Bhedabheda (difference and non-difference), as early as the 7th century CE, or even the 4th century CE. Some scholars are inclined to consider it as a "tradition" rather than a school of Vedanta.
Modern developments in Vedanta include Neo-Vedanta, and the growth of the Swaminarayan Sampradaya. All of these schools, except Advaita Vedanta and Neo-Vedanta, are related to Vaishavism and emphasize devotion, regarding Vishnu or Krishna or a related manifestation, to be the highest Reality. While Advaita Vedanta attracted considerable attention in the West due to the influence of Hindu modernists like Swami Vivekananda, most of the other Vedanta traditions are seen as discourses articulating a form of Vaishnav theology.
The word Vedanta literally means the end of the Vedas and originally referred to the Upanishads. Vedanta is concerned with the jñ?naka or knowledge section of the vedas which is called the Upanishads. The denotation of Vedanta subsequently widened to include the various philosophical traditions based on to the Prasthanatrayi.
The Upanishads may be regarded as the end of Vedas in different senses:
These were the last literary products of the Vedic period.
Brahman, c.q. Ishvara (God), exists as the unchanging material cause and instrumental cause of the world. The only exception here is that Dvaita Vedanta does not hold Brahman to be the material cause, but only the efficient cause.
The self (?tman/Jiva) is the agent of its own acts (karma) and the recipient of the consequences of these actions.
Belief in rebirth and the desirability of release from the cycle of rebirths, (mok?a).
The Upanishads,[b] or ?ruti prasth?na; considered the Sruti, the "heard" (and repeated) foundation of Vedanta.
The Brahma Sutras, or Nyaya prasthana / Yukti prasthana; considered the reason-based foundation of Vedanta.
The Bhagavad Gita, or Smriti prasth?na; considered the Smriti (remembered tradition) foundation of Vedanta.
The Brahma Sutras attempted to synthesize the teachings of the Upanishads. The diversity in the teaching of the Upanishads necessitated the systematization of these teachings. This was likely done in many ways in ancient India, but the only surviving version of this synthesis is the Brahma Sutras of Badarayana.
Prakriti/Jagat: the empirical world, ever-changing physical universe, body and matter
Brahman / Ishvara - Conceptions of the Supreme Reality
Shankara, in formulating Advaita, talks of two conceptions of Brahman: The higher Brahman as undifferentiated Being, and a lower Brahman endowed with qualities as the creator of the universe.
Par? or Higher Brahman: The undifferentiated, absolute, infinite, transcendental, supra-relational Brahman beyond all thought and speech is defined as par?Brahman, nirvi?e?a Brahman or nirgu?a Brahman and is the Absolute of metaphysics.
Apar? or Lower Brahman: The Brahman with qualities defined as apar?Brahman or sagu?aBrahman. The sagu?aBrahman is endowed with attributes and represents the personal God of religion.
Ramanuja, in formulating Vishishtadvaita Vedanta, rejects nirgu?a - that the undifferentiated Absolute is inconceivable - and adopts a theistic interpretation of the Upanishads, accepts Brahman as Ishvara, the personal God who is the seat of all auspicious attributes, as the One reality. The God of Vishishtadvaita is accessible to the devotee, yet remains the Absolute, with differentiated attributes.
Madhva, in expounding Dvaita philosophy, maintains that Vishnu is the supreme God, thus identifying the Brahman, or absolute reality, of the Upanishads with a personal god, as Ramanuja had done before him. Nimbarka, in his dvaitadvata philosophy, accepted the Brahman both as nirgu?a and as sagu?a. Vallabha, in his shuddhadvaita philosophy, not only accepts the triple ontological essence of the Brahman, but also His manifestation as personal God (Ishvara), as matter and as individual souls.
Relation between Brahman and Jiva / Atman
The schools of Vedanta differ in their conception of the relation they see between ?tman / Jiv?tman and Brahman / Ishvara:
According to Advaita Vedanta, ?tman is identical with Brahman and there is no difference.
According to Vishishtadvaita, J?v?tman is different from Ishvara, though eternally connected with Him as His mode. The oneness of the Supreme Reality is understood in the sense of an organic unity (vishistaikya). Brahman / Ishvara alone, as organically related to all J?v?tman and the material universe is the one Ultimate Reality.
According to Dvaita, the J?v?tman is totally and always different from Brahman / Ishvara.
According to Shuddhadvaita (pure monism), the J?v?tman and Brahman are identical; both, along with the changing empirically-observed universe being Krishna.
Epistemology in Dvaita and Vishishtadvaita Vedanta. Advaita and some other Vedanta schools recognize six epistemic means.
