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Thursday, October 24, 2024

Brahman

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Brahman
Om signifies the essence of Brahman, the ultimate reality.
A drop in the ocean: an analogy for Ātman merging into Brahman.

In Hinduism, Brahman (Sanskrit: ब्रह्मन्; IAST: Brahman) connotes the highest universal principle, the Ultimate Reality of the universe. In major schools of Hindu philosophy, it is the non-physical, efficient, formal and final cause of all that exists. It is the pervasive, infinite, eternal truth, consciousness and bliss which does not change, yet is the cause of all changes. Brahman as a metaphysical concept refers to the single binding unity behind diversity in all that exists.

Brahman is a Vedic Sanskrit word, and it is conceptualized in Hinduism, states Paul Deussen, as the "creative principle which lies realized in the whole world". Brahman is a key concept found in the Vedas, and it is extensively discussed in the early Upanishads. The Vedas conceptualize Brahman as the Cosmic Principle. In the Upanishads, it has been variously described as Sat-cit-ānanda (truth-consciousness-bliss) and as the unchanging, permanent, Highest Reality.

Brahman is discussed in Hindu texts with the concept of Atman (Sanskrit: आत्मन्, 'Self'), personal, impersonal or Para Brahman, or in various combinations of these qualities depending on the philosophical school. In dualistic schools of Hinduism such as the theistic Dvaita Vedanta, Brahman is different from Atman (Self) in each being. In non-dual schools such as the Advaita Vedanta, the substance of Brahman is identical to the substance of Atman, is everywhere and inside each living being, and there is connected spiritual oneness in all existence.

Sanskrit (ब्रह्मन्) Brahman (an n-stem, nominative bráhma, from a root bṛh- "to swell, expand, grow, enlarge") is a neuter noun to be distinguished from the masculine brahmán—denoting a person associated with Brahman, and from Brahmā, the creator God in the Hindu Trinity, the Trimurti. Brahman is thus a gender-neutral concept that implies greater impersonality than masculine or feminine conceptions of the deity. Brahman is referred to as the supreme self. Puligandla states it as "the unchanging reality amidst and beyond the world", while Sinar states Brahman is a concept that "cannot be exactly defined".

In Vedic Sanskrit:

  • Brahma (ब्रह्म) (nominative singular), brahman (ब्रह्मन्) (stem) (neuter gender) from root bṛh-, means "to be or make firm, strong, solid, expand, promote".
  • Brahmana (ब्रह्मन) (nominative singular, never plural), from stems brha (to make firm, strong, expand) + Sanskrit -man- which denotes some manifest form of "definite power, inherent firmness, supporting or fundamental principle".

In later Sanskrit usage:

  • Brahma (ब्रह्म) (nominative singular), brahman (ब्रह्मन्) (stem) (neuter gender) means the concept of the transcendent and immanent ultimate reality, Supreme Cosmic Spirit in Hinduism. The concept is central to Hindu philosophy, especially Vedanta; this is discussed below.
  • Brahmā (ब्रह्मा) (nominative singular), Brahman (ब्रह्मन्) (stem) (masculine gender), means the deity or deva Prajāpati Brahmā. He is one of the members of the Hindu trinity and associated with creation, but does not have a cult in present-day India. This is because Brahmā, the creator-god, is long-lived but not eternal i.e. Brahmā gets absorbed back into Purusha at the end of an aeon, and is born again at the beginning of a new kalpa.

These are distinct from:

  • A brāhmaa (ब्राह्मण) (masculine, pronounced [ˈbɽaːɦmɐɳɐ]), (which literally means "pertaining to prayer") is a prose commentary on the Vedic mantras—an integral part of the Vedic literature.
  • A brāhmaa (ब्राह्मण) (masculine, same pronunciation as above), means priest; in this usage the word is usually rendered in English as "Brahmin". This usage is also found in the Atharva Veda. In neuter plural form, Brahmāṇi. See Vedic priest.
  • Ishvara, (lit., Supreme Lord), in Advaita, is identified as a partial worldly manifestation (with limited attributes) of the ultimate reality, the attributeless Brahman. In Visishtadvaita and Dvaita, however, Ishvara (the Supreme Controller) has infinite attributes and the source of the impersonal Brahman.
  • Devas, the expansions of Brahman/God into various forms, each with a certain quality. In the Vedic religion, there were 33 devas, which later became exaggerated to 330 million devas. In fact, devas are themselves regarded as more mundane manifestations of the One and the Supreme Brahman (See Para Brahman). The Sanskrit word for "ten million" also means group, and 330 million devas originally meant 33 types of divine manifestations.

History and literature

Vedic

Brahman is a concept present in Vedic Samhitas, the oldest layer of the Vedas dated to the late 2nd millennium BCE. For example,

The Ṛcs are limited (parimita),
The Samans are limited,
And the Yajuses are limited,
But of the Word Brahman, there is no end.

— Taittiriya Samhita VII.3.1.4, Translated by Barbara Holdrege

The concept Brahman is referred to in hundreds of hymns in the Vedic literature. The word Brahma is found in Rig veda hymns such as 2.2.10, 6.21.8, 10.72.2 and in Atharva veda hymns such as 6.122.5, 10.1.12, and 14.1.131. The concept is found in various layers of the Vedic literature; for example: Aitareya Brahmana 1.18.3, Kausitaki Brahmana 6.12, Satapatha Brahmana 13.5.2.5, Taittiriya Brahmana 2.8.8.10, Jaiminiya Brahmana 1.129, Taittiriya Aranyaka 4.4.1 through 5.4.1, Vajasaneyi Samhita 22.4 through 23.25, Maitrayani Samhita 3.12.1:16.2 through 4.9.2:122.15. The concept is extensively discussed in the Upanishads embedded in the Vedas (see next section), and also mentioned in the vedāṅga (the limbs of Vedas) such as the Srauta sutra 1.12.12 and Paraskara Gryhasutra 3.2.10 through 3.4.5.

Jan Gonda states that the diverse reference of Brahman in the Vedic literature, starting with Rigveda Samhitas, convey "different senses or different shades of meaning". There is no one single word in modern Western languages that can render the various shades of meaning of the word Brahman in the Vedic literature, according to Jan Gonda. In verses considered as the most ancient, the Vedic idea of Brahman is the "power immanent in the sound, words, verses and formulas of Vedas". However, states Gonda, the verses suggest that this ancient meaning was never the only meaning, and the concept evolved and expanded in ancient India.

Barbara Holdrege states that the concept Brahman is discussed in the Vedas along four major themes: as the Word or verses (Sabdabrahman), as Knowledge embodied in Creator Principle, as Creation itself, and a Corpus of traditions. Hananya Goodman states that the Vedas conceptualize Brahman as the Cosmic Principles underlying all that exists. Gavin Flood states that the Vedic era witnessed a process of abstraction, where the concept of Brahman evolved and expanded from the power of sound, words and rituals to the "essence of the universe", the "deeper foundation of all phenomena", the "essence of the self (Atman, Self)", and the deeper "truth of a person beyond apparent difference".

