The Arithmetic of Concepts
In all religious philosophies, God is the original person, Who creates all else. If we were to count things, then God would represent 1. In Vedic philosophy, additionally, all that is created is also a part of God, Who is then described as the complete truth. In effect, since God is the complete truth, everything that follows is a partial truth. Similarly, since God is the original truth, everything that follows is a relative truth compared to the original truth. The partial truth represents a fraction of God, and the relative truth represents an order or succession among the fractions, which can be counted as 1, 2, 3, etc. Two ideas—(1) that God is the origin, and (2) God is the whole truth—can thus be used to construct a theory of natural numbers and fractions, which this post discusses. Once this foundation is established, then we also talk about other types such as complex and irrational numbers. This post discusses how a new understanding of numbers can be built based on God’s existence.
Table of Contents
- 1 Philosophical Background
- 1.1 The Invocation of Sri Īśopaniṣad
- 1.2 The Addition and Subtraction of Knowledge
- 1.3 God is the Completion of Knowledge
- 1.4 The Creation of Parts from the Whole
- 1.5 Three Kinds of Expansions
- 1.6 The Construction of a Tree
- 1.7 How Addition Becomes Subtraction
- 1.8 The Role of Māyā
- 1.9 Yoga-Māyā vs. Mahā-Maya
- 1.10 Unity and Diversity in a Tree
- 2 Fundamental Principles of Counting
- 2.1 Cardinal and Ordinal Counting
- 2.2 The Meaning of Zero
- 2.3 The Reality of Negative Numbers
- 2.4 Masculine and Feminine
- 2.5 The Yin-Yang Principle
- 2.6 The Logic of Three Parts
- 2.7 The Role of Concepts and Logic
- 2.8 The Ideal Thing
- 2.9 The Absolute Truth
- 2.10 The Mechanism of Counting
- 2.11 Finite vs. Infinite
- 3 Different Classes of Numbers
- 3.1 The Reality of Fractions
- 3.2 The Problem of Missing Fractions
- 3.3 Cantor’s Diagonal Argument
- 3.4 A Precise Definition of the Tree
- 3.5 Atomism to Time and Operations
- 3.6 The Universe is a Half-Empty Glass
- 3.7 Space and Time Trees
- 3.8 Sub-Structure in the Tree
- 3.9 Varieties of Number Types
- 3.10 The Use of Complex Numbers
- 3.11 The Creation and Destruction of the Universe
- 4 Mathematics and Conscious Experience
- 4.1 The Philosophy of Perception
- 4.2 The Description of Dṛṣṭa and Dṛṣya
- 4.3 The Use of Logic in Creation
- 4.4 The Genesis of Logic
- 4.5 The Three Principles of Logic
- 4.6 The Five Principles of Logic
- 4.7 Two Kinds of Restrictions
- 4.8 Logically Sizing the Universes
- 4.9 The Significance of Four Directions
- 4.10 The Creation of the Alphabet
- 4.11 The Nature of Mathematics
- 4.12 Share this:
The Invocation of Sri Īśopaniṣad
The first verse in the Sri Īśopaniṣad presents a paradox which, if understood, can open a new thinking about numbers. The invocation states the following.
oṁ pūrṇam adaḥ pūrṇam idaṁ
pūrṇāt pūrṇam udacyate
pūrṇasya pūrṇam ādāya
Śrīla Prabhupāda provides a devotional translation to this verse here. But I will provide a more literal translation to the above verse below, before returning to the devotional translation in a subsequent section.
That is complete. This is complete. From the complete emerged another complete. After the complete emerged from the complete the remainder is certainly complete.
This verse presents a paradox because if you remove something from another thing, the remainder must reduce. It is supposed that the thing being removed is a part of the whole, and if that part is taken out then the whole should have become incomplete. Then how do we understand this paradox?
The Addition and Subtraction of Knowledge
Suppose you know Newton’s three laws of motion, and I then tell you about the first law. The information I provide you doesn’t add to your knowledge as you already knew the three laws. By adding the knowledge of the first law to the preexisting knowledge of the three laws, the total knowledge is unchanged. If the knowledge of three laws is X and the knowledge of one law is Y, then
X + Y = X
Magic has happened. This equation is possible in the case of concepts. When you have a more complete concept, a partial concept doesn’t add to the more complete concept. Rather, the complete concept is like a big box, and you just put the incomplete concept like a smaller box inside that bigger box. Ultimately you still get the bigger box as the more complete concept.
Now, suppose that X means “full knowledge”, and you add “full knowledge” to that “full knowledge”. The conclusion applies again. We replace Y by X.
X + X = X.
Since full knowledge doesn’t add to itself, we can move X to the other side:
X = X – X
If adding X to X gives us X, then subtracting X from X must also leave X as the remainder. This is precisely what Īśopaniṣad says in the invocation. You take out complete from the complete and the balance is still complete. How does it happen? You impart your complete knowledge to someone and by such imparting, your knowledge is not reduced. You still have complete knowledge, and the other person also has complete knowledge. That’s how X – X = X. Again, remember, we are talking about concepts.
This idea is partially encompassed in the union of sets where adding the same set to itself doesn’t add to the set, and X U X = X. But subtraction from the set removes elements from it. Therefore, the set theory construal is incomplete because the subtraction from the whole leaves the whole reduced.
God is the Completion of Knowledge
The Vedic texts describe that God is the completion of knowledge. When He imparts His knowledge to others, He is not reduced. And yet a world is created from that knowledge. Similarly, when this material world is absorbed back into God’s person, all the knowledge remains within God from where it can emerge again. That is complete and this is complete; from complete emerges the complete. Now that we have a grasp of the literal translation above, we can appreciate Śrīla Prabhupāda’s devotional translation.
The Personality of Godhead is perfect and complete, and because He is completely perfect, all emanations from Him, such as this phenomenal world, are perfectly equipped as complete wholes. Whatever is produced of the Complete Whole is also complete in itself. Because He is the Complete Whole, even though so many complete units emanate from Him, He remains the complete balance.
Śrīla Prabhupāda’s translation combines the Īśopaniṣad verse with one additional fact, namely, that the complete is the Personality of Godhead. My explanation adds another fact to it namely that the complete person is also the fullness of knowledge because He imparts and expands His knowledge into everything else. This method of bringing additional ideas to explain a particular verse has been used by Ācharyas especially when the meaning of the verse may present a paradox not easily understood by everyone.
The Creation of Parts from the Whole
We are dealing with an anti-reductionist paradigm in which the parts are not combined to build the whole. Rather, the whole is divided into parts. In order to create the parts, the whole must precede the parts. This idea is presented in Śrimad Bhāgavatam 1.3.28 when fullness expands into subsidiary forms.
ete cāṁśa-kalāḥ puṁsaḥ
kṛṣṇas tu bhagavān svayam
mṛḍayanti yuge yuge
All of the above-mentioned incarnations are either plenary portions or portions of the plenary portions of the Lord, but Lord Śri Kṛṣṇa is the original Personality of Godhead. All of them appear on planets whenever there is a disturbance created by the atheists. The Lord incarnates to protect the theists.
This verse further adds to the Sri Īśopaniṣad invocation noted above and identifies that the complete is Kṛṣṇa. The conclusion is described after enumerating various incarnations such as Nara and Narayana, Kapila, Rishabhadeva, Narasiṃha, Vāmana, Vyasadeva, and so forth. In other words, while the aforementioned forms are parts, there is a complete form which is the source of these manifestations. Accordingly, these are subsidiary forms who are classified into three categories—aṁsa, kalā, and puruṣa.
Three Kinds of Expansions
There is deeper philosophical significance to these subdivision forms which I will briefly discuss. My forthcoming book Cosmic Theogony discusses this at greater length. The basic idea is that God has three aspects—chit, sat, and ananda—which manifest as part, function, and purpose, respectively. (If you are interested in this topic, I would suggest reading the book Emotion).
The aṁsa are the parts of the whole and manifest from chit; the chit has 24 primary divisions which manifest as 24 incarnations in this world (described prior to noting that Kṛṣṇa is the origin of all – SB 1.3.1 to 1.3.27). These 24 incarnations are as follows: (1) Ādi Puruṣa, (2) The Four Kumara, (3) Vāraha, (4) Nārada, (5) Nara, (6) Narayana, (7) Kapila, (8) Dattatreya, (9) Yajña, (10) Rishabh, (11) Prithu, (12) Matsya, (13) Kūrma, (14) Dhanvantari, (15) Mohini, (16) Narasiṃha, (17) Hayagriva, (18) Vāmana, (19) Parshurama, (20) Vyās, (21) Rāma, (22) Balarama, (23) Buddha, (24) Kalki. Within the spiritual creation too, there are 24 Viṣṇu forms described as follows in the purport to SB 2.2.8: (1) Puruṣottama, (2) Acyuta, (3) Narasiṁha, (4) Trivikrama, (5) Hṛṣīkeśa, (6) Keśava, (7) Mādhava, (8) Aniruddha, (9) Pradyumna, (10) Saṅkarṣaṇa, (11) Śrīdhara, (12) Vāsudeva, (13) Dāmodara, (14) Janārdana, (15) Nārāyaṇa, (16) Hari, (17) Padmanābha, (18) Vāmana, (19) Madhusūdana, (20) Govinda, (21) Kṛṣṇa, (22) Viṣṇu-mūrti, (23) Adhokṣaja and (24) Upendra.
Aside from these 24 forms is the fullness called Kṛṣṇa described in Śrimad Bhāgavatam 1.3.28. Similarly, Sāńkhya philosophy describes 24 elements with the soul as the 25th element. There is hence a similarity between the two: the 24 incarnations of Kṛṣṇa are partial expressions of the 25th form; quite like the 24 elements of Sāńkhya are partial expressions of the soul’s chit or cognition. This chit is divided into 3 parts called ādidaivika, ādibhautika, and ādiatmika. The 24 Sāńkhya elements include ādibhautika and ādiatmika but not the ādidaivika—which constitute the demigods.
As a result, the Sāńkhya elements are 24 but the demigods are only 12. These 12 parts are called Aditya and are represented by the 12 signs of the zodiac. 12 forms are also marked on the body by applying Tilaka at 12 places—they are called Keśava, Nārāyaṇa, Mādhava, Govinda, Viṣṇu-mūrti, Madhusūdana, Trivikrama, Vāmana, Śrīdhara, Hṛṣīkeśa, Padmanābha, and Dāmodara. These 12 forms are included in the above 24 forms, but the other 12 forms are left out. We can understand that for God, we are ādibhautika and the other 12 forms are ādidaivika as God Himself. The forms that we mark on our body are the ādibhautika forms; by marking the ādibhautika on our body, we are spiritualizing the body and making it God’s known.