Prama (Sanskrit: ) literally means "proof", "that which is the means of valid knowledge". It refers to epistemology in Indian philosophies, and encompasses the study of reliable and valid means by which human beings gain accurate, true knowledge. The focus of Pramana is the manner in which correct knowledge can be acquired, how one knows or does not know, and to what extent knowledge pertinent about someone or something can be acquired. Ancient and medieval Indian texts identify six[c]pramanas as correct means of accurate knowledge and truths:
?abda (scriptural testimony/ verbal testimony of past or present reliable experts).
The different schools of Vedanta have historically disagreed as to which of the six are epistemologically valid. For example, while Advaita Vedanta accepts all six pramanas, Vishishtadvaita and Dvaita accept only three pramanas (perception, inference and testimony).
Advaita considers Pratyak?a (perception) as the most reliable source of knowledge, and ?abda, the scriptural evidence, is considered secondary except for matters related to Brahman, where it is the only evidence.[d] In Vishistadvaita and Dvaita, ?abda, the scriptural testimony, is considered the most authentic means of knowledge instead.
Theories of cause and effect
All schools of Vedanta subscribe to the theory of Satk?ryav?da, which means that the effect is pre-existent in the cause. But there are two different views on the status of the "effect", that is, the world. Most schools of Vedanta, as well as Samkhya, support Parinamavada, the idea that the world is a real transformation (parinama) of Brahman. According to Nicholson (2010, p. 27), "the Brahma Sutras espouse the realist Parinamavada position, which appears to have been the view most common among early Vedantins". In contrast to Badarayana, Adi Shankara and Advaita Vedantists hold a different view, Vivartavada, which says that the effect, the world, is merely an unreal (vivarta) transformation of its cause, Brahman.[e]
Overview of the main schools of Vedanta
The Upanishads present an associative philosophical inquiry in the form of identifying various doctrines and then presenting arguments for or against them. They form the basic texts and Vedanta interprets them through rigorous philosophical exegesis to defend the point of view of their specific sampradaya. Varying interpretations of the Upanishads and their synthesis, the Brahma Sutras, led to the development of different schools of Vedanta over time.
According to Gavin Flood, while Advaita Vedanta is the "most famous" school of Vedanta, and "often, mistakenly, taken to be the only represntative of Vedantic thought," and Shankara a Saivite, "Vedanta is essentially a Vaisnava theological articulation," a discourse broadly within the parameters of Vaisnavism." Within the Vaishnava traditions four sampradays have special status, while different scholars have classified the Vedanta schools ranging from three to six[f] as prominent ones.[g]
Bhedabheda, as early as the 7th century CE, or even the 4th century CE. Some scholars are inclined to consider it as a "tradition" rather than a school of Vedanta.
Bhed?bheda means "difference and non-difference" and is more a tradition than a school of Vedanta. The schools of this tradition emphasize that the individual self (J?vatman) is both different and not different from Brahman. Notable figures in this school are Bhartriprapancha, Nimb?rka (7th century) who founded the Dvaitadvaita school, Bh?skara (8th-9th century), Ramanuja's teacher Y?davapraka, Caitanya (1486-1534) who founded the Achintya Bheda Abheda school, and Vijñ?nabhik?u (16th century).[h]
Nimbarkacharya's icon at Ukhra, West Bengal
Nimb?rka (7th century) sometimes identified with Bh?skara, propounded Dvait?dvaita.Brahman (God), souls (chit) and matter or the universe (achit) are considered as three equally real and co-eternal realities. Brahman is the controller (niyanta), the soul is the enjoyer (bhokta), and the material universe is the object enjoyed (bhogya). The Brahman is Krishna, the ultimate cause who is omniscient, omnipotent, all-pervading Being. He is the efficient cause of the universe because, as Lord of Karma and internal ruler of souls, He brings about creation so that the souls can reap the consequences of their karma. God is considered to be the material cause of the universe because creation was a manifestation of His powers of soul (chit) and matter (achit); creation is a transformation (parinama) of God's powers. He can be realized only through a constant effort to merge oneself with His nature through meditation and devotion. 
Chaitanya Mahaprabhu (1486 - 1533) was the prime exponent of Achintya-Bheda-Abheda. In Sanskritachintya means 'inconceivable'.Achintya-Bheda-Abheda represents the philosophy of "inconceivable difference in non-difference", in relation to the non-dual reality of Brahman-Atman which it calls (Krishna), svayam bhagavan. The notion of "inconceivability" (acintyatva) is used to reconcile apparently contradictory notions in Upanishadic teachings. This school asserts that Krishna is Bhagavan of the bhakti yogins, the Brahman of the jnana yogins, and has a divine potency that is inconceivable. He is all-pervading and thus in all parts of the universe (non-difference), yet he is inconceivably more (difference). This school is at the foundation of the Gaudiya Vaishnava religious tradition.