Upanishads

Swan (Hansa, हंस) is the symbol for Brahman-Atman in Hindu iconography.

The central concern of all Upanishads is to discover the relations between ritual, cosmic realities (including gods), and the human body/person. The texts do not present a single unified theory, rather they present a variety of themes with multiple possible interpretations, which flowered in post-Vedic era as premises for the diverse schools of Hinduism.

Paul Deussen states that the concept of Brahman in the Upanishads expands to metaphysical, ontological and soteriological themes, such as it being the "primordial reality that creates, maintains and withdraws within it the universe", the "principle of the world", the "absolute", the "general, universal", the "cosmic principle", the "ultimate that is the cause of everything including all gods", the "divine being, Lord, distinct God, or God within oneself", the "knowledge", the "Self, sense of self of each human being that is fearless, luminuous, exalted and blissful", the "essence of liberation, of spiritual freedom", the "universe within each living being and the universe outside", the "essence and everything innate in all that exists inside, outside and everywhere".

Gavin Flood summarizes the concept of Brahman in the Upanishads to be the "essence, the smallest particle of the cosmos and the infinite universe", the "essence of all things which cannot be seen, though it can be experienced", the "Self within each person, each being", the "truth", the "reality", the "absolute", the "bliss" (ananda).

According to Radhakrishnan, the sages of the Upanishads teach Brahman as the ultimate essence of material phenomena that cannot be seen or heard, but whose nature can be known through the development of self-knowledge (atma jnana).

The Upanishads contain several mahā-vākyas or "Great Sayings" on the concept of Brahman:

Text Upanishad Translation
अहं ब्रह्म अस्मि
aham brahmāsmi
Brihadaranyaka Upanishad 1.4.10 "I am Brahman"
अयम् आत्मा ब्रह्म
ayam ātmā brahma
Brihadaranyaka Upanishad 4.4.5 "The Self is Brahman"
सर्वं खल्विदं ब्रह्म
sarvam khalvidam brahma
Chandogya Upanishad 3.14.1 "All this is Brahman"
एकमेवाद्वितीयम्
ekam evadvitiyam
Chandogya Upanishad 6.2.1 "That [Brahman] is one, without a second"
तत्त्वमसि
tat tvam asi
Chandogya Upanishad 6.8.7 et seq. "Thou art that" ("You are Brahman")
प्रज्ञानं ब्रह्म
prajnānam brahma
Aitareya Upanishad 3.3.7 "Wisdom is Brahman"

The Upanishad discuss the metaphysical concept of Brahman in many ways, such as the Śāṇḍilya doctrine in Chapter 3 of the Chandogya Upanishad, among of the oldest Upanishadic texts. The Śāṇḍilya doctrine on Brahman is not unique to Chandogya Upanishad, but found in other ancient texts such as the Satapatha Brahmana in section 10.6.3. It asserts that Atman (the inner essence, Self inside man) exists, the Brahman is identical with Atman, that the Brahman is inside man—thematic quotations that are frequently cited by later schools of Hinduism and modern studies on Indian philosophies.

This whole universe is Brahman. In tranquility, let one worship It, as Tajjalan (that from which he came forth, as that into which he will be dissolved, as that in which he breathes).

— Chandogya Upanishad 3.14.1

Man is a creature of his Kratumaya (क्रतुमयः, will, purpose). Let him therefore have for himself this will, this purpose: The intelligent, whose body is imbued with life-principle, whose form is light, whose thoughts are driven by truth, whose self is like space (invisible but ever present), from whom all works, all desires, all sensory feelings encompassing this whole world, the silent, the unconcerned, this is me, my Self, my Soul within my heart.

— Chandogya Upanishad 3.14.1 – 3.14.3

This is my Soul in the innermost heart, greater than the earth, greater than the aerial space, greater than these worlds. This Soul, this Self of mine is that Brahman.

— Chandogya Upanishad 3.14.3 – 3.14.4

Paul Deussen notes that teachings similar to above on Brahman, re-appeared centuries later in the words of the 3rd century CE Neoplatonic Roman philosopher Plotinus in Enneades 5.1.2.

Discussion

The concept Brahman has a lot of undertones of meaning and is difficult to understand. It has relevance in metaphysics, ontology, axiology (ethics & aesthetics), teleology and soteriology.

Brahman as a metaphysical concept

Brahman is the key metaphysical concept in various schools of Hindu philosophy. It is the theme in its diverse discussions to the two central questions of metaphysics: what is ultimately real, and are there principles applying to everything that is real? Brahman is the ultimate "eternally, constant" reality, while the observed universe is a different kind of reality but one which is "temporary, changing" Maya in various orthodox Hindu schools. Maya pre-exists and co-exists with Brahman—the Ultimate Reality, The Highest Universal, the Cosmic Principles.

Atman: the ultimate reality

In addition to the concept of Brahman, Hindu metaphysics includes the concept of Atman—or Self, which is also considered ultimately real. The various schools of Hinduism, particularly the dual and non-dual schools, differ on the nature of Atman, whether it is distinct from Brahman, or same as Brahman. Those that consider Brahman and Atman as distinct are theistic, and Dvaita Vedanta and later Nyaya schools illustrate this premise. Those that consider Brahman and Atman as same are monist or pantheistic, and Advaita Vedanta, later Samkhya and Yoga schools illustrate this metaphysical premise. In schools that equate Brahman with Atman, Brahman is the sole, ultimate reality. The predominant teaching in the Upanishads is the spiritual identity of Self within each human being, with the Self of every other human being and living being, as well as with the supreme, ultimate reality Brahman.

Maya: the perceived reality

In the metaphysics of the major schools of Hinduism, Maya is perceived reality, one that does not reveal the hidden principles, the true reality—the Brahman. Maya is unconscious, Brahman-Atman is conscious. Maya is the literal and the effect, Brahman is the figurative Upādāna—the principle and the cause. Maya is born, changes, evolves, dies with time, from circumstances, due to invisible principles of nature. Atman-Brahman is eternal, unchanging, invisible principle, unaffected absolute and resplendent consciousness. Maya concept, states Archibald Gough, is "the indifferent aggregate of all the possibilities of emanatory or derived existences, pre-existing with Brahman", just like the possibility of a future tree pre-exists in the seed of the tree.

Nirguna and Saguna Brahman

Brahman, the ultimate reality, is both with and without attributes. In this context, Para Brahman is formless and omniscient Ishvara - the god or Paramatman and Om, where as Saguna Brahman is manifestation or avatara of god in personified form.

While Hinduism sub-schools such as Advaita Vedanta emphasize the complete equivalence of Brahman and Atman, they also expound on Brahman as saguna Brahman—the Brahman with attributes, and nirguna Brahman—the Brahman without attributes. The nirguna Brahman is the Brahman as it really is, however, the saguna Brahman is posited as a means to realizing nirguna Brahman, but the Hinduism schools declare saguna Brahman to be a part of the ultimate nirguna Brahman The concept of the saguna Brahman, such as in the form of avatars, is considered in these schools of Hinduism to be a useful symbolism, path and tool for those who are still on their spiritual journey, but the concept is finally cast aside by the fully enlightened.