God is both ādidaivika and ādiatmika from His perspective. Therefore, if we look at the world from God’s perspective, there is only ādibhautika (the world) and ādiatmika (God Himself) denoted by the 24 forms. But if we look at things from our perspective, then there are 24 elements of Sāńkhya—which denote ādibhautika (objects other than God) and ādiatmika (our body)—in addition to the 12 ādidaivika representing God. Owing to this fact, the 24 elements of Sāńkhya can be compared to the 24 forms of God as they embody the ādibhautika and ādiatmika of God, quite like the 24 elements of Sāńkhya embody the ādibhautika and ādiatmika for the soul.
The 24 forms are incredibly important not just as the study of matter (as many people may accept) but also as the study of God Himself. They should be regarded as meaning forms and the world as the byproduct of combining these elementary meanings to create complex meanings. The study of the 24 elements of Sāńkhya is almost identical to the study of the 24 Viṣṇu forms and incarnations. Nevertheless, it is a scientific study where the form of the meaning is treated as a personality. It is not ‘material’ science because God’s meaning is reflected in matter.
The kalā constitute the divisions of purpose, or pleasure. They are manifest from ananda and have 16 parts which are expressed as the 16 phases of the moon called ṣoḍaśa-kalaḥ in Śrimad Bhāgavatam 5.22.10. Of these 16, the first part is the fullness of happiness, and the other 15 are subdivisions of that fullness. Therefore, while we speak about the 16 parts, we often express only the 15 parts. Our happiness is totally subjective; that is, it doesn’t depend on material objects or a demigod. The happiness is created by our reaction to the surroundings, and demigods can help us change the surroundings but not to make us happy or unhappy. Owing to this, the spiritual advancement of the soul is considered more important than the material prosperity—i.e. changing the circumstances. Hence, happiness is not divided into ādidaivika, ādibhautika, and ādiatmika. However, happiness has an associated gender (male and female). These are also represented as rising and falling forms of happiness. Thus, the 15 parts of the moon are doubled to create a lunar month of 30 days. Of these, half the days denote rising moon and the other half a setting moon. In this regard, we might note narrations in the Purana regarding how Chandra was originally a male but then became a female and then stayed as male for 15 days and female as 15 days.
Kṛṣṇa as the embodiment of all varieties of pleasures is said to be the personality comprised of 16 kalā. In the Tantra system the Sakti who represents this pleasure in the material world has 16 different forms beginning with Pārvati. In several places (e.g. SB 3.11.40) the material energy is described to have 16 different aspects. These 16 forms of pleasure are enjoyed through 4 different relationships (child, master, lover, and friend) owing to which the 16 pleasures are also divided into 4 parts, thereby creating 64 kalā according to CC Madhya 24.289 – 24.231. When Kṛṣṇa studies with Sāndipani Muni he masters these 64 kalā which are then described as 64 different art forms employed for enjoyment.
The meaning of sat is awareness or relation to other things. This awareness has 8 parts—the first being self-awareness. The other 7 parts of awareness are expressed at different levels in a hierarchy. These levels are morality, ego, intellect, mind, senses, sensations, and sense objects. Each type of awareness helps us connect to something else at a deeper or shallower level. These 8 forms of awareness are taken by the 8 puruṣa forms. These forms constitute the coverings of each universe, and are described by Kṛṣṇa as prakriti astadha or the nature with 8 parts. These 8 parts are sometimes called Earth, Water, Fire, Air, Ether, Mind, Intelligence, and Ego, but they are not the same as the elements of Sāńkhya. The significance of these elements is that in the innermost part of the universe, one can obtain all the 8 levels of experience, but in successively outer coverings of the universe, one type of experience would be missing. Ultimately, one would be left only with self-awareness.
We noted 4 types of relationships earlier, and these divide the 7 lower parts of awareness to form 28 divisions of awareness. When these divisions of awareness are combined with the three modes of nature—sattva, rajas, and tamas, three parts of the universe are created—28 Nakṣatra or all the stars under the mode of sattva, 28 Naraka or hells under the mode of tamas, and 28 divisions of planetary systems (14 planetary systems divided by 14 oceans and islands) under the mode of rajas. This is the summary of the Vedic cosmology discussed in Mystic Universe.
The division by 8, 16, and 24 are described in the Śrimad Bhāgavatam. The confusion is that the same words are used to describe each of these categories. For instance, among the 8 elements are Earth, Water, Fire, etc. The same is the case with the 16 elements, and the 24 elements. This leads to confusion about their true nature especially if we disregard the 8 and 16 divisions as being already subsumed under the 24 elements of Sāńkhya.
SB 3.11.40. This phenomenal material world is expanded to a diameter of four billion miles, as a combination of eight material elements transformed into sixteen further categories, within and without, as follows.
SB 3.11.41. The layers or elements covering the universes are each ten times thicker than the one before, and all the universes clustered together appear like atoms in a huge combination.
This confusion can be cleared by understanding that 8 forms are created from sat, 16 forms from ananda, and 25 forms from chit, of which the first form denotes self-awareness, fullness of pleasure, and fullness of cognition, respectively. Together these form 49 categories, and God is the 49th category. Beyond these 49 categories is śakti which constitutes the 50th category, and represents the logic or rationality of the world. We will see later how there are 8 such types of rationalities and worlds, each of which divides into 3 parts. These three parts of material rationality are represented by three puruṣa called Karanodakaśāyī, Garbhodakaśāyī, and Kṣīrodakaśāyī. In the material world, additionally, there are two other principles associated with material rationality (we will discuss them later) which embody time and karma, and they are represented by Saṅkarṣaṇa and Śeṣa. Thus, the 50th category called śakti has 5 parts denoted by a separate form.
All these 50 categories constitute a typology, which as we will see later creates the alphabets of the Sanskrit language. If we can understand the forms of God, then we can understand the meaning of alphabets. In that process, we will also understand how all meaning is created in order, due to which this meaning can also be counted as numbers. The type, alphabet, form, and number are identical; they are all manifest from the three features of the soul, which is originally embodied in Kṛṣṇa. Once the basic pattern of creating parts from the whole is understood, then the same pattern is used to subdivide the previously created parts. As a result, now, we get even more numbers (fractions), although the succession of divisions appears as additions (we will discuss the reasons for this shortly). They constitute the basis of language, objects, and numbers. The study of numbers is identical to the study of language, meaning, and God’s forms.
In the case of demigods, the 12 divisions of chit are called Aditya, the 16 divisions of ananda are called Rudra, and the 8 divisions of sat are called Vasu. Together they form the 36 gods of which the first person is a representation of Viṣṇu, Shiva, and Brahma, respectively. If we keep the above said trinity aside, then Vedic texts speak about 33 demigods. The system of demigods therefore also reflects the same typology, with the exception that the Aditya are 12 instead of 24. The understanding of this typology is the essential preliminary step required to understand everything else. As mentioned earlier, Cosmic Theogony discusses these types.
The Construction of a Tree
Just as the whole divides into parts, the parts subdivide into further parts. This successive subdivision produces a ‘tree’ in which the whole is the root, and the smallest parts the leaves and fruits. The successive emanations from the root are fractions of the whole, and yet they can be ordered and counted. In several places in Vedic texts a tree of diversification from the origin is noted. The same idea is described in Śrimad Bhāgavatam as rivulets from a source of water after completing the narration of various incarnations.
O brāhmaṇas, the incarnations of the Lord are innumerable, like rivulets flowing from inexhaustible sources of water. (SB 1.3.26).
The analogy is different than a tree, but the essential idea of branching is the same. However, while we use the analogy of trees and rivulets, we must remember these are analogies. We don’t expect to see a real tree or river. The reality is that the root is the whole the branches are the parts. We should keep this hierarchical picture of whole-part in mind, while also remembering that the hierarchy of such divisions creates a tree-like structure.
How Addition Becomes Subtraction
Since the whole is the complete, the diversities are created by removing something from the complete to create parts. Unlike reductionism, where things have to be added to create the whole, here they are removed from the whole. This removal is represented by the negation called māyā.
Māyā hides parts of the whole such that only the part of it is visible; the successive steps of hiding appear as additions to the tree but they are in fact subtractions from the whole. Thus with additional branches it seems that the tree is growing but in reality parts of the whole are gradually hidden. Thus, we have to think of the operation of addition as the operation of subtraction from the whole, because what is being added to the whole is negation or māyā which hides some aspects of the whole to create parts.
Owing to this fact, a ‘chair’ is an object in its own right, separate from the ‘legs’ of the chair which are created by hiding some parts of ‘chair’. In one sense, the ‘legs’ were added to the chair. But in reality the legs are manifest by hiding parts of the full chair. The hiding comprises opposites or duality due to which you can have a left and right leg of the chair. Therefore, the leg also has the full chair in it, although the leg is itself not the full chair (because parts of the whole are being hidden in order to create the part). Thus, God as the whole is in everything, but everything (created by hiding different aspects of God in order to create the parts of the whole) is not equal to God.
The Role of Māyā
The process of hiding the whole leads to a problem that is often a source of confusion. Is God being covered by māyā? Or is māyā covering our vision of God? If māyā were covering God, then God would gradually become ignorant about His own self through this covering. If instead māyā covers the soul, then the soul gradually becomes ignorant of God. It is not hard to guess why Vedic texts say that māyā covers the soul’s vision of God.
Māyā covers the three aspects of the soul, creating the soul’s goggles through which he sees. In this seeing, the soul is always seeing God, and yet because of wearing the goggles he only sees the parts instead of the whole. Now that the whole is missed from the vision, the parts are misinterpreted as being independent things. Furthermore, the soul gets so absorbed in the vision that he forgets that he is different from the thing he is seeing.
The situation is like that of a fan at a rock concert. When he sees the rock star playing the guitar, he imagines himself to be a rock star and starts acting just like the rock star. The fan pretending to be the rock star is an imitation of the real rock star. It is unreal as the fan is not the rock star. In another sense it is real because there is indeed a rock star. The actions of the rock star are not the illusion; the fan’s identification with the rock star is the illusion.
Yoga-Māyā vs. Mahā-Maya
God is the rock star and the soul is a fan of that rock star. Sometimes the soul remains a fan and appreciates the rock star. But sometimes the soul imagines himself to be the rock star. These two are called yoga-māyā and mahā-māyā, respectively. In the vision of yoga-māyā the soul has the identification with the thing he is seeing (as he is absorbed in seeing), but the whole is not missed in the vision. As a result, the soul sees the whole and identifies with a part. Through that identification and vision he becomes a part of God. In mahā-māyā the soul identifies with the part but doesn’t see the whole. So he thinks he is the body and that body exists independent of God.
Both are called māyā because the function of māyā is to create parts from the whole. The soul naturally identifies with a part and considers it his identity. The crucial difference is only that the vision of the whole is missed in mahā-māyā and the vision of the whole is present in yoga-māyā.
In order to pretend to be a rock star, the fan has to forget a deeper level difference with the rock star, and just see some superficial similarities like both having the same body parts. The fan will ignore the fact that the rock star has friends, relatives, duties, property, commitments, and personality different from the fan. After forgetting many differences, and remembering only the superficial similarities, the fan pretends to be the rock star.