Advaita Vedanta (non-dualism)
Advaita Vedanta (IASTAdvaita Ved?nta; Sanskrit: ?), propounded by Gaudapada (7th century) and Adi Shankara (8th century), espouses non-dualism and monism. Brahman is held to be the sole unchanging metaphysical reality and identical to the individual Atman. The physical world, on the other hand, is always-changing empirical Maya.[i] The absolute and infinite Atman-Brahman is realized by a process of negating everything relative, finite, empirical and changing.
The school accepts no duality, no limited individual souls (Atman / Jivatman), and no separate unlimited cosmic soul. All souls and their existence across space and time are considered to be the same oneness.  Spiritual liberation in Advaita is the full comprehension and realization of oneness, that one's unchanging Atman (soul) is the same as the Atman in everyone else, as well as being identical to Brahman.
Vishishtadvaita (qualified non-dualism)
Ramanujacharya depicted with Vaishnava Tilaka and Vishnu statue.
Vishishtadvaita, propounded by Ramanuja (11-12th century), asserts that Jivatman (human souls) and Brahman (as Vishnu) are different, a difference that is never transcended. With this qualification, Ramanuja also affirmed monism by saying that there is unity of all souls and that the individual soul has the potential to realize identity with the Brahman.Vishishtadvaita, like Advaita, is a non-dualistic school of Vedanta in a qualified way, and both begin by assuming that all souls can hope for and achieve the state of blissful liberation. On the relation between the Brahman and the world of matter (Prakriti), Vishishtadvaita states both are two different absolutes, both metaphysically true and real, neither is false or illusive, and that sagunaBrahman with attributes is also real. Ramanuja states that God, like man, has both soul and body, and the world of matter is the glory of God's body. The path to Brahman (Vishnu), according to Ramanuja, is devotion to godliness and constant remembrance of the beauty and love of the personal god (bhakti of sagunaBrahman).
Dvaita, propounded by Madhvacharya (13th century), is based on the premise of dualism. Atman (soul) and Brahman (as Vishnu) are understood as two completely different entities.Brahman is the creator of the universe, perfect in knowledge, perfect in knowing, perfect in its power, and distinct from souls, distinct from matter.[j] In Dvaita Vedanta, an individual soul must feel attraction, love, attachment and complete devotional surrender to Vishnu for salvation, and it is only His grace that leads to redemption and salvation. Madhva believed that some souls are eternally doomed and damned, a view not found in Advaita and Vishishtadvaita Vedanta. While the Vishishtadvaita Vedanta asserted "qualitative monism and quantitative pluralism of souls", Madhva asserted both "qualitative and quantitative pluralism of souls".
Shuddh?dvaita (pure nondualism)
Shuddhadvaita (pure non-dualism), propounded by Vallabhacharya (1479-1531 CE), states that the entire universe is real and is subtly Brahman only in the form of Krishna. Vallabhacharya agreed with Advaita Vedanta's ontology, but emphasized that prakriti (empirical world, body) is not separate from the Brahman, but just another manifestation of the latter. Everything, everyone, everywhere - soul and body, living and non-living, jiva and matter - is the eternal Krishna. The way to Krishna, in this school, is bhakti. Vallabha opposed renunciation of monistic sannyasa as ineffective and advocates the path of devotion (bhakti) rather than knowledge (jnana). The goal of bhakti is to turn away from ego, self-centered-ness and deception, and to turn towards the eternal Krishna in everything continually offering freedom from samsara.
The history of Vedanta can be divided into two periods: one prior to the composition of the Brahma Sutras and the other encompassing the schools that developed after the Brahma Sutras were written.
Before the Brahma Sutras (before the 5th century)
Little is known of schools of Vedanta existing before the composition of the Brahma Sutras (400-450 CE).[k] It is clear that Badarayana, the writer of Brahma Sutras, was not the first person to systematize the teachings of the Upanishads, as he quotes six Vedantic teachers before him - Ashmarathya, Badari, Audulomi, Kashakrtsna, Karsnajini and Atreya. References to other early Vedanta teachers - Brahmadatta, Sundara, Pandaya, Tanka and Dravidacharya - are found in secondary literature of later periods. The works of these ancient teachers have not survived, but based on the quotes attributed to them in later literature, Sharma postulates that Ashmarathya and Audulomi were Bhedabheda scholars, Kashakrtsna and Brahmadatta were Advaita scholars, while Tanka and Dravidacharya were either Advaita or Vishistadvaita scholars.
Brahma Sutras (completed in the 5th century)
Badarayana summarized and interpreted teachings of the Upanishads in the Brahma Sutras, also called the Vedanta Sutra,[l] possibly "written from a Bhed?bheda Ved?ntic viewpoint." Badarayana summarized the teachings of the classical Upanishads[m] and refuted the rival philosophical schools in ancient India.Nicholson 2010, p. 26 The Brahma Sutras laid the basis for the development of Vedanta philosophy.