Brahman as an ontological concept

Brahman, along with Self (Atman) are part of the ontological premises of Indian philosophy. Different schools of Indian philosophy have held widely dissimilar ontologies. Buddhism and Carvaka school of Hinduism deny that there exists anything called "a Self" (individual Atman or Brahman in the cosmic sense), while the orthodox schools of Hinduism, Jainism and Ajivikas hold that there exists "a Self".

Brahman as well the Atman in every human being (and living being) is considered equivalent and the sole reality, the eternal, self-born, unlimited, innately free, blissful Absolute in schools of Hinduism such as the Advaita Vedanta and Yoga. Knowing one's own self is knowing the God inside oneself, and this is held as the path to knowing the ontological nature of Brahman (universal Self) as it is identical to the Atman (individual Self). The nature of Atman-Brahman is held in these schools, states Barbara Holdrege, to be as a pure being (sat), consciousness (cit) and full of bliss (ananda), and it is formless, distinctionless, nonchanging and unbounded.

In theistic schools, in contrast, such as Dvaita Vedanta, the nature of Brahman is held as eternal, unlimited, innately free, blissful Absolute, while each individual's Self is held as distinct and limited which can at best come close in eternal blissful love of the Brahman (therein viewed as the Godhead).

Other schools of Hinduism have their own ontological premises relating to Brahman, reality and nature of existence. Vaisheshika school of Hinduism, for example, holds a substantial, realist ontology. The Carvaka school denied Brahman and Atman, and held a materialist ontology.

Brahman as an axiological concept

Brahman and Atman are key concepts to Hindu theories of axiology: ethics and aesthetics. Ananda (bliss), state Michael Myers and other scholars, has axiological importance to the concept of Brahman, as the universal inner harmony. Some scholars equate Brahman with the highest value, in an axiological sense.

The axiological concepts of Brahman and Atman is central to Hindu theory of values. A statement such as 'I am Brahman', states Shaw, means 'I am related to everything', and this is the underlying premise for compassion for others in Hinduism, for each individual's welfare, peace, or happiness depends on others, including other beings and nature at large, and vice versa. Tietge states that even in non-dual schools of Hinduism where Brahman and Atman are treated ontologically equivalent, the theory of values emphasizes individual agent and ethics. In these schools of Hinduism, states Tietge, the theory of action are derived from and centered in compassion for the other, and not egotistical concern for the self.

The axiological theory of values emerges implicitly from the concepts of Brahman and Atman, states Bauer. The aesthetics of human experience and ethics are one consequence of self-knowledge in Hinduism, one resulting from the perfect, timeless unification of one's Self with the Brahman, the Self of everyone, everything and all eternity, wherein the pinnacle of human experience is not dependent on an afterlife, but pure consciousness in the present life itself. It does not assume that an individual is weak nor does it presume that he is inherently evil, but the opposite: human Self and its nature is held as fundamentally unqualified, faultless, beautiful, blissful, ethical, compassionate and good. Ignorance is to assume it evil, liberation is to know its eternal, expansive, pristine, happy and good nature. The axiological premises in the Hindu thought and Indian philosophies in general, states Nikam, is to elevate the individual, exalting the innate potential of man, where the reality of his being is the objective reality of the universe. The Upanishads of Hinduism, summarizes Nikam, hold that the individual has the same essence and reality as the objective universe, and this essence is the finest essence; the individual Self is the universal Self, and Atman is the same reality and the same aesthetics as the Brahman.

Brahman as a teleological concept

Brahman and Atman are very important teleological concepts. Teleology deals with the apparent purpose, principle, or goal of something. In the first chapter of the Shvetashvatara Upanishad, these questions are addressed. It says:

"People who make inquiries about brahman say:
What is the cause of Brahman? Why were we born? By what do we live? On what are we established? Governed by whom, O you who know Brahman, do we live in pleasure and in pain, each in our respective situation?

— Shvetashvatara Upanishad, Hymns 1.1

According to the Upanishads, the main purpose/meaning of anything or everything can be explained or achieved/understood only through the realization of the Brahman. The apparent purpose of everything can be grasped by obtaining the Brahman, as the Brahman is referred to that when known, all things become known.

"What is that my lord, by which being known, all of this becomes known?"
Angiras told him, "Two types of knowledge a man should learn, those who know Brahman tell us — the higher and the lower. The lower of the two consists of the Rgveda, Yajurveda, Samaveda (...), whereas, the higher is that by which one grasps the imperishable (Brahman)."

— Mundaka Upanishad, Hymns 1.1

Elsewhere in the Upanishads, the relationship between Brahman & all knowledge is established, such that any questions of apparent purpose/teleology are resolved when the Brahman is ultimately known. This is found in the Aitareya Upanishad 3.3 and Brihadaranyaka Upanishad 4.4.17.

Knowledge is the eye of all that, and on knowledge it is founded. Knowledge is the eye of the world, and knowledge, the foundation. Brahman is knowing.

— Aitereya Upanishad, Hymns 3.3

One of the main reasons why Brahman should be realized is because it removes suffering from a person's life. Following on Advaita Vedanta tradition, this is because the person has the ability and knowledge to discriminate between the unchanging (Purusha; Atman-Brahman) and the ever-changing (Prakriti; maya) and so the person is not attached to the transient, fleeting & impermanent. Hence, the person is only content with their true self and not the body or anything else. Further elaborations of Brahman as the central teleological issue are found in Shankara's commentaries of the Brahma Sutras & his Vivekachudamani.

In Brihadaranyaka Upanishad 3.9.26 it mentions that the atman 'neither trembles in fear nor suffers injury' and the Isha Upanishad 6-7 too talks about suffering as non-existent when one becomes the Brahman as they see the self in all beings and all beings in the self. The famous Advaita Vedanta commentator Shankara noted that Sabda Pramana (scriptural epistemology) & anubhava (personal experience) is the ultimate & only source of knowing/learning the Brahman, and that its purpose or existence cannot be verified independently because it is not an object of perception/inference (unless one is spiritually advanced, thereby it's truth becomes self-evident/intuitive) & is beyond conceptualizations. But he does note the Upanishads themselves are ultimately derived from use of the various pramanas to derive at ultimate truths (as seen in Yalnavalkya's philosophical inquires). All Vedanta schools agree on this. These teleological discussions inspired some refutations from competing philosophies about the origin/purpose of Brahman & avidya (ignorance) and the relationship between the two, leading to variant schools like Kashmiri Shaivism & others.

Brahman as a soteriological concept: Moksha

The orthodox schools of Hinduism, particularly Vedanta, Samkhya and Yoga schools, focus on the concept of Brahman and Atman in their discussion of moksha. The Advaita Vedanta holds there is no being/non-being distinction between Atman and Brahman. The knowledge of Atman (Self-knowledge) is synonymous to the knowledge of Brahman inside the person and outside the person. Furthermore, the knowledge of Brahman leads to a sense of oneness with all existence, self-realization, indescribable joy, and moksha (freedom, bliss), because Brahman-Atman is the origin and end of all things, the universal principle behind and at source of everything that exists, consciousness that pervades everything and everyone.