Unity and Diversity in a Tree
Since both mahā-māyā and yoga-māyā are trees, the unity is the root of the tree, and the diversity is the many leaves on the tree. In mahā-māyā only the leaves are seen while the deeper level branches and the root are ignored. The deeper realities are the unity in the diversity. When they are ignored, only the diversity is seen, while the unity is forgotten. Therefore, any knowledge that reveals the whole even through gradual steps constitutes a spiritual path. Mahā-māyā is the vision in which the universe appears to be comprised of individual particles which are operating independently. Yoga-māyā is the vision in which the unity is seen along with the diversity. Therefore, the full hierarchy from leaves to root is visible. In this vision, things do not move independently; they are rather controlled by the deeper level reality, and ultimately by God.
When the deeper level reality is missed, then the theory of nature by which motion is controlled is not understood. One speculates on the theory that controls nature, and there are many possible speculative theories, which seem to work temporarily but fail in other situations. This working and failure constitute our ignorant lives in which an incorrect theory sometimes works and gives us happiness, but at other times this theory fails and then we become unhappy. When the theory fails we try to change it. Similarly, when we are unhappy, we are compelled to revise our view of the world.
A crucial part of this theory is the nature of the self; from that self-identity springs the understanding of the world around us. If the true identity is not found, we keep inventing new ideas about the self and the world—i.e. pretending to be different rock stars. In each case, the illusion is broken; just dressing up like the rock star doesn’t make one a rock star. This pretention thus leads to change in pretension. However, if one is situated in the true identity then the changes to personal identity also cease to occur.
Fundamental Principles of Counting
Cardinal and Ordinal Counting
When the tree expands into branches and leaves, we begin counting the tree from the root. The root is the first element in the tree, so it denotes 1. As successive branches come out of the tree, they are designated as 2, 3, 4, etc. In effect, the complete at the start is designated by 1 and all the fractions that follow are the successive numbers. This presents a confusion because in one sense they are fractions and not whole numbers, and yet they are ordered or sequenced as if they were whole numbers. To understand this paradox, we need to distinguish between ordinals and cardinals.
Numbers are defined in two ways—cardinal and ordinal. Ordinal numbering is represented by the words first, second, third, fourth, etc. Cardinal numbering is denoted by words such as one, two, three, four, etc. The parts of the whole are fractions as cardinals and sequenced as ordinals or whole numbers. There is no paradox between describing the same thing as a fraction and as a whole number if we remember that the fraction is a cardinal and the whole number is an ordinal. Thus, we can interchangeably call the whole numbers as fractions or vice versa as long as we remember this distinction.
A curious property however arises when we note that the arithmetic operations—when performed on ordinals—produce different results. For example, when 1, 2, 3, 4 are ordinals, then 1 + 2 = 2, because the addition says: add a second element to first. The result of addition is that you get a second element beyond the first. Thus:
1 (ordinal) + 2 (ordinal) = 2 (ordinal)
Conversely, when 1, 2, 3, 4 are cardinals, then 1 + 2 = 3, which is the more familiar type of addition. It says, add two elements to one element due to which we now get three (cardinal) elements. Thus:
1 (cardinal) + 2 (cardinal) = 3 (cardinal)
This distinction is very important because—as we noted above—the same thing is a fraction (cardinally) and a whole number (ordinally). Therefore, in principle, we can replace the entire mathematical description of the world (which is presently based on cardinals) to a description based on ordinals. That will not refute the validity of the cardinal addition, but the successes of the cardinal addition will also not contradict the truth of the ordinal system. This is very important because it means that everything in mathematics can be replaced by an ordinal arithmetic without disproving anything in modern cardinal mathematics. There can be an alternative description of the world based on ordinal arithmetic without disproving cardinal usefulness.
Based on the cardinal-ordinal distinction, we can revisit the addition operation we discussed at the start of this post. The whole or the complete is the original object, which is designated as 1. Since nothing else exists right now, we can only add 1 to itself, which is like adding complete knowledge to itself, and it only produces the same complete knowledge. Therefore, 1 + 1 = 1. Remember we are talking about ordinal (first) here. Adding first to first gives us first. Similarly, when the complete expands into the other completes, we are talking about subtracting the original from the original, and the balance of subtraction is still the original complete. Therefore, 1 – 1 = 1. Clearly, if ‘1’ denoted a cardinal then the familiar rules of addition and subtraction will apply (1 + 1 = 2, and 1 – 1 = 0). The crucial point is that the first element can never be removed, indicated by the fact that 1 – 1 = 1. Thus, you cannot go prior to the root of the tree, and 0 is not before 1. Logically, you cannot assert the origin to be void. The paradox in Sri Īśopaniṣad is thus mathematically equivalent to the claim that the origin is not void.
The Meaning of Zero
Now, if 0 is not before 1, then where is it? If we cannot get to a zero before 1, then it would appear that we cannot also go to the negative ordinals such as -1, -2, -3, etc. That is both good and bad. It is good because if negative ordinals were possible then there would simply be no starting point in the process of counting because we would have to start counting at -∞ which cannot be defined as an ordinal. It is also bad because if we cannot get to 0 then we cannot get to negative numbers which would mean they are unreal, when the fact is that negative directions in space are denoted by opposites such as east and west. Therefore, we need to get the reality of 0 and that of negative numbers, without claiming that 1 – 1 = 0. This requires a different approach which—as we have seen previously—can be easily understood if we describe 0 as a cardinal and view it as the smallest fraction.
Remember that we are dividing the whole into parts, and those parts are fractions of the whole. Therefore, if we took that division to its logical conclusion, we will get 0, and it won’t be a 0 before 1. It will be a 0 that follows 1 after the successive application of hiding parts of the whole. Effectively we are saying that the whole (1) is not created by summing up the smallest parts (0)—which is what modern mathematics tries to do when it constructs length by adding up infinitesimal points. We adopt the opposite approach in which the length is divided into smaller and smaller parts and the limit of that division (the smallest obtainable part) represents 0. This 0 is not actually an infinitesimal point. It is the smallest part of the whole.
To better understand this idea we need to discuss the three modes of nature called sattva, rajas, and tamas, in conjunction with two other ideas—everything and nothing. Let me illustrate these with the example of colors, and then we will try to formalize this understanding logically thereafter.
The color white represents everything; it is the complete color, because it contains all other colors. This complete color is divided into three parts—yellow, magenta, and cyan (loosely called yellow, red and blue). When all the colors have been removed from white, we are left with nothing denoted by black. If you are familiar with color schemes, then 4 colors (cyan, magenta, yellow, and black) are used for printing on a white paper. Each color hides the background white, and if everything is hidden, then the result is black, which (from the standpoint of colors) constitutes hiding all the colors.
These five categories are sometimes called pañca-tattva and sometimes the five deities of Vaishnavism—Narayana, Vasudeva, Saṅkarṣaṇa, Pradyumna, and Aniruddha. These are not merely theological topics; they are in fact the basis of expanding, distinguishing, and counting. Counting here simply means the succession of things emanated from the original Absolute Truth.
The color white denotes śuddha-sattva or 1. Yellow is sattva. Then magenta is rajas. Then cyan is tamas. Finally, the negation of white is black; it denotes 0. It is obtained when everything in the whole has been hidden. If hiding is achieved in atomic steps, then the last step in hiding produces 0. We can thus equate 0 with the last step of hiding, and the result with total obfuscation. White and black are opposites. And yet, we are not using positive and negative numbers to denote these opposites. Rather, we denote the positive by 1 and its negation by 0. The reason is that the absence of light is darkness. You just remove all the light and you are left with darkness. Once you attain total darkness, you cannot take more light out of that darkness.
Therefore, we can talk about opposites like light and dark, but darkness is simply the absence of light. You cannot ever get beyond that absence and imagine something darker than complete darkness. The same holds true of opposites like hot and cold. You remove all the heat and you get cold, but you cannot make it any colder. Therefore, there are opposite directions in space and yet in the examples above the negation is simply 0. With this process we have established the reality of 0, but in the act of doing so we also have seemed to imply that negative numbers are impossible.
The Reality of Negative Numbers
These negatives are found in many types of concepts. Take the example of money. If you have the fullness of wealth then we can denote that by 1. If we remove all the wealth, then we get 0 which is poverty. But there is a state beyond poverty—that of debt, where you don’t have any money and instead you owe someone money. The money you have in debt is negative money. Similarly, we can have fame and if fame is removed we will get anonymity. But beyond anonymity is disrepute and to escape from disrepute, anonymity would be progress. Similarly, there can be knowledge and if this knowledge is removed then we get ignorance. But beyond ignorance are false beliefs where you deny reality or believe in something that doesn’t exist.
Thus, there are certain cases—e.g. light and dark—where removal of light is itself darkness, and you can’t go beyond that absence of light. But there are other cases—e.g. knowledge, fame, wealth, etc.—where removal leads to ignorance, anonymity, and poverty, but it is not as bad as falsity, disrepute, and debt. We have to understand how both scenarios are real.
The short answer is māyā which hides some parts of the whole and thereby reveals some other parts. The hidden parts are negated and the revealed parts are asserted. The prāna or śakti that hides represents the negation while the vāk which is revealed through such hiding denotes assertions. These respectively constitute the negative and positive numbers. We might say that the negatives are being added to the whole to subtract from it. Owing to the successive application of negations—which in turn produce the assertions, the negatives and positives must be alternated.
Masculine and Feminine
The positive represents the masculine and the negative, the feminine. The positive is the whole, and the negative is the restriction on the whole. The whole is the manas, the restriction on that manas is prāna, and the product of restriction is another positive called the vāk. The restriction creates a boundary for choice; things within the boundary are possible and things outside the boundary are impossible. Things within the boundary constitute our freedom; things outside that boundary are forbidden. Choice is an assertion within the boundary and what lies out of bounds is the negation. We make the choice after we know the possibilities. Therefore, the assertion follows the negation. Possibility is feminine and choice is masculine.
The female is superior to the male because she asserts boundaries on the male. The male is superior to the female because he chooses within the bounds. The basic need in a female is to control; the basic need in a male is to be free. When a male exists without a female, he sees infinite possibilities but without a constraint he is unable to make a choice; the male languishes in indecision. Restriction accelerates choice; if the alternatives are limited by the female then the male chooses something definite over the endless possibilities. By that choice he becomes something real instead of an unlimited potential. A child is an unlimited potential; he could be anything. But as the child grows up, it has to sacrifice that unlimited possibility to be some definite reality. The process of growing up involves association with a female. The female that restricts a male’s choices accelerates a choice. Both enjoy in that combination because the male chooses something definite over mere potentiality, and the female restricts the potentiality to something definite. The choice is masculine and the restriction is feminine.
The language used to define the restriction is different from that used to describe the choice because the restrictions are what lies outside the boundary and choices are what lies inside the boundary. The choices are relative to the boundary which means two complementary methods have to be employed in order to describe the choice. For instance, you can say that a certain choice was the best choice given the circumstances. Therefore, even to define the ideal, we need the definition of all that is possible. As a result, all masculine forms of God are coupled with a feminine form because the masculine represents choice and the feminine denotes possibility.