Though attributed to Badarayana, the Brahma Sutras were likely composed by multiple authors over the course of hundreds of years. The estimates on when the Brahma Sutras were complete vary, with Nakamura in 1989 and Nicholson in his 2013 review stating, that they were most likely compiled in the present form around 400-450 CE.[n] Isaeva suggests they were complete and in current form by 200 CE, while Nakamura states that "the great part of the Sutra must have been in existence much earlier than that."
The book is composed of four chapters, each divided into four-quarters or sections. These sutras attempt to synthesize the diverse teachings of the Upanishads. However, the cryptic nature of aphorisms of the Brahma Sutras have required exegetical commentaries. These commentaries have resulted in the formation of numerous Vedanta schools, each interpreting the texts in its own way and producing its own commentary.
Between the Brahma Sutras and Adi Shankara (5th-8th centuries)
Little with specificity is known of the period between the Brahma Sutras (5th century CE) and Adi Shankara (8th century CE). Only two writings of this period have survived: the V?kyapad?ya, written by Bhart?hari (second half 5th century,) and the K?rik? written by Gaudapada (early 6th or 7th century CE).
Shankara mentions 99 different predecessors of his school in his commentaries. A number of important early Vedanta thinkers have been listed in the Siddhitraya by Yamun?c?rya (c. 1050), the Ved?rthasamgraha by R?m?nuja (c. 1050-1157), and the Yat?ndramatad?pik? by ?r?niv?sa D?sa. At least fourteen thinkers are known to have existed between the composition of the Brahma Sutras and Shankara's lifetime.[o]
A noted scholar of this period was Bhartriprapancha. Bhartriprapancha maintained that the Brahman is one and there is unity, but that this unity has varieties. Scholars see Bhartriprapancha as an early philosopher in the line who teach the tenet of Bhedabheda.
Gaudapada, Adi Shankara (Advaita Vedanta) (6th-9th centuries)
Influenced by Buddhism, Advaita vedanta departs from the bhedabheda-philosophy, instead postulating the identity of Atman with the Whole (Brahman),
Gaudapada (c. 6th century CE), was the teacher or a more distant predecessor of Govindapada, the teacher of Adi Shankara. Shankara is widely considered as the apostle of Advaita Vedanta. Gaudapada's treatise, the K?rik? - also known as the Mukya K?rik? or the ?gama stra - is the earliest surviving complete text on Advaita Vedanta.[p]
Gaudapada's K?rik? relied on the Mandukya, Brihadaranyaka and ChhandogyaUpanishads. In the K?rik?, Advaita (non-dualism) is established on rational grounds (upapatti) independent of scriptural revelation; its arguments are devoid of all religious, mystical or scholastic elements. Scholars are divided on a possible influence of Buddhism on Gaudapada's philosophy.[q] The fact that Shankara, in addition to the Brahma Sutras, the principal Upanishads and the Bhagvad Gita, wrote an independent commentary on the K?rik? proves its importance in Ved?ntic literature.
Adi Shankara (788-820), elaborated on Gaudapada's work and more ancient scholarship to write detailed commentaries on the Prasthanatrayi and the K?rik?. The Mandukya Upanishad and the K?rik? have been described by Shankara as containing "the epitome of the substance of the import of Vedanta". It was Shankara who integrated Gaudapada work with the ancient Brahma Sutras, "and give it a locus classicus" alongside the realistic strain of the Brahma Sutras.[r] His interpretation, including works ascribed to him, has become the normative interpretation of Advaita Vedanta.[s]
A noted contemporary of Shankara was Maana Mi?ra, who regarded Mimamsa and Vedanta as forming a single system and advocated their combination known as Karma-jnana-samuchchaya-vada.[t] The treatise on the differences between the Vedanta school and the Mimamsa school was a contribution of Adi Shankara. Advaita Vedanta rejects rituals in favor of renunciation, for example.
Early Vaishnavism Vedanta (7th-9th centuries)
Early Vaishnava Vedanta retains the tradition of bhedabheda, equating Brahman with Vishnu or Krishna.
Bh?skara (8th-9th century) also taught Bhedabheda. In postulating Upadhika, he considers both identity and difference to be equally real. As the causal principle, Brahman is considered non-dual and formless pure being and intelligence. The same Brahman, manifest as events, becomes the world of plurality. J?va is Brahman limited by the mind. Matter and its limitations are considered real, not a manifestation of ignorance. Bhaskara advocated bhakti as dhyana (meditation) directed toward the transcendental Brahman. He refuted the idea of Maya and denied the possibility of liberation in bodily existence.