The theistic sub-school such as Dvaita Vedanta of Hinduism, starts with the same premises, but adds the premise that individual Self and Brahman are distinct, and thereby reaches entirely different conclusions where Brahman is conceptualized in a manner similar to God in other major world religions. The theistic schools assert that moksha is the loving, eternal union or nearness of one's Self with the distinct and separate Brahman (Vishnu, Shiva or equivalent henotheism). Brahman, in these sub-schools of Hinduism is considered the highest perfection of existence, which every Self journeys towards in its own way for moksha.

Hindu schools of thought

Vedanta

The concept of Brahman, its nature and its relationship with Atman and the observed universe, is a major point of difference between the various sub-schools of the Vedanta school of Hinduism.

Advaita Vedanta

The universe does not simply come from Brahman, it is Brahman. According to Adi Shankara, a proponent of Advaita Vedanta, the knowledge of Brahman that shruti provides cannot be obtained by any other means besides self inquiry.

In Advaita Vedanta, nirguna Brahman is held to be the ultimate and sole reality. Consciousness is not a property of Brahman but rather its very nature. In this respect, Advaita Vedanta differs from other Vedanta schools.

Example verses from Bhagavad-Gita include:

The offering is Brahman; the oblation is Brahman;
offered by Brahman into the fire of Brahman.
Brahman will be attained by him,
who always sees Brahman in action. – Hymn 4.24
He who finds his happiness within,
His delight within,
And his light within,
This yogin attains the bliss of Brahman, becoming Brahman. – Hymn 5.24

— Bhagavad Gita

Dvaita Vedanta

Brahman of Dvaita is a concept similar to God in major world religions. Dvaita holds that the individual Self is dependent on God, but distinct. Dvaita philosophy argues against the concept of a shared existence between Brahman and finite beings. It sees any concept of shared existence or non-dualism (Advaita) as incompatible with the nature of Brahman's transcendent perfection. Madhva places importance on the unique individuality of each entity (vishesha).

Dvaita propounds Tattvavada which means understanding differences between Tattvas (significant properties) of entities within the universal substrate as follows:

  1. Jîva-Îshvara-bheda — difference between the Self and the Supreme God
  2. Jada-Îshvara-bheda — difference between the insentient and the Supreme God
  3. Mitha-jîva-bheda — difference between any two Selves
  4. Jada-jîva-bheda — difference between insentient and the Self
  5. Mitha-jada-bheda — difference between any two insentients

Vishishtadvaita

In Vishishtadvaita, Ramanuja asserts that Brahman is God, and that this God is Narayana. In his commentary on the Brahma Sutras 1.1.1, Ramanuja defines Brahman as the "'highest person,' one who by his own nature is free from all imperfections and in possession of host of innumerable auspicious qualities of unsurpassable excellence." Using this definition, Ramanuja argues that Brahman must be God because Brahman's qualities are unsurpassably superior to all, and thus "only the Lord of all can thus be denoted, and 'Brahman' primarily denotes him alone". Ramanuja asserts that the relationship between God and the individual selves must be one of devotion, and moksha or liberation is said by him to be caused by the selves' worship of Brahman: "The cessation of bondage...is to be obtained only through the grace of the highest Person who is pleased by worshipper's meditation, which is devotion".

Achintya Bheda Abheda

The Acintya Bheda Abheda philosophy is similar to Dvaitadvaita (differential monism). In this philosophy, Brahman is not just impersonal, but also personal. That Brahman is Supreme Personality of Godhead, though on first stage of realization (by process called jnana) of Absolute Truth, He is realized as impersonal Brahman, then as personal Brahman having eternal Vaikuntha abode (also known as Brahmalokah sanatana), then as Paramatma (by process of yogameditation on Superself, Vishnu-God in heart)—Vishnu (Narayana, also in everyone's heart) who has many abodes known as Vishnulokas (Vaikunthalokas), and finally (Absolute Truth is realized by bhakti) as Bhagavan, Supreme Personality of Godhead, who is source of both Paramatma and Brahman (personal, impersonal, or both).

Bhakti movement

The Bhakti movement of Hinduism built its theosophy around two concepts of Brahman—Nirguna and Saguna. Nirguna Brahman was the concept of the Ultimate Reality as formless, without attributes or quality. Saguna Brahman, in contrast, was envisioned and developed as with form, attributes and quality. The two had parallels in the ancient pantheistic unmanifest and theistic manifest traditions, respectively, and traceable to Arjuna-Krishna dialogue in the Bhagavad Gita. It is the same Brahman, but viewed from two perspectives, one from Nirguni knowledge-focus and other from Saguni love-focus, united as Krishna (an 8th incarnation of Lord Vishnu) in the Gita. Nirguna bhakta's poetry were Jnana-shrayi, or had roots in knowledge. Saguna bhakta's poetry were Prema-shrayi, or with roots in love. In Bhakti, the emphasis is reciprocal love and devotion, where the devotee loves God, and God loves the devotee.

Jeaneane Fowler states that the concepts of Nirguna and Saguna Brahman, at the root of Bhakti movement theosophy, underwent more profound development with the ideas of Vedanta school of Hinduism, particularly those of Adi Shankara's Advaita Vedanta, Ramanuja's Vishishtadvaita Vedanta, and Madhvacharya's Dvaita Vedanta. Two 12th-century influential treatises on bhakti were Sandilya Bhakti Sutra—a treatise resonating with Nirguna-bhakti, and Narada Bhakti Sutra—a treatise that leans towards Saguna-bhakti.

Nirguna and Saguna Brahman concepts of the Bhakti movement has been a baffling one to scholars, particularly the Nirguni tradition because it offers, states David Lorenzen, "heart-felt devotion to a God without attributes, without even any definable personality". Yet given the "mountains of Nirguni bhakti literature", adds Lorenzen, bhakti for Nirguna Brahman has been a part of the reality of the Hindu tradition along with the bhakti for Saguna Brahman. These were two alternate ways of imagining God during the bhakti movement.

Buddhist understanding of Brahman

Buddhism rejects the Upanishadic doctrine of Brahman and Atman (permanent Self, essence). According to Damien Keown, "the Buddha said he could find no evidence for the existence of either the personal Self (atman) or its cosmic counterpart (brahman)". The metaphysics of Buddhism rejects Brahman (ultimate being), Brahman-like essence, Self and anything metaphysically equivalent through its Anatta doctrine, but their belief of Trikaya is very similar to the sat-cit-ananda characteristics of Brahman.