The Yin-Yang Principle
Since the whole is successively restricted, there is an alternating cycle of restriction and assertion. The picture below expresses this fact in which the left side negates the complete and as this negation proceeds, the right side denotes the successively reducing parts of the whole. Thus, the progression from 1 to 0 involves successive restrictions and revelations.
Now we have the reality of positive and negative numbers; these are simply fractions lying between 1 and 0. There is no ordinal 0 prior to 1. There is only a cardinal 0 which follows 1. There is no ordinal -1 which is prior to 1 because there is no ordinal 0. But, we can talk about positive and negative fractions if we allow for a cardinal negative that subtracts from the cardinal 1. At the bottom of this expansion is the cardinal -1 which represents complete constraint on the choice, and hence freedom is completely denied. The extent of this negation is equal to the extent of the whole, which means that you can deny the existence of the whole truth but nothing beyond that can be denied, because that denial constitutes the negation of everything.
Owing to the successive application of the negation to create an assertion, and the alternating nature of the male and female, the tree can also be described through the yin-yang picture in which the yin is the dark part that negates and yang is the light part which is revealed. Once something has been negated, something is still left asserted. Therefore, within the negation denoted by the dark, there is a circle of white. Similarly, whatever is left asserted will subsequently be negated. As a result, within the assertion denoted by white, there is a circle of black. It is not a coincidence that these black and white parts are called feminine and masculine respectively.
Care must be exercised in distinguishing between the opposites everything and nothing which lie at the top and bottom respectively and are denoted by white and black respectively, and assertion created from the negation which exist in the middle. In the above example, the masculine and the feminine have been denoted as white and black which is inaccurate. An accurate symbol of the feminine is magenta, and that of masculine is yellow and cyan.
The Logic of Three Parts
There is often confusion between the śuddha-sattva denoted by white and sattva denoted by yellow. Similarly, there is confusion between tamas denoted by cyan and nothing denoted by black. This confusion can be cleared when we look at the full spectrum of colors. It can also be cleared logically. If choices are represented as the selection between alternatives A and B, then white represents the category called both while black represents the category called neither. Each globe in space exists in between the logical extremes called everything and nothing; the top of the globe represents everything and the bottom represents nothing.
At the top of the universe, choice means both or everything. Similarly, at the bottom of the universe, choice means neither or nothing. These appear to be logically non-permissible from the middle but they become possible when we recognize that they were split from an originally unified state. Similarly, the rejection of all alternatives at the bottom becomes possible when you cannot make out the difference between the two. We will see shortly that this inability to make choices is caused by the inability to distinguish, which is caused because the ‘distance’ between things reduces to a point much smaller than the cognitive ability in the observer to make a distinction.
In conventional dualistic logic, two extremes—everything and nothing—are mutual opposites. In between everything and nothing lies something. This something is comprised of three aspects, which are called the three modes of nature. Thus, everything built from the three modes of nature is something; we cannot call it everything and we cannot say it is nothing. Each something embodies an idea or manas. The thing is vāk, and by embodying the idea it becomes a symbol of the idea, produced from the idea by a process or prāna. Therefore, the same word can denote the idea, process, and thing, although they are not the same. The concept is yellow. The activity or process is magenta. And the object that embodies the concept is cyan.
The Role of Concepts and Logic
In modern science we see objects, but we don’t recognize the reality of concepts and processes. Without the existence of concepts, and their unique implications to arithmetic, we won’t have a hierarchical tree structure. We don’t need to deny the objects; we must realize that this objectivity is the last step in production; the first two steps are concept and process. The concept is the entity that is being hidden and the process is selecting the part to be hidden. The result of that process is that we see a part of the whole.
The part that we see as an object is an effect of the concept and the process of hiding. In that sense, vāk is less relevant to the explanation relative to manas and prāna. We might also say that vāk is the final empirical evidence that the process has been completed. It is required for empirical verification but it is totally irrelevant from the standpoint of explanation.
The irrelevance of vāk arises as the same empirical evidence can be produced by taking a different whole and hiding different parts in it. For instance, your vision of a square object can be created from hiding different parts of a triangle and a circle. Just because you see a square doesn’t mean you know how it was created. However, if you know the whole and the process of hiding, then you will know the outcome to be empirically measured. In that sense, the science of objects (i.e. modern science) can be completely replaced by a science of concepts and processes. Modern science sees squares and thinks that they are real. A new science will say that some part of a circle was hidden to create a square. Both can be empirically confirmed but the latter is more powerful because you can keep the same reality (manas) and apply a different process (prāna) to see something else.
The Ideal Thing
The modern notion of numbers came from set theory. A set is an unordered collection of objects, which means that you can claim that there are N objects but you don’t know which object is first, second, third, etc. For that claim to be substantiated, you have to actually order them from 1 to N, which means that the number of objects in a set is known only after ordering.
Ordering requires concepts and processes. For instance, if you have a collection of horses which you call the set ‘horse’, then to order the objects in that set you would typically begin with the best horse as the first member of the set, the next best horse as the second member of the set, and so forth. In this act, you need to know two things—(a) the definition of horse which identifies the ideal horse as the concept of ‘horse’, and (b) and a comparison procedure by which the ideal is compared against the other non-ideal horses.
In simple terms, you need to establish an origin of the set which constitutes the ideal horse, and a metric from the origin which identifies how far an actual horse is relative to the ideal horse. If the origin and metric have been defined, then ordering is straightforward, and once ordering has been done then we actually know how many objects exist, which leads to the number of members in the set. In essence, a set by itself cannot tell us how many members it has; we need to add a geometry with an origin and a metric in order to distinguish and order the members in the set before counting them.
Thus, we require three things to count—(a) origin, (b) metric, and (c) objects. The first is the definition of the ideal horse, the second is a process of measuring horses, and the third is the actual horse. To know an individual object, we must know the origin and metric prior. These three things are known in Sāńkhya as manas or concept, prāna or process, and vāk or object.
The Absolute Truth
When set theory talks about collections, it speaks of the complete before counting. To actually count, we need three additional things as noted above. The first of the three things is the ideal thing. There is a difference between the collection of all horses and the first ideal horse—origin. Śrīla Prabhupāda explains this idea in the introduction of Śrimad Bhāgavatam as follows:
The conception of God and the conception of Absolute Truth are not on the same level. The Śrīmad-Bhāgavatam hits on the target of the Absolute Truth. The conception of God indicates the controller, whereas the conception of the Absolute Truth indicates the summum bonum or the ultimate source of all energies.
The Absolute Truth is the complete, or what mathematicians call the ‘set’ of all things. However, God is the first member of this set, or the ideal thing which becomes the origin in relation to which all subsequent things are measured. Thus, there are two ideas—(1) everything, and (2) the ideal thing. The ideal thing is the first member of everything, but everything is logically prior to the ideal thing. Therefore, Śrimad Bhāgavatam is speaking about everything. Theologically, Kṛṣṇa is everything, and Balarama is the ideal thing. Balarama is the first manifestation from Kṛṣṇa which means the ideal thing manifests from everything. Since Kṛṣṇa is everything, He has darkness as everything includes the dark side. Balarama as the ideal thing is yellow.
The Mechanism of Counting
The ideal horse has the same three aspects. It is a definition of the ideal, which constitutes the manas; it measures against itself which is the prāna, and it is the instantiation of the ideal which is vāk. When only the ideal exists, then manas, prāna, and vāk are not separated as the definition, process of measurement, and the object being measured are the very same thing. As successive objects are manifest, then we know that the first object is manas, the second object is vāk, and the relation or distance between these two objects is prāna. The three modes were inseparable in the ideal object and they are separated when subsequent objects are created.
The ideal object by itself combines manas, prāna, and vāk. But, when we speak of the ideal in relation to the subsequent objects coming from the ideal, then it is only manas because the other objects are vāk and their relation to the ideal is prāna. In this way, God is described to be fully satisfied in Himself, and yet He engages in relationships with other persons.
Every successive object is also a combination of manas, prāna, and vāk. It embodies an idea, which becomes the origin of subdivisions, in relation to which the subdivisions are located. In this way, each thing is a combination of manas, prāna, and vāk, and yet they are organized hierarchically.
Finite vs. Infinite
Everything is inexhaustible. However, parts of everything are finite and can therefore see an end. This idea can be expressed very simply by saying the tree has infinite branches but each of the branches of the tree is itself finite.
A universe is a branch and it is finite. But there are infinite universes. Therefore, we can say that the full thing is unlimited, and yet each part of that thing is limited and has an end. The soul is also a branch of the tree. Since the branch is finite, the soul can know the path to God. The soul cannot, however, know all the branches. In that sense, while the soul can know the complete, he is not omniscient. The difference between the soul and God is that the soul knows one branch and God knows all branches.
Different Classes of Numbers
The Reality of Fractions
All positive and negative numbers are defined between 1 and 0. However, the whole numbers only represent the order among the fractions. Fractions are therefore real, and negative numbers are also fractions. The fraction is defined in relation to the whole; i.e. we can say how small the fraction is relative to the whole. Now, we can map the natural numbers to fractions. For instance, 2 can be mapped to 1/2, 3 to 1/3, 4 to 1/4, etc. As the natural numbers get larger, the fractions get smaller, but we can never reach 0 through the successive subdivisions. In that sense the limit of natural numbers—infinity—is equivalent to the limit of fractions, namely 0. While infinity is unbounded, zero is bounded. If you were asked to go till infinity you will not know the destination. But if you have to go to zero you will at least know the destination even though you can never reach it.
In one sense, therefore, even the present universe is infinite because you can never reach the boundary. And in another sense, the universe is finite because there is a definite boundary of the universe. Therefore, when we say that each branch is finite, we simply mean that it is bounded.
This idea is clearly demonstrated by Zeno’s Paradox in which you start from 1 and head towards 0, but there are infinite steps because you subdivide the steps to the destination through fractions. If the distance between 1 and 0 was indeed infinitely divisible, then you can never reach 0 (which is what Zeno argued) because there are infinite steps to that destination. In modern mathematics this is sometimes called a bounded limit; to reach it infinite steps are needed even though there is a definite boundary.
The Problem of Missing Fractions
Through this method of mapping natural numbers to fractions, we miss out on many fractions. For example, we cannot reach 2/3 = 0.666 through such a method because 1/2, which is the first fraction in the list, is 0.5, and it is already smaller than 0.666. Similarly, 4/9 = 0.444 is smaller than 1/2 but larger than 1/3. The conclusion is that there are many more fractions between 1 and 0 than all the natural numbers. Therefore, if this universe were countable as 1, 2, 3, etc. and it began in 1, then many numbers will never be possible. This now constitutes the problem of missing fractions.