R?m?nuja (1017-1137 CE) was the most influential philosopher in the Vishishtadvaita tradition. As the philosophical architect of Vishishtadvaita, he taught qualified non-dualism. Ramanuja's teacher, Yadava Prakasha, followed the Advaita monastic tradition. Tradition has it that Ramanuja disagreed with Yadava and Advaita Vedanta, and instead followed Nathamuni and Y?muna. Ramanuja reconciled the Prasthanatrayi with the theism and philosophy of the Vaishnava Alvars poet-saints. Ramanuja wrote a number of influential texts, such as a bhasya on the Brahma Sutras and the Bhagavad Gita, all in Sanskrit.
Ramanuja presented the epistemological and soteriological importance of bhakti, or the devotion to a personal God (Vishnu in Ramanuja's case) as a means to spiritual liberation. His theories assert that there exists a plurality and distinction between Atman (souls) and Brahman (metaphysical, ultimate reality), while he also affirmed that there is unity of all souls and that the individual soul has the potential to realize identity with the Brahman. Vishishtadvaiata provides the philosophical basis of Sri Vaishnavism.
Ramanuja was influential in integrating Bhakti, the devotional worship, into Vedanta premises.
Madhva (Dvaita Vedanta)(13th-14th centuries)
Dvaita Vedanta was propounded by Madhvacharya (1238-1317 CE).[u] He presented the opposite interpretation of Shankara in his Dvaita, or dualistic system. In contrast to Shankara's non-dualism and Ramanuja's qualified non-dualism, he championed unqualified dualism. Madhva wrote commentaries on the chief Upanishads, the Bhagavad Gita and the Brahma Sutra.
Madhva started his Vedic studies at age seven, joined an Advaita Vedanta monastery in Dwarka (Gujarat), studied under guru Achyutrapreksha, frequently disagreed with him, left the Advaita monastery, and founded Dvaita. Madhva and his followers Jayatirtha and Vyasatirtha, were critical of all competing Hindu philosophies, Jainism and Buddhism, but particularly intense in their criticism of Advaita Vedanta and Adi Shankara.
Dvaita Vedanta is theistic and it identifies Brahman with Narayana, or more specifically Vishnu, in a manner similar to Ramanuja's Vishishtadvaita Vedanta. But it is more explicitly pluralistic. Madhva's emphasis for difference between soul and Brahman was so pronounced that he taught there were differences (1) between material things; (2) between material things and souls; (3) between material things and God; (4) between souls; and (5) between souls and God. He also advocated for a difference in degrees in the possession of knowledge. He also advocated for differences in the enjoyment of bliss even in the case of liberated souls, a doctrine found in no other system of Indian philosophy.
Swaminarayan and Akshar-Purushottam Darshan (19th century)
The Akshar-Purushottam Darshan, which is philosophically related to Ramanuja's Vishishtadvaita,[v] was founded in 1801 by Swaminarayan (1781-1830 CE), and is contemporarily most notably propagated by BAPS. Due to the commentarial work of Bhadreshdas Swami, the Akshar-Purushottam teachings were recognized as a distinct school of Vedanta by the Shri Kashi Vidvat Parishad in 2017 and by members of the 17th World Sanskrit Conference in 2018.[w] Swami Paramtattvadas describes the Akshar-Purushottam teachings as "a distinct school of thought within the larger expanse of classical Vedanta," presenting the Akshar-Purushottam teachings as a seventh school of Vedanta.
Neo-Vedanta (19th century)
Neo-Vedanta, variously called as "Hindu modernism", "neo-Hinduism", and "neo-Advaita", is a term that denotes some novel interpretations of Hinduism that developed in the 19th century, presumably as a reaction to the colonial British rule.King (2002, pp. 129-135) writes that these notions accorded the Hindu nationalists an opportunity to attempt the construction of a nationalist ideology to help unite the Hindus to fight colonial oppression. Western orientalists, in their search for its "essence", attempted to formulate a notion of "Hinduism" based on a single interpretation of Vedanta as a unified body of religious praxis. This was contra-factual as, historically, Hinduism and Vedanta had always accepted a diversity of traditions. King (1999, pp. 133-136) asserts that the neo-Vedantic theory of "overarching tolerance and acceptance" was used by the Hindu reformers, together with the ideas of Universalism and Perennialism, to challenge the polemic dogmatism of Judaeo-Christian-Islamic missionaries against the Hindus.