According to Merv Fowler, some forms of Buddhism have incorporated concepts that resemble that of Brahman. As an example, Fowler cites the early Sarvastivada school of Buddhism, which "had come to accept a very pantheistic religious philosophy, and are important because of the impetus they gave to the development of Mahayana Buddhism". According to William Theodore De Bary, in the doctrines of the Yogacara school of Mahayana Buddhism, "the Body of Essence, the Ultimate Buddha, who pervaded and underlay the whole universe [...] was in fact the World Self, the Brahman of the Upanishads, in a new form". According to Fowler, some scholars have identified the Buddhist nirvana, conceived of as the Ultimate Reality, with the Hindu Brahman/atman; Fowler claims that this view "has gained little support in Buddhist circles." Fowler asserts that the authors of a number of Mahayana texts took pains to differentiate their ideas from the Upanishadic doctrine of Brahman.

Brahma as a surrogate for Brahman in Buddhist texts

The spiritual concept of Brahman is far older in the Vedic literature, and some scholars suggest deity Brahma may have emerged as a personal conception and icon with form and attributes (saguna version) of the impersonal, nirguna (without attributes), formless universal principle called Brahman. In the Hindu texts, one of the earliest mentions of deity Brahma along with Vishnu and Shiva is in the fifth Prapathaka (lesson) of the Maitrayaniya Upanishad, probably composed in late 1st millennium BCE, after the rise of Buddhism.

The early Buddhists attacked the concept of Brahma, states Gananath Obeyesekere, and thereby polemically attacked the Vedic and Upanishadic concept of gender neutral, abstract metaphysical Brahman. This critique of Brahma in early Buddhist texts aims at ridiculing the Vedas, but the same texts simultaneously call metta (loving-kindness, compassion) as the state of union with Brahma. The early Buddhist approach to Brahma was to reject any creator aspect, while retaining the value system in the Vedic Brahmavihara concepts, in the Buddhist value system. According to Martin Wiltshire, the term "Brahma loka" in the Buddhist canon, instead of "Svarga loka", is likely a Buddhist attempt to choose and emphasize the "truth power" and knowledge focus of the Brahman concept in the Upanishads. Simultaneously, by reformulating Brahman as Brahma and relegating it within its Devas and Samsara theories, early Buddhism rejected the Atman-Brahman premise of the Vedas to present its own Dhamma doctrines (anicca, dukkha and anatta).

Brahman in Chinese thought

L. S. Vasil’ev, writing for the University of Pennsylvania, argued that syncretic Taoists used Brahman in their worship as a substitute for Tao.

Victor H. Mair thought that Taoists in the early history of the faith had positive "cultural relations" with Hindu groups and that the Tao Te Ching was written in reaction to Indian philosophy and that the author(s) viewed Brahman as being the same as Tao.

Brahman in Sikhism

Ik Onkar (left) is part of the Mul Mantar in Sikhism, where it means "Onkar [God, Reality] is one". The Onkar of Sikhism is related to Om—also called Omkāra—in Hinduism. The ancient texts of Hinduism state Om to be a symbolism for the Highest Reality, Brahman.

The metaphysical concept of Brahman, particularly as nirguni Brahman—attributeless, formless, eternal Highest Reality—is at the foundation of Sikhism. This belief is observed through nirguni Bhakti by the Sikhs.

In Gauri, which is part of the Guru Granth Sahib, Brahman is declared as "One without a second", in Sri Rag "everything is born of Him, and is finally absorbed in Him", in Var Asa "whatever we see or hear is the manifestation of Brahman". Nesbitt states that the first two words, Ik Onkar, in the twelve-word Mul Mantar at the opening of the Sikh scripture Guru Granth Sahib, has been translated in three different ways by scholars: "There is one god", "This being is one", and as "One reality is".

Similar emphasis on "One without a second" for metaphysical concept of Brahman, is found in ancient texts of Hinduism, such as the Chandogya Upanishad's chapter 6.2. The ideas about God and Highest Reality in Sikhism share themes found in the Saguna and Nirguna concepts of Brahman in Hinduism.

The concept of Ultimate Reality (Brahman) is also referred in Sikhism as Nam, Sat-naam or Naam, and Ik Oankar like Hindu Om symbolizes this Reality.

Brahman in Jainism

Scholars contest whether the concept of Brahman is rejected or accepted in Jainism. The concept of a theistic God is rejected by Jainism, but Jiva or "Atman (Self) exists" is held to be a metaphysical truth and central to its theory of rebirths and Kevala Jnana.

Bissett states that Jainism accepts the "material world" and "Atman", but rejects Brahman—the metaphysical concept of Ultimate Reality and Cosmic Principles found in the ancient texts of Hinduism. Goswami, in contrast, states that the literature of Jainism has an undercurrent of monist theme, where the self who gains the knowledge of Brahman (Highest Reality, Supreme Knowledge) is identical to Brahman itself. Jaini states that Jainism neither accepts nor rejects the premise of Ultimate Reality (Brahman), instead Jain ontology adopts a many sided doctrine called Anekantavada. This doctrine holds that "reality is irreducibly complex" and no human view or description can represent the Absolute Truth. Those who have understood and realized the Absolute Truth are the liberated ones and the Supreme Self (Paramatman), with Kevala Jnana.

Comparison of Brahma, Brahman, Brahmin and Brahmanas

Brahma is distinct from Brahman. Brahma is a male deity, in the post-Vedic Puranic literature, who creates but neither preserves nor destroys anything. He is envisioned in some Hindu texts to have emerged from the metaphysical Brahman along with Vishnu (preserver), Shiva (destroyer), all other gods, goddesses, matter and other beings.

Brahman is a metaphysical concept of Hinduism referring to the ultimate unchanging reality, that is uncreated, eternal, infinite, transcendent, the cause, the foundation, the source and the goal of all existence. It is envisioned as either the cause or that which transforms itself into everything that exists in the universe as well as all beings, that which existed before the present universe and time, which exists as current universe and time, and that which will absorb and exist after the present universe and time ends. It is a gender neutral abstract concept. The abstract Brahman concept is predominant in the Vedic texts, particularly the Upanishads; while the deity Brahma finds minor mention in the Vedas and the Upanishads. In the Puranic and the Epics literature, the deity Brahma appears more often, but inconsistently.

Some texts suggest that the god Vishnu created Brahma (Vaishnavism), others suggest god Shiva created Brahma (Shaivism), yet others suggest goddess Devi created Brahma (Shaktism), and these texts then go on to state that Brahma is a secondary creator of the world working respectively on their behalf. A similarity between Brahma and Brahman is that Brahman is said to be an anchor for the world and the relations between all things, including opposites, in it, whereas Brahma is a creator god who aids the world in many Hindu and Buddhist traditions.

Further, the medieval era texts of these major theistic traditions of Hinduism assert that the saguna Brahman is Vishnu, is Shiva, or is Devi respectively, they are different names or aspects of the Brahman, and that the Atman (Self) within every living being is the same or part of this ultimate, eternal Brahman.

Brahmin is a varna in Hinduism specialising in theory as priests, preservers and transmitters of sacred literature across generations.

The Brahmanas are one of the four ancient layers of texts within the Vedas. They are primarily a digest incorporating myths, legends, the explanation of Vedic rituals and in some cases philosophy.They are embedded within each of the four Vedas, and form a part of the Hindu śruti literature.