This is however not a real problem because we are counting within a branch, but there are infinite branches. A fraction such as 2/3 can be understood if the numerator denoted the order of the branches while the denominator denoted the fractional part of that branch. To allow for this, we will have to allow for infinite branches (as we saw above) and yet each branch having a limiting end in 0. This would mean there is no location called 2/3 in the first universe, because that location exists in the second universe.
|1||DEPTH WITHIN A BRANCH|
|BREADTH OF BRANCHES||1||2||3||4||5||6||7||8||9|
We can see how everything is 1 because it is the first thing, and all the other things that follow are considered fractional parts of that whole. Thus, everything represents both cardinal and ordinal 1. Thereafter, there is the ordinal 1, which is a part of the whole, but the ideal part. From this ideal part we proceed horizontally and vertically. The vertical process creates the non-ideal from the ideal, and the horizontal process creates subparts of the ideal and non-ideal parts. The vertical process is creating numerous branches of the tree, and the horizontal process sub-branches of each branch.
Both contribute to the fractionalization of the whole—the horizontal process changes the numerator and the vertical path changes the denominator. As a result of applying these two processes we obtain all the fractions.
Cantor’s Diagonal Argument
In 1891 Georg Cantor produced a similar (though not identical) argument depicted below. Let’s look at the similarities and the differences.
The point of Cantor’s argument was that all the rational numbers can be sequenced in an order, and thereby 1-1 mapped to the sequence of natural numbers. He claimed that the set of rational numbers has a size equal to the set of natural numbers. This seems rather unintuitive because we know that there are many fractions between any two natural numbers.
In Cantor’s argument this problem rears its head because each number—such as 1—if viewed cardinally is counted infinite times: 1/1, 2/2, 3/3, etc. Similarly, 2 is counted infinite times as 2/1, 4/2, 6/3, etc. In effect, all the whole numbers are counted infinite times. Similarly, all the fractions are counted infinite times—e.g. 1/2, 2/4, 3/6, etc. So the list above is ∞ * ∞ for cardinal numbers. Similarly each natural number creates an infinite fractions (numerator and denominator), so it is also ∞ * ∞ although the whole number infinite is contained in the infinity of the fractions.
Cantor’s claim was that all these numbers can be counted like infinity, so in a weird sense of sizing the set, collectively they have the same ‘size’ as ∞. Cantor then showed that the irrational numbers (e.g. π) were uncountable and therefore had a bigger size—2∞. Clearly it turns out that 2∞ would be much bigger than ∞ * ∞, but this is a distinction that can wait for a bit.
A Precise Definition of the Tree
To address the problem of including the same fraction many times, we have to say that 1/2 is actually different from 2/4. The number 1/2 is the second member of the first branch, while the number 2/4 is the fourth member of the second branch. In other words, while 1/2 and 2/4 are fractions of the whole, they are in different places on the tree. Similarly, all the branches are parts of the complete whole. While the members of the branches are denoted by using numerator and denominator, to depict all the branches as parts of the complete whole, we need a third level that denotes the whole.
This is eminently possible if we use the triad of manas, prāna, and vāk, wherein the whole is the original idea, each branch becomes the metric relative to the whole (which denotes the numerator), while the individual parts on a given branch become the individual objects distinguished using that metric (which represents the denominator). The diversification of the root would then be a consequence of defining multiple metrics which produce many branches, and then many leaves on each branch.
We can now give a very precise definition to the tree. There is a whole which we call its root. Then we have many individual metrics which are used to divide this root. Then we have the individual objects produced as a consequence of this division of the whole using the different metrics.
Atomism to Time and Operations
Since manas, prāna, and vāk are hierarchical, as we traverse this hierarchy from top to bottom, the ideas, methods, and objects get smaller. The incremental reduction constitutes the march from everything to nothing. There is a limit to this reduction which constitutes atomism of ideas, methods, and objects. Depending on the scale employed, atomism may be attained quickly (if the scale is large) and slowly (if the scale is small). The variety in the universe depends on how small the scale is. If we measure something with a foot, we get more feet than if we measure with a meter.
Accordingly, the universe is larger when our scale of measurement is small, and smaller when the scale is large. Each of the branches of the tree can be viewed as a universe; the growing order in the universe implies the use of successively smaller scale, and therefore the successive universes are also larger. A smaller scale means a greater capacity to zoom in, but a limited capacity to zoom out. A larger scale means a limited capacity to zoom in but greater capacity to zoom out. Thus, in the universe with larger scale, it is easier to see the big picture and harder to see the detailed picture. Conversely, in the universe with the smaller scale, it is easier to see the detailed picture but much harder to perceive the full reality.
The present universe is described as the smallest universe, which means it is the first in the order of universes, and therefore has the largest scale of measurement. Using this largest scale, it is easier to know the whole (which is why the universe will appear to be the smallest). In this universe, it is harder to know the smallest possible atomic reality because the senses are ‘blunt’ instruments to measure atoms. But the same senses can understand the whole reality much better. In other universes, they can understand the atomic reality much better, but it is harder to know the whole truth.
The Universe is a Half-Empty Glass
The universe in Vedic cosmology is described as half filled with water and the other half is planets floating like particles in the air. The planets are vāk, the air is prāna, and planets hang like light bulbs of a chandelier from a roof called manas. But this is half the universe, which comprises all the possible ways in which the whole can be hidden to create individual objects. Not every method of hiding is however real at all times. For instance if you see some empty land, there is possibility that it can get a building in the future. Similarly, a place where there are buildings can become an empty ground later.
All the elements of the tree can be counted, but they are not always visible which means these branches are not occupied by any soul. Time causes them to manifest and unmanifest. If there are N individual possibilities, then there are 2N combinations of possibilities, which are manifest slowly over time. The lower half of the universe constitutes the 2N irrational numbers (as we discussed from Cantor’s theorem earlier), and each such number represents the state of the entire universe at a given time. You can imagine a glass half full from which one particle of water evaporates every moment and creates the entire visible universe. The very next moment, the particle condenses and goes back into the lower half and a new particle evaporates.
There is hence a cycle of rainfall and evaporation inside the half-empty glass. When we reach the point we call ‘nothing’ we have reached the limit of spatial division. Beyond this is a world of irrational numbers, equal to all the combinations in the upper half. These numbers are real, but we cannot perceive them because knowing each such number is equivalent to knowing the complete state of the universe at any given moment.
Space and Time Trees
The upper half of the universe represents locations in space and the lower half represents instances in time. Both space and time are hierarchical, so the bottom half is also like a tree in which the higher nodes represent larger spans in time while the lower half denotes parts of these larger time spans. The universe is spherical, which can be understood to indicate that the time tree is upright while the space tree is inverted. Thus in the time tree the lowest element represents the entire duration of the universe, whereas the lower nodes in the same tree represent smaller time durations.
The entire past, present, and future of the universe are therefore real. But we cannot experience this past and future unless we have a sense perception capable of seeing beyond the countable individual objects. The view from top to bottom represents all the things that will ever exist at any given time. The view from bottom to top denotes all things existing at any given time. By knowing both we know the extension and duration of the universe.
Sub-Structure in the Tree
So far I have depicted the tree simply—a single inverted structure. The real structure is like a Christmas tree with many tiers called planetary systems. It comprises a central ‘stem’ or the axis mundi from which many individual stems protrude. As one descends downward, more of reality is hidden. Similarly, as one goes outward even more reality is hidden. Therefore, the path of advancement is inward and upward—inward on a single plane, and then upward through the higher planes. This model is remarkably similar to the nerves coming out of the spinal cord in the human body.
Śrīla Prabhupāda writes in the purport to SB 3.26.34: Form rests in subtle existence in the sky, and internally it is perceived as the veins within the body and the circulation of the vital air (emphasis mine). This idea allows us to see the planetary systems as analogues of chakra in the body, the number of petals in the chakra representing the number of divisions of the space. While the yoga system speaks about the 7 upper chakra because the goal of the system is to rise, Vedic cosmology speaks about 7 upper and 7 lower planetary systems. Correspondingly there are 7 lower chakra, which together constitute the 14 levels corresponding to 14 planetary systems. Mystic Universe discusses the division of flat plane at greater length.
Varieties of Number Types
Students of number theory know that mathematicians classify numbers into many classes such as Triangular numbers, Square numbers, Pentagonal numbers, Hexagonal numbers, Heptagonal numbers, Octagonal numbers, Nonagonal numbers, Decagonal numbers, Dodecagonal numbers, etc. There are many kinds of integer sequences such as the Fibonacci numbers. The different levels of hierarchy in the tree can be thought of as corresponding to some unique number types, which may include the above mentioned types or other types that have not yet been discovered.
These worlds will be mathematically describable but quite different in their properties from each other because they will correspond to different number types. A detailed study of these number systems, and organizing them along a hierarchy, will constitute a hierarchical number theory, which will also give us an understanding of the differences in the different tiers.
The Use of Complex Numbers
The flat planes that we see above are not physical space because each of sat, chit, and ananda can be further divided into two parts—knowing and acting, denoted by the senses of knowledge and action. Factually the division is only within chit but it can also appear in sat and ananda. Thus there are roles of knowing and acting, objects of knowing and acting, and purposes of knowing and acting. The hierarchy in a tree looked at simplistically is just two dimensions—vertical height and horizontal plane. But the horizontal plane has to be further divided into two dimensions—knowing and acting.
This plane needs complex numbers as each object has two features—it is some properties and it can be used in certain ways. A knife can be used to cut and pierce, and a saw can also be used to cut, while a needle can be used to pierce. If we change the properties perceived by the senses of knowledge (e.g. hardness and sharpness) we also change the uses to which the knife can be put to (e.g. cutting and piercing). However, changes to other properties (e.g. color or smell) may not change the use of a knife. Knowing and acting are hence correlated but they are not identical. Similarly, a plastic knife may be useful in cutting butter, but not in sawing through wood.
Mystic Universe discusses this correlation between the two properties at greater length, and uses them to describe the subdivision of a flat surface into many subparts which correspond to different types of objects and the different uses they can be put to. These divisions correspond (in Vedic cosmology) to different body types which look different and can be used for different types of activities. The body has two kinds of senses (of knowledge and action) which perform individual functions. These two types of senses are unified by the mind which is able to hold both the concepts of knowing and acting. The flat surface is a mental or conceptual plane, not a sensually perceivable surface. The property of this plane is that if you add two numbers you are performing two operations—translation and rotation. This is unlike natural numbers where addition simply translates the number.
The imaginary part of the number is action. As you apply the same action repeatedly, the object returns to its original state, which means that actions rotate the object. Why does this happen? The reason is that we created the object by hiding different parts of the whole. As we perform actions on this object, that action simply hides another part and reveals a different part. This hiding and revealing can happen at various levels (due to hierarchy) but all change appears to be rotation because there are finite number of aspects of the whole which are being hidden and revealed. The situation is like throwing a dice or tossing a coin. Depending on how many ‘faces’ a thing has (based on how small we divided it) each action will reveal one face.
A complex number combines prāna and vāk or action and thing; it says that the object was produced by a process, but it could have been produced by another process. This becomes important when you want to know the location in space and how one reached that place (because you could have used other paths to reach the same place). In hierarchical space, the element on the tree is the place, and the branch connecting it to the higher element is the path. You can never have more than two paths to a destination and hence two numbers are not needed. The confusion arises in flat space as we only see the leaves and by looking at them we don’t know which branch they are connected to. As a result we suppose that they could be connected to any branch. Complex numbers are simply saying that there is an element and there is a connection to the center of that space (which constitutes the higher node). A complex number forces a hierarchical view of the world because you have to represent a path along with an object. It is a caricature of hierarchy and if one employed hierarchy this caricature would not be needed.