The neo-Vedantins argued that the six orthodox schools of Hindu philosophy were perspectives on a single truth, all valid and complementary to each other.Halbfass (2007, p. 307) sees these interpretations as incorporating western ideas into traditional systems, especially Advaita Vedanta. It is the modern form of Advaita Vedanta, states King (1999, p. 135), the neo-Vedantists subsumed the Buddhist philosophies as part of the Vedanta tradition[x] and then argued that all the world religions are same "non-dualistic position as the philosophia perennis", ignoring the differences within and outside of Hinduism. According to Gier (2000, p. 140), neo-Vedanta is Advaita Vedanta which accepts universal realism:
Ramakrishna, Vivekananda and Aurobindo have been labeled neo-Vedantists (the latter called it realistic Advaita), a view of Vedanta that rejects the Advaitins' idea that the world is illusory. As Aurobindo phrased it, philosophers need to move from 'universal illusionism' to 'universal realism', in the strict philosophical sense of assuming the world to be fully real.
Nicholson (2010, p. 2) writes that the attempts at integration which came to be known as neo-Vedanta were evident as early as between the 12th and the 16th century-
... certain thinkers began to treat as a single whole the diverse philosophical teachings of the Upanishads, epics, Puranas, and the schools known retrospectively as the "six systems" (saddarsana) of mainstream Hindu philosophy.[y]
Matilal criticizes Neo-Hinduism as an oddity developed by West-inspired Western Indologists and attributes it to the flawed Western perception of Hinduism in modern India. In his scathing criticism of this school of reasoning, Matilal (2002, pp. 403-404) says:
The so-called 'traditional' outlook is in fact a construction. Indian history shows that the tradition itself was self-conscious and critical of itself, sometimes overtly and sometimes covertly. It was never free from internal tensions due to the inequalities that persisted in a hierarchical society, nor was it without confrontation and challenge throughout its history. Hence Gandhi, Vivekananda and Tagore were not simply 'transplants from Western culture, products arising solely from confrontation with the west.
...It is rather odd that, although the early Indologists' romantic dream of discovering a pure (and probably primitive, according to some) form of Hinduism (or Buddhism as the case may be) now stands discredited in many quarters; concepts like neo-Hinduism are still bandied about as substantial ideas or faultless explanation tools by the Western 'analytic' historians as well as the West-inspired historians of India.
According to Nakamura (2004, p. 3), the Vedanta school has had a historic and central influence on Hinduism:
The prevalence of Vedanta thought is found not only in philosophical writings but also in various forms of (Hindu) literature, such as the epics, lyric poetry, drama and so forth. ... the Hindu religious sects, the common faith of the Indian populace, looked to Vedanta philosophy for the theoretical foundations for their theology. The influence of Vedanta is prominent in the sacred literatures of Hinduism, such as the various Puranas, Samhitas, Agamas and Tantras ... 
Frithjof Schuon summarizes the influence of Vedanta on Hinduism as follows:
The Vedanta contained in the Upanishads, then formulated in the Brahma Sutra, and finally commented and explained by Shankara, is an invaluable key for discovering the deepest meaning of all the religious doctrines and for realizing that the Sanatana Dharma secretly penetrates all the forms of traditional spirituality.
... the most influential school of theology in India has been Vedanta, exerting enormous influence on all religious traditions and becoming the central ideology of the Hindu renaissance in the nineteenth century. It has become the philosophical paradigm of Hinduism "par excellence".
Vedanta, adopting ideas from other orthodox (?stika) schools, became the most prominent school of Hinduism. Vedanta traditions led to the development of many traditions in Hinduism.Sri Vaishnavism of south and southeastern India is based on Ramanuja's Vishishtadvaita Vedanta. Ramananda led to the Vaishnav Bhakti Movement in north, east, central and west India. This movement draws its philosophical and theistic basis from Vishishtadvaita. A large number of devotional Vaishnavism traditions of east India, north India (particularly the Braj region), west and central India are based on various sub-schools of Bhedabheda Vedanta.Advaita Vedanta influenced Krishna Vaishnavism in the northeastern state of Assam. The Madhva school of Vaishnavism found in coastal Karnataka is based on Dvaita Vedanta.
?gamas, the classical literature of Shaivism, though independent in origin, show Vedanta association and premises. Of the 92 ?gamas, ten are (dvaita) texts, eighteen (bhedabheda), and sixty-four (advaita) texts. While the Bhairava Shastras are monistic, Shiva Shastras are dualistic.Isaeva (1995, pp. 134-135) finds the link between Gaudapada's Advaita Vedanta and Kashmir Shaivism evident and natural. Tirumular, the Tamil Shaiva Siddhanta scholar, credited with creating "Vedanta-Siddhanta" (Advaita Vedanta and Shaiva Siddhanta synthesis), stated, "becoming Shiva is the goal of Vedanta and Siddhanta; all other goals are secondary to it and are vain."
Shaktism, or traditions where a goddess is considered identical to Brahman, has similarly flowered from a syncretism of the monist premises of Advaita Vedanta and dualism premises of Samkhya-Yoga school of Hindu philosophy, sometimes referred to as Shaktadavaitavada (literally, the path of nondualistic Shakti).