Ultimate reality

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Ultimate_reality

Ultimate reality is "the supreme, final, and fundamental power in all reality". It refers to the most fundamental fact about reality, especially when it is seen as also being the most valuable fact. This may overlap with the concept of the Absolute in certain philosophies.

Greek philosophy

Anaximander (c. 610–546 BCE) believed that the ultimate substance of the universe, generally known as arche, was apeiron, an infinite and eternal substance that is the origin of all things.

Aristotle (384–322 BCE) held that the unmoved mover "must be an immortal, unchanging being, ultimately responsible for all wholeness and orderliness in the sensible world" and that its existence is necessary to support everyday change.

Democritus (c. 460–370 BCE) and Epicureanism (c. 307 BCE) rejected the idea of ultimate reality, saying that only atoms and void exist, but they do have the eternal, unbounded, and self-caused nature of non-materialistic views of the concept.

In Neoplatonism (3rd century CE), the first principle of reality is "the One" which is a perfectly simple and ineffable principle which is the source of the universe, and exists without multiplicity and beyond being and non-being.

Stoic physics (c. 300 BCE–3rd century CE) called the primitive substance of the universe pneuma or God, which is everything that exists and is a creative force that develops and shapes the cosmos.

Buddhism

In Theravada Buddhism, Nirvana is ultimate reality. Nirvana is described in negative terms; it is unconstructed and unconditioned.

In some strands of Mahayana Buddhism, the Buddha-nature or the Dharmakaya is seen as ultimate reality.

Other strands of Buddhism reject the notion of ultimate reality, regarding any existent as empty (sunyata) of inherent existence (svabhava).

Hinduism

In Hinduism, Brahman connotes the highest universal principle, the ultimate reality in the universe. In major schools of Hindu philosophy, it is the material, efficient, formal and final cause of all that exists. It is the pervasive, genderless, infinite, eternal truth and bliss which does not change, yet is the cause of all changes. Brahman as a metaphysical concept is the single binding unity behind diversity in all that exists in the universe.

Taoism

In Taoism, the Tao is the impersonal principle that underlies reality. It is a metaphysical principle and process that refers to how nature develops, being an enigmatic process of transformation. It is described as the source of existence, an ineffable mystery, and something that can be individually harnessed for the good. It is thought of as being "the flow of the universe" and the source of its order and its qi, but it is not considered a deity to be worshipped, even if some interpretations believed it had the power to bless or illuminate.

Abrahamic religions

Abrahamic conceptions of ultimate reality show diversity, in which some perspectives consider God to be a personal deity, while others have taken more abstract views. John Scotus Eriugena held that God's essence is uncaused and incomprehensible. Similarly, Maimonides believed that God is a perfect unity and is indescribable with positive attributes, and that anthropomorphic imagery in the Bible is metaphorical.

Modern philosophy

Baruch Spinoza believed that God is the natural world, existing eternally and necessarily, and that everything is an effect of God's nature. He defined God as a metaphysical substance rather than a personal being, and wrote in Ethics that "blessedness" comes from the love of God, meaning knowledge of reality as it is.

Contemporary philosophy notes the possibility that reality has no fundamental explanation and should be seen as a brute fact. Adherents of the principle of sufficient reason reject this, holding that everything must have a reason.

Representation

According to Dadosky, the concept of "ultimate reality" is difficult to express in words, poetry, mythology, and art. Paradox or contradiction is often used as a medium of expression because of the "contradictory aspect of the ultimate reality".

According to Mircea Eliade, ultimate reality can be mediated or revealed through symbols. For Eliade the "archaic" mind is constantly aware of the presence of the Sacred, and for this mind all symbols are religious (relinking to the Origin). Through symbols human beings can get an immediate "intuition" of certain features of the inexhaustible Sacred. The mind makes use of images to grasp the ultimate reality of things because reality manifests itself in contradictory ways and therefore can't be described in concepts. It is therefore the image as such, as a whole bundle of meaning, that is "true" (faithful, trustworthy). Eliade says :

the sacred is equivalent to a power, and, in the last analysis, to reality. The sacred is saturated with being. Sacred power means reality and at the same time enduringness and efficacy. The polarity sacred-profane is often expressed as opposition between real and unreal or pseudoreal. [...] Thus it is easy to understand that religious man deeply desires to be, to participate in reality, to be saturated with power.

Common symbols of ultimate reality include world trees, the tree of life, microcosm, fire, children.

Paul Tillich held that God is the ground of being and is something that precedes the subject and object (philosophy) dichotomy. He considered God to be what people are ultimately concerned with, existentially, and that religious symbols can be recovered as meaningful even without faith in the personal God of traditional Christianity.

Absolute (philosophy)

From Wikipedia, the free encyclopedia

In philosophy (often specifically metaphysics), the absolute, in most common usage, is a perfect, self-sufficient reality that depends upon nothing external to itself. In theology, the term is also used to designate the supreme being.

Hegel

Hegel used the term das Absolute in his German literary works. Contrary to some popular accounts, the term is not specific to Hegel. It first occurs in the work of Nicholas of Cusa, and Hegel's own usage was developed in response to that of his contemporary Friedrich Wilhelm Joseph Schelling.

Hegel's use of "absolute" is easily misunderstood. Michael Inwood, however, clarifies: derived from the Latin absolutus, it means "not dependent on, conditional on, relative to or restricted by anything else; self-contained, perfect, complete." In the words of scholar Allegra de Laurentiis, this means that absolute knowing can only denote "an 'absolute relation' in which the ground of experience and the experiencing agent are one and the same: the object known is explicitly the subject who knows." That is, the only "thing" (which is really an activity) that is truly absolute is that which is entirely self-conditioned, and according to Hegel, this only occurs when spirit takes itself up as its own object. In some respects, this view of Hegel was anticipated by Johann Gottlieb Fichte's theory of the absolute self. The final section of Hegel's Philosophy of Spirit presents the three modes of such absolute knowing: art, religion, and philosophy.

For Hegel, as understood by Martin Heidegger, the absolute is "spirit, that which is present to itself in the certainty of unconditional self-knowing". As Hegel is understood by Frederick Copleston, "[l]ogic studies the absolute 'in itself'; the philosophy of nature studies the absolute 'for itself'; and the philosophy of spirit studies the absolute 'in and for itself'."

In British philosophy, self-identified neo-Hegelian F. H. Bradley distinguishes the concept of absolute from God, whereas Josiah Royce, another neo-Hegelian and founder of the American idealism school of philosophy, has equated them.

Indian religions

The concept of the Absolute has been used to interpret the early texts of the Indian religions such as those attributed to Yajnavalkya, Nagarjuna and Adi Shankara.

According to Glyn Richards, the early texts of Hinduism state that the Brahman or the nondual Brahman–Atman is the Absolute.