The Creation and Destruction of the Universe
We noted earlier how the tree expands when the whole is covered partially creating fractions of the whole. We also discussed at the beginning how adding partial knowledge to the whole knowledge doesn’t change the whole, and since the addition of the part doesn’t increase the whole, the removal of the part also doesn’t reduce it. Thus, during the creation of the universe, some form of śakti is used to expand the universe. This śakti originally existed in the whole as its prāna and constituted the full knowledge of the whole. However, as the śakti divides into smaller parts, the whole is also known and represented partially thus causing the expansion of the tree. Similarly, as the śakti merges the parts into the whole, the entire creation is terminated.
The creation and dissolution of the universe don’t change the whole because they are created as parts without dividing the whole itself, and they are merged back into the whole without adding anything to it. Thus the whole is unchanged by creation and dissolution of the universe. The change is entirely caused by śakti which constitutes a method of knowing the reality.
Mathematics and Conscious Experience
The Philosophy of Perception
Manas is the seer, prāna is the process of seeing, and vāk is the seen. The original seer is one. But there are many methods of seeing due to which many visions are seen. Yoga-māyā and mahā-māyā are two methods of seeing that create two visions. The soul is also a method of seeing that produces a vision. These are respectively also called bahiranga-śakti, antaranga-śakti, and taṭasthā-śakti. These three methods of seeing are like mirrors in which God sees Himself. The result of that reflection is an image inside the mirror and the seer can perceive what is within the mirror. Thus, God enters inside the soul as His own reflection in the mirror. And then God appreciates His own reflection as it is being reflected in the mirror.
Sri Chaitanya calls the soul a mirror when He exhorts the soul by asking for cheto darpan mārjanam, or to clean the mirror such that God can be reflected in this mirror. Here, chitta is the soul, darpan is the mirror, and mārjanam is the cleansing of the mirror for reflection. By comparing the soul to the mirror and its cleansing as the process of spiritual reinstatement, spiritual life is succinctly summarized as the path in which the soul reflects an image of God by holding up a mirror to Him for His self-perception.
The Description of Dṛṣṭa and Dṛṣya
In this process, the real seer or dṛṣṭa is God and the reflection in the soul is the dṛṣya or vision. However, the reflection is also within the soul, because that reflection is precisely what God sees. Therefore, the soul is also a dṛṣṭa because he creates the vision of God within himself. Similarly, the material world is the reflection of God inside bahiranga-śakti, and the spiritual world inside antaranga-śakti. They are different processes of seeing; they see the same object but create a different picture. The reflected object and the seer of that object are the same entity—God—and the process of seeing is simply an instrument for God’s seeing. This instrument has the same three properties as God, and yet the instrument is meant for God’s use.
The rock star is reflected inside the fan. The rock star sees that reflection and says—that’s me! The ideal fan sees the same reflection and says—that’s you! The material perversion is that the fan sees a reflection of the rock star in himself and says—that’s me! The soul has the ability to see himself (instead of God) and that vision of self-consciousness is true, but it is not God’s vision. The ability to have the self-vision and God’s vision are truth. The ability to impute God’s vision on the self is false. In that sense, the material world where the fan considers himself the rock star is an illusion. However, there is indeed a rock star so only the identification with reality is the illusion.
The personal image of the rock star is ādibhautika, the self-image formed in the process of perception is ādiatmika. Similarly, when the rock star sees the fan enjoying, he has a vision of the fan called ādibhautika and the self-image of the rock star in relation to the fan is ādiatmika. Under yoga-māyā there is a distinction between the self and the vision, but under mahā-māyā this distinction is blurred. In mahā-māyā there is a separate ādidaivika which connects the rock star and the fan, but in yoga-māyā this person is always visible as the root of the tree and hence everything is known in relation to ādidaivika.
The Use of Logic in Creation
The śakti represents the different ways to know the whole. However, remember that this knowing is dividing the whole into parts. We can say that it is knowing by analyzing the whole into its various parts; the whole is known to the self, but the parts of the whole are not known unless the whole is analyzed. This analysis is like saying that I have wealth worth millions but whether this wealth is held in property, gold, cash, or land is not known. The purpose of śakti is to detail how all the wealth is actually held, and there are many ways to convert millions into property, gold, cash, or land. In one sense, śakti analyzes the whole into parts. In another sense, the parts are being created from the whole by applying a process of measurement on the whole. In yet another sense, the whole is only taking stock of itself.
Logic is the fundamental tool for analysis, and the division of the whole into parts is the outcome of analysis. The śakti or prāna thus acts as logic, analyzes the whole into parts like cutting up a picture into a jigsaw puzzle. There are infinite ways in which the picture can be cut-up, each of which constitutes a branch of the whole which presents the act of detailing the whole into parts. There are hence infinite śakti each creating a different branch; each branch is a different jigsaw puzzle of the same picture.
The Genesis of Logic
We can divide the infinite representations of the whole into 8 distinct classes if we understand that the division of the whole into parts is a choice and this choice is a property of the soul. The basic mechanism of choice is that the three aspects of the soul are often in conflict, and the conflict is resolved by creating a hierarchy or a dominant-subordinate structure. Sometimes emotion dominates cognition and cognition dominates relation. At other times, this order is changed. Based on the dominant-subordinate patterns, 6 personality types are created, which the book Emotion discusses at greater length. Then there are two other personality types—(1) where we know we have choice but we don’t use it, and (2) we believe we have no choice and hence we don’t use it. The former is called oneness and the latter is called nothingness. The former accepts choice but rejects its use; the latter denies choice itself. The denial of choice is also a choice. Oneness, nothingness, and the 6 personality types create the 8 fundamental types of choice. To divide the whole into parts, we must employ these 8 methods.
When choice is denied or not used, the whole is not divided into parts but it can still be experienced as emptiness and fullness, respectively. The same reality is thus expressed in two ways which correspond to Brahman and a destination of voidism or emptiness pursued by the Buddhists. Far more interesting uses of choice involve the prioritization of the three aspects of the soul which represent right (sat), good (ananda), and truth (chit) and the prioritization creates 6 types of choices, logic, methods of division.
In the material world, truth is subordinated to good, which means that truth is relative to our goals; we choose the goals, which helps us choose the beliefs, and those beliefs confirm/deny the truth. However, the good is subordinate to right; which means that I have to choose the happiness within the bounds of my role; I cannot enjoy by violating my duty. This is well-known as the dilemma between choice and responsibility and becomes the cause of karma: if you violate your duties to enjoy you will have to suffer. This structure of choice is reflected in matter when the moral sense is placed higher than ego or desire, which is higher than intellect or beliefs. In essence, sat dominates over ananda, and ananda dominates over chit.
Now imagine another type of world in which truth dominates over desire and desire dominates over duty. The truth would indicate all that exists. We will perceive this truth which would then lead to a desire within us. Whatever that desire is, it then becomes our duty. For example, in such a world, you could walk into a fashion store where the clothes displayed are the truth. As you look at these clothes you develop a desire for them. But desire fulfillment is now your duty, which means that you can rightfully take whichever clothes you like and walk out of the store, without paying for them in return, because you are just performing the duty of fulfilling your desires and if you don’t desire to pay the shop owner any money then you are not morally obligated to do so because your duty is whatever you desire. Does this sound illogical? It is not illogical to imagine such a world, although it is not the logic of this world. Vedic texts describe that there are worlds where you can take whatever you like without having to pay for it in return, because there is no moral obligation above and beyond the fulfillment of desire itself.
Therefore, the logic of that world is different from that of this world. In this world, duty rules over desire and desire rules over truth. In another world, desire can rule over duty, and duty can be ruled by truth. In effect, what we call śakti is simply choice of how we decide to order the three facets of the soul. When the soul is in the material world, he has to prioritize duty over enjoyment, and enjoyment over truth; this prioritization is reflected when morality is higher than ego and the ego is higher than the intellect. However, in other worlds—governed by different logics—the order can change.
Of course, this doesn’t deny the possibility that you could have a different personality that prioritizes truth over good and good over right in this world. But it is not the ideal personality within the material world because you will often neglect your duties in order to pursue enjoyment. Similarly, you might sometimes deny that you have choice, or accept that you have choice but refuse to use it, and remain in an indecisive state. The soul is however not independent of the world; the world is governed by a particular type of logic and the soul has to follow that logic too. If he desires a different logic, then he has to finish the pending business of this logic—the balance of accounts—and then go to another world. All these worlds can be described logically because logic is a choice, and choice is a property of the soul which is used to resolve the inner contradiction between sat, chit, and ananda.
The above 8 methods of seeing the same reality constitute the different worlds, of which the entire material world is just one. The whole truth is manas, the type of logic is prāna, and the world created by that logic is vāk.
The Three Principles of Logic
Material logic is defined as a set of three rules—identity, non-contradiction, and mutual-exclusion. The identity in material logic is the counterpart of sat, which creates the “I am” of the soul—the soul enters a ‘role’ such as parent, employee, citizen, etc. Then, non-contradiction constitutes the chit which creates the “I have” of the soul—the soul gets a body comprised of parts which are also logical opposites (e.g. front and back, head and foot) and yet these opposites are separated by space as different locations and parts. Finally, mutual-exclusion constitutes the ananda of the soul which creates the “I want” of the soul—the soul develops desire and need for pleasure, owing to which choice becomes a necessity and you cannot rule out all the alternatives. If you are presented with vanilla and chocolate ice creams, then non-contradiction means you cannot choose both at the same time, and mutual-exclusion means you cannot choose neither of the alternatives. Once both and neither are rejected, you must choose one. This choice may not be compatible with your identity—i.e. your role—and if there is inconsistency between the role and the choice, then a consequence will be created.
Effectively, you must make a choice (not choosing is not an alternative), and you must choose one alternative (choosing everything is not allowed), and whatever you choose will be compared to your identity. In modern logic, identity is defined as the truism ‘A is A’ which is to say that you can never deny your identity or claim something contrary to that identity. This is an erroneous claim because the choices you make may be incompatible with your identity, when that identity is defined not as your personality but as a role. The fundamental question is: What is identity? Is the identity of an entity defined by itself (which would mean that the object is independent of other objects)? Or, is the identity of something defined in relation to the original thing? In modern logic ‘A is A’ entails that my identity is defined by me, and hence I can never contradict myself. That is false on two grounds. First, my identity is defined in relation to others (e.g. parent, citizen, employee, etc.). Second, I can contradict myself in the future; or I can make self-contradicting statements right now: “This sentence is false”.