Influence on Western thinkers
An exchange of ideas has been taking place between the western world and Asia since the late 18th century as a result of colonization of parts of Asia by Western powers. This also influenced western religiosity. The first translation of Upanishads, published in two parts in 1801 and 1802, significantly influenced Arthur Schopenhauer, who called them the consolation of his life. He drew explicit parallels between his philosophy, as set out in The World as Will and Representation, and that of the Vedanta philosophy as described in the work of Sir William Jones. Early translations also appeared in other European languages. Influenced by ?a?kara's concepts of Brahman (God) and m?y? (illusion), Lucian Blaga often used the concepts marele anonim (the Great Anonymous) and cenzura transcendent? (the transcendental censorship) in his philosophy.
Similarities with Spinoza's philosophy
German Sanskritist Theodore Goldstücker was among the early scholars to notice similarities between the religious conceptions of the Vedanta and those of the Dutch Jewish philosopher Baruch Spinoza, writing that Spinoza's thought was
... so exact a representation of the ideas of the Vedanta, that we might have suspected its founder to have borrowed the fundamental principles of his system from the Hindus, did his biography not satisfy us that he was wholly unacquainted with their doctrines [...] comparing the fundamental ideas of both we should have no difficulty in proving that, had Spinoza been a Hindu, his system would in all probability mark a last phase of the Vedanta philosophy.
Max Müller noted the striking similarities between Vedanta and the system of Spinoza, saying,
The Brahman, as conceived in the Upanishads and defined by Sankara, is clearly the same as Spinoza's 'Substantia'."
As to Spinoza's Deity - natura naturans - conceived in his attributes simply and alone; and the same Deity - as natura naturata or as conceived in the endless series of modifications or correlations, the direct outflowing results from the properties of these attributes, it is the Vedantic Deity pure and simple.
^Historically, Vedanta has been called by various names. The early names were the Upanishadic ones (Aupanisada), the doctrine of the end of the Vedas (Vedanta-vada), the doctrine of Brahman (Brahma-vada), and the doctrine that Brahma is the cause (Brahma-karana-vada).
^The Upanishads were many in number and developed in the different schools at different times and places, some in the Vedic period and others in the medieval or modern era (the names of up to 112 Upanishads have been recorded). All major commentators have considered twelve to thirteen oldest of these texts as the Principal Upanishads and as the foundation of Vedanta.
^A few Indian scholars such as Vedvyasa discuss ten; Krtakoti discusses eight; six is most widely accepted: see Nicholson (2010, pp. 149-150)
^Anantanand Rambachan (1991, pp. xii-xiii) states, "According to these [widely represented contemporary] studies, Shankara only accorded a provisional validity to the knowledge gained by inquiry into the words of the ?ruti (Vedas) and did not see the latter as the unique source (pramana) of Brahmajnana. The affirmations of the ?ruti, it is argued, need to be verified and confirmed by the knowledge gained through direct experience (anubhava) and the authority of the ?ruti, therefore, is only secondary." Sengaku Mayeda (2006, pp. 46-47) concurs, adding Shankara maintained the need for objectivity in the process of gaining knowledge (vastutantra), and considered subjective opinions (purushatantra) and injunctions in ?ruti (codanatantra) as secondary. Mayeda cites Shankara's explicit statements emphasizing epistemology (pramana-janya) in section 1.18.133 of Upadesasahasri and section 1.1.4 of Brahmasutra-bhasya.
^Nicholson (2010, p. 27) writes of Advaita Vedantin position of cause and effect - Although Brahman seems to undergo a transformation, in fact no real change takes place. The myriad of beings are essentially unreal, as the only real being is Brahman, that ultimate reality which is unborn, unchanging, and entirely without parts.
^Proponents of other Vedantic schools continue to write and develop their ideas as well, although their works are not widely known outside of smaller circles of followers in India.
^According to Nakamura and Dasgupta, the Brahmasutras reflect a Bhedabheda point of view, the most influential tradition of Vedanta before Shankara. Numerous Indologists, including Surendranath Dasgupta, Paul hacker, Hajime Nakamura, and Mysore Hiriyanna, have described Bhedabheda as the most influential school of Vedanta before Shankara.
^Doniger (1986, p. 119) says "that to say that the universe is an illusion (m?y?) is not to say that it is unreal; it is to say, instead, that it is not what it seems to be, that it is something constantly being made. Maya not only deceives people about the things they think they know; more basically, it limits their knowledge."
^The concept of Brahman in Dvaita Vedanta is so similar to the monotheistic eternal God, that some early colonial-era Indologists such as George Abraham Grierson suggested Madhva was influenced by early Christians who migrated to India,  but later scholarship has rejected this theory.