According to Takeshi Umehara, some ancient texts of Buddhism state that the "truly Absolute and the truly Free must be nothingness", the "void". Yet, the early Buddhist scholar Nagarjuna, states Paul Williams, does not present "emptiness" as some kind of Absolute; rather, it is "the very absence (a pure non-existence) of inherent existence" in Mādhyamaka school of the Buddhist philosophy.

 

Wednesday, October 23, 2024

Taiji (philosophy)

From Wikipedia, the free encyclopedia
A diagram illustrating the concept of taiji, called a taijitu. The above design, depicting interlocking swirls of yin and yang around a central void, is the symbol's original form as introduced by Ming-era philosopher Lai Zhide.
 
Chinese name
Traditional Chinese
Simplified Chinese
Literal meaning"Supreme pole/goal"

Etymology

Taiji (太極) is a compound of tai ( 'great', 'supreme') and ji ( 'pole', 'extremity'). Used together, taiji may be understood as 'source of the world'. Common English translations of taiji in the cosmological sense include "Supreme Ultimate", "Supreme Pole", and "Great Absolute".

Core concept

Scholars Zhang and Ryden explain the ontological necessity of taiji.

Any philosophy that asserts two elements such as the yin-yang of Chinese philosophy will also look for a term to reconcile the two, to ensure that both belong to the same sphere of discourse. The term 'supreme ultimate' performs this role in the philosophy of the Book of Changes. In the Song dynasty it became a metaphysical term on a par with the Way.

Taiji is understood to be the highest conceivable principle from which existence flows. This is very similar to the Daoist idea "reversal is the movement of the Dao". The "supreme ultimate" creates yang and yin. Movement generates yang, and when its activity reaches its limit, it becomes tranquil. Through tranquility the supreme ultimate generates yin. When tranquility has reached its limit, there is a return to movement. Movement and tranquility, in alternation, become each the source of the other. The distinction between the yin and yang is determined and the two forms (that is, the yin and yang) stand revealed. By the transformations of the yang and the union of the yin, the 4 directions then the 5 phases (wuxing) of wood, fire, earth, metal and water.

Taiji is, ometal ond then water. larity, revealing opposing features as in expanding/contracting, rising/falling, clockwise/ anticlockwise. However, taiji has sometimes been thought of as a monistic concept similar to wuji, as in the Wujitu diagram. Wuji literally translates as "without roof pole", but means without limit, polarity, and/or opposite. Compared with wuji, taiji describes movement and change wherein limits do arise. While wuji is undifferentiated, timeless, absolute, infinite potential, taiji is often wrongly portrayed as conflictual, differentiated and dualistic, where as the core to this philosophy is their harmonious, relative and complementary natures.

Yin and yang are reflections and originate from wuji to become taiji.

In Chinese texts

Zhuangzi

The Daoist classic Zhuangzi introduced the taiji concept. One of the (ca. 3rd century BCE) "Inner Chapters" contrasts taiji (here translated as "zenith") with the liuji (六極). Liuji literally means "six ultimates; six cardinal directions", but here it is translated as "nadir".

The Way has attributes and evidence, but it has no action and no form. It may be transmitted but cannot be received. It may be apprehended but cannot be seen. From the root, from the stock, before there was heaven or earth, for all eternity truly has it existed. It inspirits demons and gods, gives birth to heaven and earth. It lies above the zenith but is not high; it lies beneath the nadir but is not deep. It is prior to heaven and earth, but is not ancient; it is senior to high antiquity, but it is not old.

Huainanzi

The 2nd century BCE Huainanzi mentions a zhenren ("true person; perfected person") and the taiji that transcends categories like yin and yang, exemplified with the fusui and fangzhu mirrors.

The fu-sui 夫煫 (burning mirror) gathers fire energy from the sun; the fang-chu 方諸 (moon mirror) gathers dew from the moon. What are [contained] between Heaven and Earth, even an expert calculator cannot compute their number. Thus, though the hand can handle and examine extremely small things, it cannot lay hold of the brightness [of the sun and moon]. Were it within the grasp of one's hand (within one's power) to gather [things within] one category from the Supreme Ultimate (t'ai-chi 太極) above, one could immediately produce both fire and water. This is because Yin and Yang share a common ch'i and move each other.

I Ching

Taiji also appears in the Xici, a commentary to the I Ching. It is traditionally attributed to Confucius but more likely dates to about the 3rd century BCE.

Therefore there is in the Changes the Great Primal Beginning. This generates the two primary forces. The two primary forces generate the four images. The four images generate the eight trigrams. The eight trigrams determine good fortune and misfortune. Good fortune and misfortune create the great field of action.

This sequence of powers of two includes taiji → yin and yang (two polarities) → Sixiang (Four Symbols) → Bagua (eight trigrams).

The fundamental postulate is the "great primal beginning" of all that exists, t'ai chi – in its original meaning, the "ridgepole". Later Indian philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t'ai chi was represented by the circle divided into the light and the dark, yang and yin, ☯. This symbol has also played a significant part in India and Europe. However, speculations of a Gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back – in a word, the world of the opposites.

Song dynasty

Zhou's Taijitu diagram

In the Neo-Confucianism philosophy that developed during the Song dynasty, taiji was viewed "as a microcosm equivalent to the structure of the human body." The Song-era philosopher Zhou Dunyi (1017-1073 CE) wrote the Taijitushuo (太極圖說) "Explanation of the Diagram of the Supreme Ultimate", which became the cornerstone of Neo-Confucianist cosmology. Zhou's brief text synthesized aspects of Chinese Buddhism and Daoism with the metaphysical discussions in the I Ching. Zhou's opening lines are:

Non-polar (wuji) and yet Supreme Polarity (taiji)! The Supreme Polarity in activity generates yang; yet at the limit of activity it is still. In stillness it generates yin; yet at the limit of stillness it is also active. Activity and stillness alternate; each is the basis of the other. In distinguishing yin and yang, the Two Modes are thereby established. The alternation and combination of yang and yin generate water, fire, wood, metal, and earth. With these five [phases of] qi harmoniously arranged, the Four Seasons proceed through them. The Five Phases are simply yin and yang; yin and yang are simply the Supreme Polarity; the Supreme Polarity is fundamentally Non-polar. [Yet] in the generation of the Five Phases, each one has its nature.

In tai chi

The martial art tai chi draws heavily on Chinese philosophy, especially the concept of the taiji. The Chinese name of tai chi, taijiquan, literally translates as "taiji boxing" or "taiji fist". Early tai chi masters such as Yang Luchan promoted the connection between their martial art and the concept of the taiji. The twenty-fourth chapter of the "Forty Chapter" tai chi classic that Yang Banhou gave to Wu Quanyou says the following about the connect between tai chi and spirituality:

If the essence of material substances lies in their phenomenological reality, then the presence of the ontological status of abstract objects shall become clear in the final culmination of the energy that is derived from oneness and the Real. How can man learn this truth? By truly seeking that which is the shadow of philosophy and the charge of all living substances, that of the nature of the divine.