Thus, the fundamental problem in modern logic is the rule of identity which is used to supposedly create a consistency when the fact is that we are often inconsistent with ourselves. We are also—as the conflicts and wars of this world suggest—often contradicting others. To understand this inconsistency we have to go back to the issue of internal contradictions in the soul. The ananda or the desire of the soul makes a choice out of competing options. However, this choice is not necessarily compatible with our role or duty. Identity is true when our choices are subordinated to the role. However, the identity rule fails when the choices are incompatible with the role.
Everything in logic changes when this incompatibility has to be accounted for. For instance, the concept of moral consequence emerges when choices are incompatible with our duty. This moral consequence changes our role (e.g. as a punishment) and even more incompatibilities may then be created between what we want and what our situation or role will allow.
Modern logic views the world as one big consistent reality disregarding the fact that meaning is created through opposites. This error exists in the principle of non-contradiction where everything in the universe must be consistent, which entails the rejection of meaning created by opposites.
Thus out of the three principles in modern logic, two of them are patently false. The failure of non-contradiction represents the resurrection of meaning, and the failure of identity denotes the idea of moral consequences based on the interaction between choice and role. Even the third principle of mutual-exclusion—which leads to choice—is presented as a necessity but it is not necessary because choices are based on goals, and by discarding goals and the quest for happiness we can in fact stop making choices. Thus, every fundamental principle in modern logic is flawed—partially or completely. The rejection of meaning and morality is ensconced in modern logic itself.
The rule of non-contradiction must be modified to mean that while opposite alternatives will exist in the universe (and hence the universe as a whole is logically contradictory), I cannot choose the contradictions simultaneously. I may choose them one after another, or different persons may choose them simultaneously. The former entails that non-contradiction is subject to time (because contradictory choices can be made at different times) and the latter means that non-contradiction is subject to space (because contradictory choices can be made by two separate individuals simultaneously). The universe as a whole—and by that we mean different souls—can be logically inconsistent with each other, but I am not allowed to make opposite choices because I can only be situated at one place at one time. Violating that principle requires me to be present at multiple locations simultaneously and that is forbidden because the soul is atomic. Non-contradiction therefore follows from the atomicity of the soul, just as the necessity for choice follows from the desire for happiness which compels one to make choices.
Once this new foundation of non-contradiction is grasped, then we can speak about the problem of identity—namely that my choices as a human can be incompatible with my role as a citizen. My identity is the role—e.g. a citizen. In Vedic philosophy this is also called Svarūpa (sva = self, rūpa = form), and the Svarūpa is defined in relation to God as a relationship (e.g. of a servant or friend). Being situated in that identity simply means that my choices are compatible with my role. If that identity is broken, then I make choices inconsiderate of my role. A moral law must now apply. This moral law is not like modern scientific laws presented as formulae, which in turn assume all of mathematics and logic. Rather, morality is a logical principle and it emerges from the fact that ananda is subordinate to sat which means that my desires can be subordinated by my duties, but my duties cannot be subordinated to my desires. In simple terms, we must always do our duty even if we don’t like it. Not doing so constitutes the violation of the identity principle. If I enjoy my duties, then the contradiction between ananda and sat is resolved. To live free of karma therefore one has to perform their duties happily.
Modern logic is necessary, but the above mentioned logic is normative. Its individual principles can be violated—e.g. that I may not do my duty—but logic as a whole is necessary because violation creates consequences. Thus, every principle in logic is not always upheld. However, collectively logic is always upheld. The core principles of logic—namely identity, mutual-exclusion and non-contradiction are purely normative and not necessary. That is, we are expected and supposed to be logical according to these core principles but we can act illogically—that illogical action is our free will. The subsequent fallout of this illogical choice is necessary and not normative. That is, the moral consequence is not just something that is supposed to happen; it is necessary, which means that it will always happen.
To the above three core normative principle of logic, therefore we can add a fourth necessary principle which we can call the ‘action-reaction’ principle. This is a universal principle which depends on the gap between the choice and the role and effected when the normative identity principle is violated. Then the reaction to the action doesn’t act immediately, so we must add a fifth necessary principle called ‘delayed-action’ that delivers the consequence of a previous normative violation after a certain gap in time. The action-reaction principle is a feedback loop, and the delayed-action principle causes this loop to cause oscillation which manifests as various cycles.
In modern mathematics theorems are eternally true, and applicable to all times and places, and to describe change we have to separately compute the laws of nature, which requires a computer to calculate the outcome, which in turn needs some material system, which must in turn be governed by some logic, which also needs to be computed on a computer governed by logic, ad infinitum. Logic and mathematics don’t explain how things are changing because they never describe how logic and mathematics is computed. The system we describe above is self-computing. We don’t need a law which is then computed assuming input data. Rather, our choices are the input data and moral consequences are the output which again drive the input. In one sense, the driver underlying this logical system is our choice; it is not electricity or material force, and as long as the choices exist, the system can keep running forever. Therefore we need only one idea to describe why nature works perennially—namely that the soul is eternal. When the soul leaves the body, the bodily computer stops working gradually, because the logical system was being fed by choice and their consequences.
The Five Principles of Logic
Thus, we can define the scope of logic as the following 5 principles:
- The principle of the eternal possibility of choice,
- The principle of individual non-contradiction,
- The normative principle of identity,
- The action-reaction principle,
- The delayed-action principle.
The principle of mutual-exclusion is called ‘choice’ above. The principle of non-contradiction is matter. The identity principle represents God because the soul’s role is defined only in relation to God. The principle of action-reaction is called karma. And the principle of delayed-action is time. This the scientific-logical framework in which we need to think of a new logic—it is representative of five key ideas—soul, God, matter, time, and karma. This logic has normative and necessary parts, and it is free from the contradiction between necessity and choice. Similarly, logic is also self-sufficient because it doesn’t need an external computational device to find an outcome. The principle of mutual-exclusion or choice is itself the computer.
Clearly, the last two—karma and time—arise when the principle of identity is broken. If these principles were upheld, either because the morality was subordinated to desire or truth, or because one enjoyed their duties in the material world, the last two principles will cease to be operative. This cessation is the logical explanation of salvation by which one can exit the material logic. That exit is not a denial of logic; it is entirely within material logic, and it is also supported by alternative ‘transcendental’ logics.
These five principles are embodied in this world as the five prāna or ‘forces’. In some transcendental worlds (beyond the material world and different from the places of oneness and nothingness), only three of the above five principles will exist, and we can call them three divisions of śakti. In the realms of oneness the principle of śakti is understood but not differentiated into separate principles. In the realm of nothingness, śakti is not used. In that sense, the different worlds are different because of different śakti. The material world is also working according to logic, with two differences. First, this logic is different than stated in the modern understanding. Second, the logic is itself the force that drives the universe, so we don’t need a computer to emulate that logic, which in turn needs some operating logic, etc.
The prāna is divided into five parts called—prāna, apāna, samāna, udāna, and vyāna. The power of prāna is the power of choice or what we call mutual-exclusion by which we must make a selection. To live means to make choices, and therefore prāna is the essence of life; it manifests in the body selecting—e.g. in accepting food, inhaling, and hearing. The power of apāna represents individual non-contradiction; which means that if I select X then I must reject its opposite. This power manifests in the various excretory processes in the body. The power of samāna is the principle of identity by which we compare the choice against the role to find a compatibility; it is said to be the ‘digestive’ power by which the consumed food is reconciled by the digestive system. The power of udāna is the creation of a consequence if an incompatibility is found; this power manifests in our urge to speak because we are unable to hold the contradiction within ourselves. The power of vyāna is the motion or change caused when the consequence manifests; it the main cause of the transmigration of the soul, or any kind of motion.
Two Kinds of Restrictions
Once logic has been created, to reason, we must also create symbols. The role of śakti now changes from defining the type of logic to defining the origin, metric, and objects. Quite specifically, we move from doing logic to doing mathematics, thereby introducing space, time, and objects. Due to the change in role, śakti is now called prakriti or the creatrix. Manas, prāna, and vāk act again, but this time manas is the ideal object, prāna is the metric or relation to this ideal object, and vāk is the individual objects.
There are hence two kinds of prāna. The first prāna is śakti which creates one of 8 types of logics based on the nature of choice of the soul. We discussed this śakti in the previous section. Then there is the prāna which produces the negation or hiding thereby creating a manifest tree from a root. The latter form of prāna is also called prakriti in contrast to śakti. The śakti is also a type of negation in the sense that it creates a world in which only certain types of choices are idealized. The material choices for instance dominate in sat and subordinate chit and ananda. Subsequently, this restriction on choice is used to further restrict the complete truth by creating smaller parts of the whole truth such that the choices are further restricted until we get to the bottom of the universe where there is no choice—only determinism.
Logically Sizing the Universes
We must note that the ideal object can be defined only after the logic of the world has been defined. For instance, in the material world, since morality rules over desires, an ideal person is one who controls His desires through principles of morality. Since morality is defined by austerity, truthfulness, cleanliness, and kindness, the ideal person will embody renunciation, knowledge cultivation, restrictions on sexuality, and charity to others. In a different world, the definition of the ideal person will be different. The ideal person may for example simply play instead of working hard, may indulge in deception instead of being truthful, have numerous amorous partners instead of abstinence, and be competitive instead of being kind.
Sakti and prakriti are successive stages of the same power. In the spiritual world, Hara is śakti and Ramā is prakriti. In the material creation, similarly, Pārvati is śakti and Durga is prakriti. We can say that śakti is the power that controls the entire material creation (all the universes) and prakriti is the power that controls a single universe. The region outside all the universe is described as a material ‘ocean’ but it has no material objects. It is the domain in which sat is higher than ananda and ananda is higher than chit. Entering this domain means a soul wants to enjoy but also be morally responsible. Entering a particular universe, on the other hand, means giving a definition to morality—e.g. how many values constitute the ideal morality? Similarly, how many pleasures are to be considered primordial? Likewise, defining how many unique concepts are sufficient to form a complete theory?
These definitions provide the number of directions in space. When the space represents cognition, then the number of directions in space represent the number of fundamental concepts necessary to form a theory. When the space represents pleasure, then the directions denote the fundamental pleasures of life. When the space denotes relations then directions denote the basic types of moralities and duties within the society. Vedic texts describe that the present universe is the smallest, and the planar surface in this universe is fundamentally divided into four directions called—East, West, North, and South. Due to this fact, there are four basic moralities (austerity, cleanliness, charity, and truthfulness), four basic types of roles in society (Brahmana, Kshatriya, Vaisya, and Sudra), four basic divisions of time into ages (Satya, Tretā, Dvāpara, and Kali). Similarly, there are four kinds of pleasures of life, namely, eating, sleeping, mating, and defending; life as a whole is divided into four phases—Brahmacharya, Grihastha, Vānaprastha, and Saṃnyāsa. Finally, the space of concepts needs minimally four basic concepts; those familiar with atomic theory will know that these four concepts are energy, momentum, angular momentum, and spin. Life is also modeled as 4 primordial goals—dharma, artha, kāma, and moksha.
