent to the circle in order to perceive the circle, or else the circle itself will be transcendent to one’s perception.
  4. Circle in its entirety is always a transcendent shape to the immanent
  5. In order to perceive a circle in its entirety, the shape of the circle, one has to step into the third dimension, a dimension that is transcendent to the dimensions of the circle and yet contains the space of the circle.
  6. The condition for a perception of the 2-D totality of a circle is being in a space that is both transcendent to the space of the circle and yet contains the space of the circle as immanent.
  7. To perceive an N dimensional shape in its entirety one has to be placed in the N+1th

Conclusion:

  1. To perceive (experience) the N dimensionality of a space, rather than deducing and inferring it, one has to be transcendent to it, being located in an N+1 dimensional space.
  2. For an N dimensional space to be perceived (experienced) it has to be immanent to the observer’s consciousness, but this immanence is possible only if the observer is transcendent to the observed, hence enjoying a higher dimension than the observed.
  3. The dimensionality of an N dimensional space is always transcendent to the observer located in the same space.
  4. The dimensionality of a space becomes immanent only and only for an observer that is transcendent to that space.
  5. Our experience of 3-D space is immanent: We intuit the 3 dimensional character of our space immediately and don’t need to deduce or infer it by examination and investigation.
  6. The 3-d character of our space is given to us immediately; it is immanent.
  7. Since our experience of the world is already 3 dimensional and immanent, therefore we have to be transcendent to it and located outside of it.

                                                                                                      QED.

 

    ***To experience two dimensionality directly one has to be in the third dimension. Therefore to experience three dimensionality one has to be in the fourth dimension.