Entitlement (fair division)

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Entitlement_(fair_division)

In economics, philosophy, and social choice theory, a person's entitlement refers to the value of goods they are owed or deserve, i.e. the total value of the goods or resources that a player would ideally receive. For example, in party-list proportional representation, a party's seat entitlement is equal to its share of the vote, times the number of seats in the legislature. 

Even when only money is to be divided and some fixed amount has been specified for each recipient, the problem can be complex. The amounts specified may be more or less than the amount of money, and the profit or loss will then need to be shared out. The proportional rule is normally used in law nowadays, and is the default assumption in the theory of bankruptcy. However, other rules can also be used. For example:

  • The Shapley value is one common method of deciding bargaining power, as can be seen in the airport problem.
  • Welfare economics on the other hand tries to determine allocations depending on a social welfare function.
  • The people can also agree on their relative entitlements by a consensus process. For instance they could say what they think everyone else is entitled to and if the assessments agree then they have an agreed impartial consensus division.
  • Priority rules are another kind of mechanism for allocation with different entitlements.

In the Talmud

The Talmud has a number of examples where entitlements are not decided on a proportional basis.

  • The disputed garment problem. If one person claims the whole of a cloth and another half then it is divided 3/4 and 1/4.
  • The estate division problem. Three wives have claims to 100, 200 and 300 zuz. Three cases are considered, if the estate is 100 zuz then they get 33 and a third each, if 200 then 50, 75, 75, and if 300 then 50, 100 and 150.
  • Profits from a joint fund. If two people put 200 and 100 into a fund and buy an ox for ploughing and use it for that purpose, they must divide the profit evenly between them. But if they instead they slaughter the ox, they divide the profit in proportion. This is discussed in the Babylonian Talmud (just after the estate division problem).
  • Ibn Ezra's problem. This is a later problem of estate division that was solved in a different way. A man with an estate of 120 dies bequeathing 120, 60, 40 and 30 to his four sons. The recommendation was to award (120–60)/1+(60–40)/2+(40–30)/3+(30–0)/4 to the first and sums with leading terms removed for the rest ending with 30/4 for the last. This allocation is different from the previous estate division.

These solutions can all be modeled by cooperative games. The estate division problem has a large literature and was first given a theoretical basis in game theory by Robert J. Aumann and Michael Maschler in 1985. See Contested garment rule.

Dividing continuous resources

Fair cake-cutting is the problem of dividing a heterogeneous continuous resource. There always exists a proportional cake-cutting respecting the different entitlements. The two main research questions are (a) how many cuts are required for a fair division? (b) how many queries are needed for computing a division? See:

Cloud computing environments require to divide multiple homogeneous divisible resources (e.g. memory or CPU) between users, where each user needs a different combination of resources.

Fair item allocation

Identical indivisible items - dividing seats in parliaments

In parliamentary democracies with proportional representation, each party is entitled to seats in proportion to its number of votes. In multi-constituency systems, each constituency is entitled to seats in proportion to its population. This is a problem of dividing identical indivisible items (the seats) among agents with different entitlements. It is called the apportionment problem.

The allocation of seats by size of population can leave small constituencies with no voice at all. The easiest solution is to have constituencies of equal size. Sometimes, however, this can prove impossible – for instance, in the European Union or United States. Ensuring the 'voting power' is proportional to the size of constituencies is a problem of entitlement.

There are a number of methods which compute a voting power for different sized or weighted constituencies. The main ones are the Shapley–Shubik power index, the Banzhaf power index. These power indexes assume the constituencies can join up in any random way and approximate to the square root of the weighting as given by the Penrose method. This assumption does not correspond to actual practice and it is arguable that larger constituencies are unfairly treated by them.

Heterogeneous indivisible items

In the more complex setting of fair item allocation, there are multiple different items with possibly different values to different people.

Aziz, Gaspers, Mackenzie and Walsh define proportionality and envy-freeness for agents with different entitlements, when the agents reveal only an ordinal ranking on the items, rather than their complete utility functions. They present a polynomial-time algorithm for checking whether there exists an allocation that is possibly proportional (proportional according to at least one utility profile consistent with the agent rankings), or necessarily proportional (proportional according to all utility profiles consistent with the rankings).

Farhadi, Ghodsi, Hajiaghayi, Lahaie, Pennock, Seddighin, Seddighin and Yami defined the Weighted Maximin Share (WMMS) as a generalization of the maximin share to agents with different entitlements. They showed that the best attainable multiplicative guarantee for the WMMS is 1/n in general, and 1/2 in the special case in which the value of each good to every agent is at most the agent's WMMS. Aziz, Chan and Li adapted the notion of WMMS to chores (items with negative utilities). They showed that, even for two agents, it is impossible to guarantee more than 4/3 of the WMMS (Note that with chores, the approximation ratios are larger than 1, and smaller is better). They present a 3/2-WMMS approximation algorithm for two agents, and an WMMS algorithm for n agents with binary valuations. They also define the OWMMS, which is the optimal approximation of WMMS that is attainable in the given instance. They present a polynomial-time algorithm that attains a 4-factor approximation of the OWMMS.

The WMMS is a cardinal notion in that, if the cardinal utilities of an agent changes, then the set of bundles that satisfy the WMMS for the agent may change. Babaioff, Nisan and Talgam-Cohen introduced another adaptation of the MMS to agents with different entitlements, which is based only on the agent's ordinal ranking of the bundles. They show that this fairness notion is attained by a competitive equilibrium with different budgets, where the budgets are proportional to the entitlements. This fairness notion is called Ordinal Maximin Share (OMMS) by Chakraborty, Segal-Halevi and Suksompong. The relation between various ordinal MMS approximations is further studied by Segal-Halevi.

Babaioff, Ezra and Feige present another ordinal notion, stronger than OMMS, which they call the AnyPrice Share (APS). They show a polynomial-time algorithm that attains a 3/5-fraction of the APS.

Aziz, Moulin and Sandomirskiy present a strongly polynomial time algorithm that always finds a Pareto-optimal and WPROP(0,1) allocation for agents with different entitlements and arbitrary (positive or negative) valuations.

Relaxations of WEF have been studied, so far, only for goods. Chakraborty, Igarashi and Suksompong introduced the weighted round-robin algorithm for WEF(1,0). In a follow-up work, Chakraborty, Schmidt-Kraepelin and Suksompong generalized the weighted round-robin algorithm to general picking-sequences, and studied various monotonicity properties of these sequences.

Items and money

In the problem of fair allocation of items and money, monetary transfers can be used to attain exact fairness of indivisible goods.

Corradi and Corradi define an allocation as equitable if the utility of each agent i (defined as the value of items plus the money given to i) is r ti ui (AllItems), where r is the same for all agents.

They present an algorithm that finds an equitable allocation with r >= 1, which means that the allocation is also proportional.

Bargaining

Cooperative bargaining is the abstract problem of selecting a feasible vector of utilities, as a function of the set of feasible utility vectors (fair division is a special case of bargaining).

Three classic bargaining solutions have variants for agents with different entitlements. In particular:

Operator (computer programming)

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Operator_(computer_programmin...