These divisions are unique properties of this universe, but they don’t apply to other universes. In other universes, there are more divisions of society, more stages of life, more distinct moral values, more basic types of pleasure, more properties are necessary to formulate a theory, and life has more goals. The smallest universe is the simplest because it involves the least number of divisions. Cosmic Theogony discusses in greater detail the basis of this division by 4. The basic reason is that there are 4 kinds of relationships (one-to-one, one-to-many, many-to-one, and many-to-many) which create dharma.
Space can be divided into more and more directions if the ‘many’ in the above relations is treated as 2, 3, 4, etc. In the smallest universe, all these distinctions are collapsed, and hence ‘many’ includes 2, 3, 4, etc. However, the successively larger universes will divide the same plane by 8, 12, 16, or more parts. Then the same material elements will be divided into a greater number of parts, and the same pleasures will be further divided into a greater number of pleasures. The smallest universe is defined by the 4 directions because they constitute the simplest meaning of ‘many’—i.e. it includes everything greater than 1. In the next larger universe, we can separate out the relations of 1 and 2, and use ‘many’ to denote 3 and above. In the next higher universe, there will be separate relationships for 1, 2, and 3, and ‘many’ will be used for 4 and beyond. Each universe therefore simply redefines the meaning of ‘many’ and thereby expands gradually.
The Significance of Four Directions
We saw above how the smallest universe has four directions, and the ideal must therefore have four facets. The meaning of the ideal having many facets is that these facets can be self-contradictory. For example, in this universe both truthfulness and kindness are moral values, but they can easily conflict. Thus if a killer asks you of the whereabouts of a person who he intends to kill, should you demonstrate truthfulness and tell him about the person’s location, or should you show compassion and lie about the person’s location in order to mislead the killer? If compassion were always superior to truth, then you could save all criminals out of compassion by lying to the police. If truth were always superior to compassion, then being truthful to a killer and telling him of the whereabouts of the person he intends to kill would be morality. Thus, truth and kindness are often contradictory moral values; you cannot practice them always as they are going to conflict. In such cases, you must prioritize them—e.g. compassion over truth, or vice versa.
The ideal situation is one in which all the four moral values are present, which lies at the center of the space. As we move out of the center in any of the four directions, we will find truth without compassion, or compassion without truth. This indicates how the imperfect is created from the perfect. The combination of truthfulness, cleanliness, austerity, and kindness is perfect; however, parts of that are imperfect. The world appears to have evil, and this evil is created from good. The method of creating evil is simply to take parts of the good in isolation which is achieved by moving away from the center. Now, if we embody truth and neglect compassion, the fact that compassion is missing can be construed as cruelty. Similarly, if we show compassion and neglect the truth, the fact that truth is missing can be viewed as cheating. As we saw earlier, the missing ingredient can be regarded as the denial of the ingredient. As more things go missing, more and more of the ideal can potentially be denied. As a result, some truthful people will argue against compassion, while other compassionate people will argue against truth. They forget that both positions are not ideal because the ideal lies not at the edges or even in the middle of the flat space, but its center.
As we move out of the center, we increase our distance from the ideal. This is commonly seen in many religions where compassion may be advocated but truth may be neglected. Alternately, one may be compassionate to some and cruel to others; truthful in some cases and deceitful in others.
The Creation of the Alphabet
The chit comprises of six qualities: knowledge, beauty, fame, power, wealth, and renunciation. The chit is also divided into acting and knowing, which are also called prāna and vāk. As we saw earlier, the prāna acts as negation and the vāk as assertion; they are respectively the feminine and masculine. The 12 divisions of the zodiac represent the six qualities of chit in both assertion and negation. This completes one circle, and the circle represents knowledge and action, assertion and negation, masculine and feminine. The 12 divisions of the zodiac, however, constitute the ādidaivika portion of chit which becomes the controller or administrator of cognition or chit.
The chit also divides into ādiatmika and ādibhautika or knower and known, which are the administrated parts of cognition or chit. The known reality is represented within the knower as a symbol of reality, which means that there is a real thing externally and there is a representation of the known in the knower. The known thing and its representation are not identical, as a result of which we must distinguish between the reality and its knowledge, although the real thing is also knowledge (if it is completely known). Thus, aside from the fact that the whole is itself dividing into parts to create the various knowns, the knower is also looking at the known through different relationships or perspectives in order to obtain a subset of the knowledge. Owing to this separation, there is a further distinction by the knower and known, which doubles the number of divisions from 12 to 24.
Then the 16 types of pleasures are enjoyed through the 8 levels and the 24 divisions of each knower into ādiatmika and ādibhautika. The sum of these categories—8 + 24 + 16 = 48—constitute the elementary alphabets in terms of which everything has to be described. Aside from these are two other categories—the śakti and God—which make a total of 50. God is the ultimate male and His śakti is the ultimate female. As seen above, this śakti is the five-fold logic, which binds God into a cosmic ‘egg’ restricting His choices.
Out of the 50 alphabets, 25 alphabets represent the 24 elements of Sāńkhya and the 25th alphabet represents the whole truth. These 25 elements are represented by the 25 consonants in Sanskrit. As we have seen above, this pattern of 25 categories was seen previously in the 24 transcendent forms and the 24 incarnations in the material universe, followed by Kṛṣṇa as the complete whole. The 24 elements of Sāńkhya are therefore not just material elements. They are actually semantic categories or divisions of chit which constitutes all that can be known and done by the conscious observer.
The 16 forms of pleasure are denoted by the 16 vowels. The 8 types of awareness are represented by the 8 semi-vowels and sibilants. This makes a total of 49 alphabets. Finally, śakti as the logic which restricts the remaining 49 categories is the 50th alphabet. This śakti is originally three principles of identity, non-contradiction, and mutual-exclusion which are represented by the 3 conjuncts. But in the material world, there are two additional principles of time and karma represented by श्र and ळ. If we keep the 5 divisions of śakti aside, then the alphabet has 50 letters. If these divisions are included, then the alphabet has 55 letters. For the sake of our discussion, we will restrict ourselves to the 50 alphabets. In the yoga system, 48 of the 50 alphabets are represented in the 6 lower chakra and the top chakra represents God and His śakti or the union of male and female. Therefore even if we understood all the material categories we would have obtained only 48 out of the 50 principles. Our knowledge will remain incomplete because the most essential two alphabets of language represent God and His śakti.
The ādidaivika or demigods represent this symbolic structure partially. They oversee the 8 roles, and 16 pleasures, but out of the 24 they only comprise the 12 divisions of chit. As a result, the demigods are 8 + 12 + 16 = 36, which are manifested from Brahma, Viṣṇu, and Shiva respectively. If these three deities are excluded, then the total count of demigods becomes 33. The 36 deities of the material world are divided into 8 Vasu (manifest from Brahma), 12 Aditya (manifest from Viṣṇu) and 16 Rudra (manifest from Shiva). Their enjoyment is in controlling the connection between knowers and knowns. Like Kshatriya in the material world control the economic activity between producers and consumers, similarly, the demigods are the Kshatriya of the universe. They are not producers and consumers directly.
Over and beyond the demigods and the three deities from whom they are manifest, there is the ultimate male and female of the material world called Param-Shiva and His consort Pārvati—She is the śakti or logic of this world who restricts the male—Shiva—to behave according to Her dictates. Her rules constitute the rationality of the material world. The combination of Shiva and Pārvati creates the other 48 categories and 36 demigods. Together, 48 and 36 constitute the 84 principles of creation, which become the basis of the 8,400,000 lakh species of life that pervade different parts of the universe.
The crown chakra is also called sahasrāra or the thousand petal lotus obtained after time and karma are removed from the logic. The 8,400,000 lakh species on the other hand are created with these principles. This is probably an indication that each logic principle creates a 10-fold division because the entity without time and karma is said to have 1,000 divisions and the one with those divisions has 100,000 divisions. This is a topic for further investigation when we delve into the nature of logic in a separate post.
The Nature of Mathematics
The collection of alphabets and the logic employed to combine them into sentences constitutes the scope of ‘mathematics’. It is contiguous with matter, conscious experience, and even religion, because its alphabets are based on the division of the whole into parts and ordering the parts in relation to the whole. Current mathematics (because it only considers cardinals) describes the world in a materialistic way—it views things as independent objects rather than parts of a whole, and then the whole is constructed by aggregating the parts. However, the same ‘mathematics’ (when it is seen as the parts of the whole) will mean logic and alphabets and will create cognition, relation, and emotion—the three facets of the soul under the influence of a particular type of logic which constitutes the material world. This is not the only possible logic, as we have seen above, although the alphabets remain unchanged. By studying the material world, therefore we can discover the alphabet, and its governing logic. Then we can retain the alphabet and employ a different logic based on the understanding of the soul. Thereby emerges a new kind of world that looks just like this world because it embodies the same alphabet, and yet it employs a different logic. That logic entails a different type of choice.
The problem of counting is identical to the problem of perception, discrimination, and ordering. Some fundamental ideas regarding counting are altered when the first object is the complete object, and everything is defined as a part of the whole, in relation to the whole. A fundamental feature of this division is that the whole itself cannot be cut up into parts, because the origin of counting would then be lost. Instead, we have to create the parts, without dividing the whole itself into parts.
This is then the philosophy of mathematics—a whole is divided into parts. The method of this division is logic (called prāna), and the results of the division are alphabets (called vāk). Once the division has occurred and the alphabets have been created, then the second-order function of logic is to create their combinations. These combinations are nothing other than further subdivisions. Karma and time play a role in the combination or subdivision—time creates the combination and karma pushes a soul into the combination. As a result of time and karma the combinations are experienced.
The alphabets are symbols of meaning, and those meanings are subdivisions of relation, cognition, and emotion. By using an alphabet that mimics the properties of the soul, we begin to understand a language that can explain everything in experience. This experience is different everywhere; in particular, the material world is identified by a different logic, and each universe is defined by a different scale of dividing, which then produce different alphabets. By adding more letters to the alphabet, the language becomes more complex but that complexity is useful in describing nuances missing in the smaller universes. By removing letters, language is simplified, and the nuances are removed. Thus, a universe with more alphabets will afford a finer subdivision of relationships, a more refined division in concepts, and more variety in pleasure. These are also counterparts of the coordinate reference frames we use to describe nature. Reference frames are not arbitrary. Each universe is a different frame, which means that if you tried to a more elaborate language you will not be able to experience the associated subtleties there.
As these divisions are increased, we get different universes. As these divisions are successively applied in a hierarchy we get smaller parts of the universe. The part-whole relationship is the geometry of space and time. There are two crucial insights in this philosophy. First, we need to understand how logic is a choice created from inner contradictions in the soul. Second, we need to know how that logic creates the ideal embodiment of the logic by choosing the directions in space in order to create the alphabets.
Our conscious experience is seeing a world around us comprised of places, times, things, and situations—i.e. geometry. Our unconscious existence is the logic that precedes the above said geometry. By understanding the different kinds of logics we can know the different philosophies of life. By knowing the different geometries we can describe different worlds based on these philosophies of life. The scope of logic and mathematics are thus different, but they are successive progressions in analyzing the complete truth.