^Nicholson (2010, p. 26) considers the Brahma Sutras as a group of sutras composed by multiple authors over the course of hundreds of years. The precise date is disputed.Nicholson (2010, p. 26) estimates that the book was composed in its current form between 400 and 450 CE. The reference shows BCE, but it´s a typo in Nicholson´s book
^The Vedanta-s?tra are known by a variety of names, including (1) Brahma-s?tra, (2) r?raka-sutra, (3) B?dar?ya?a-s?tra and (4) Uttara-m?ms?.
^Estimates of the date of B?dar?yana's lifetime differ. Pandey 2000, p. 4
^Nicholson 2013, p. 26 Quote: "From a historical perspective, the Brahmasutras are best understood as a group of sutras composed by multiple authors over the course of hundreds of years, most likely composed in its current form between 400 and 450 BCE." This dating has a typo in Nicholson's book, it should be read "between 400 and 450 CE"
^There is ample evidence, however, to suggest that Advaita was a thriving tradition by the start of the common era or even before that. Shankara mentions 99 different predecessors of his Sampradaya. Scholarship since 1950 suggests that almost all Sannyasa Upanishads have a strong Advaita Vedanta outlook. Six Sannyasa Upanishads - Aruni, Kundika, Kathashruti, Paramahamsa, Jabala and Brahma - were composed before the 3rd Century CE, likely in the centuries before or after the start of the common era; the Asrama Upanishad is dated to the 3rd Century. The strong Advaita Vedanta views in these ancient Sannyasa Upanishads may be, states Patrick Olivelle, because major Hindu monasteries of this period belonged to the Advaita Vedanta tradition.
^Scholars like Raju (1992, p. 177), following the lead of earlier scholars like Sengupta, believe that Gaudapada co-opted the Buddhist doctrine that ultimate reality is pure consciousness (vijñapti-m?tra). Raju (1992, pp. 177-178) states, "Gaudapada wove [both doctrines] into a philosophy of the Mandukaya Upanisad, which was further developed by Shankara." Nikhilananda (2008, pp. 203-206) states that the whole purpose of Gaudapada was to present and demonstrate the ultimate reality of Atman, an idea denied by Buddhism. According to Murti (1955, pp. 114-115), Gaudapada's doctrines are unlike Buddhism. Gaudapada's influential text consists of four chapters: Chapters One, Two, and Three are entirely Vedantin and founded on the Upanishads, with little Buddhist flavor. Chapter Four uses Buddhist terminology and incorporates Buddhist doctrines but Vedanta scholars who followed Gaudapada through the 17th century, state that both Murti and Richard King never referenced nor used Chapter Four, they only quote from the first three. While there is shared terminology, the doctrines of Gaudapada and Buddhism are fundamentally different, states Murti (1955, pp. 114-115)
^Nicholson (2010, p. 27) writes: "The Brahmasutras themselves espouse the realist Parinamavada position, which appears to have been the view most common among early Vedantins."
^Shankara synthesized the Advaita-v?da which had previously existed before him, and, in this synthesis, became the restorer & defender of an ancient learning. He was an unequaled commentator, due to whose efforts and contributions, Advaita Vedanta assumed a dominant position within Indian philosophy.
^According to Mishra, the sutras, beginning with the first sutra of Jaimini and ending with the last sutra of Badarayana, form one compact shastra.
^Many sources date him to 1238-1317 period, but some place him over 1199-1278 CE.
^Vishishtadvaita roots: * Supreme Court of India, 1966 AIR 1119, 1966 SCR (3) 242: "Philosophically, Swaminarayan was a follower of Ramanuja" * Hanna H. Kim: "The philosophical foundation for Swaminarayan devotionalism is the vi?idvaita, or qualified non-dualism, of R?m?nuja (1017-1137 ce)."
^"Professor Ashok Aklujkar said [...] Just as the Kashi Vidvat Parishad acknowledged Swaminarayan Bhagwan's Akshar-Purushottam Darshan as a distinct darshan in the Vedanta tradition, we are honored to do the same from the platform of the World Sanskrit Conference [...] Professor George Cardona [said] "This is a very important classical Sanskrit commentary that very clearly and effectively explains that Akshar is distinct from Purushottam."
^Vivekananda, clarifies Richard King, stated, "I am not a Buddhist, as you have heard, and yet I am"; but thereafter Vivekananda explained that "he cannot accept the Buddhist rejection of a self, but nevertheless honors the Buddha's compassion and attitude towards others".
^The tendency of "a blurring of philosophical distinctions" has also been noted by Burley. Lorenzen locates the origins of a distinct Hindu identity in the interaction between Muslims and Hindus, and a process of "mutual self-definition with a contrasting Muslim other", which started well before 1800.
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