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There are plenty of philosophical schools in India which when studied individual are so different from each other that in any other case, they will qualify as separate religion. However, this is not the case seen in India, a land where even the mutuall contradictory philosophies and ideas have coexisted with each other peacefully.
These religions/paths were identified as Dharmic Paths OR Hindu system of religions. Here are few characteristics shared by them, irrespective of differences.
1) Belief in GOD is not mandatory in Hindu system of religions.
2) Hindu is a historical term which was used by Muslims to denote the followers of all religions of Indian origin. All Shaivas, Vaishnavas, Shaktas, tribals, Buddhists, Jains, Sikhs, Indian atheists, basically everyone who is not Muslim, Christian or Jew and followed a path of Indian origin is and was known as Hindu in medieval times. Hence, whenever I use the term Hindu, I refer to person following Non-Abrahmic religion of Indian origin.
3) All the Indian religions are categorised in two classes - Those who believe in supremacy of Vedas (Aastika) and those who do not believe in supremacy of Vedas (Naastika). Paths which fall under Aastika class are - Vaishnava, Shaiva, Shaakta, Saamkhya, Nyaya, Mimansa, Vedanta, Yoga, Vaisheshika, and others which make up modern day Hinduism. Paths which fall under Naastika class are - Bauddha, Jaina, Sikh (Who are ambiguous about Vedas), Ajivika, Charvaka and few others.
In both categories, few believe in existence of god, others do not. In both categories, the emphasis is laid on being righteous and perform one's duty, rather than belief in particular deity. This differentiates Indian religions from Abrahmic religions. Hence Indian religions are also known as Dharmic religions and are perfectly in harmony with each other.
The very concept of RELIGION was unknown to India before Islam. Prior, it was duty and righteousness, irrespective of belief in supernatural. Irrespective of belief system, following two quotes of Vedas are deeply ingrained in the minds of all Hindus, however ignorant they are.
Those two ideas are -
1) God is everywhere and everything is god ( Sarvam Khalu Idam Brahma)
2) God is one, different people refer to him and worship him in different forms (Eko Sat, vipra Bahuda Vadanti).
For an atheist Hindu, the term GOD is replaced by Brahman.
All Indian religions are called Dharmic religions. There are four achievements (Purushartha) in life of man following Dharmic religions 1) Dharma (Being righteous and following one's duty) 2) Artha (Pursuing material wealth as long as conduct is in domain of Dharma or righteousness) 3) Kaama (Fulfilment of all desires as long as means are righteous or Dharmic) 4) Moksha/Nirvana (liberation/complete self knowledge All the dharmic religions are in perfect harmony about the first three Purusharthas. The differences arise in fourth Purushartha.
Those differences arise in their differential perception of divine and ways of attaining the divine or supreme knowledge. Hence, whatever differences are, are all philosophical and not practical. It was seen very frequently that different people of same family following different schools of thought and yet live together peace fully. Hence whatever differences arose amongst Indian religions, they were resolved by means of debate, and not sword and forcible conversion. The most illustrious example is of Adi Shankaracharya who brought about non-violent revolution in India in 700 AD be vanquishing the Naastika schools and few atheist Astika Schools in India.
Majority of Indian population started following the reformed version of theistic vedantic religion after this campaign by this young unarmed monk. As I said, Vedanta is just one of the 6 philosophical schools of Astika Philosophy. Vedanta perceives world in three versions - Non-Dualism (Advaita); Nondual Dualism (Vishishta-Advaita); Dualism (Dvaita).
The oldest school is Saamkhya - It is an atheist school and does not believe in existence of God. Instead it propounds that universe is made up of two forces which are equal and opposite to each other and complete each other. The two entities are -
1) Purusha (Puram Ushati sa Purusha:) - One who spends time/burns in the citadel is Purusha. Here Citadel or a fortified city (pura) refers to Body as well as Universe.
2) Prakriti - (Pra + Kruta) - One which can perform an action spontaneously and vigorously is Prakriti.
In other words, Purusha and Prakriti refer to matter and energy. Same as Yin and Yang in tao, although much older than Tao. Prakriti (Energy) is made up of three GuN (Characteristics)
1) Satva - Literally - truthness, also refers to lightness, illumination
2) Rajas - comes from word raaja and refers to tendency to experience life, excitation, activity driven by selfish interests. (note - there is nothing negative about word Selfish)
3) Tamas - Literally darkness - refers to intense activity, coarseness, sloth
(Note - It is really difficult to express what these three GuN refer to in English. I recommend trying to understand them at least in Hindi or in any Indian language derived from Sanskrit)
Different things in universe including human beings have different combination of these three characteristics varying in different levels which makes up their Dharma or Basic nature and shapes their intellect and mind. Hence universe is how it is.
In theistic version of Samkhya, Purush and Prakriti are identified as Shiva and Shakti. It must be noted that although Samkhya talks of two entities which are equal and opposite, IT IS NOT A DUALIST school of thought.. Purusha has some Prakriti in it and Prakriti has some Purusha in it. They are not separate entities and cannot exist on their own. They always exist in pair, hence calling Samkhya a Dual school is fallacy.
According to Samkhya, Ignorance is root cause which needs to be eradicated. From Wiki article on Samkhya - [i]The Sankhya system is based on Satkaryavada. According to Satkaryavada, the effect pre-exists in the cause. Cause and effect are seen as different temporal aspects of the same thing - the effect lies latent in the cause which in turn seeds the next effect. More specifically, Sankhya system follows the Prakriti-Parinama Vada. Parinama denotes that the effect is a real transformation of the cause. The cause under consideration here is Prakriti or more precisely Mula-Prakriti (Primordial Matter).
The Sankhya system is therefore an exponent of an evolutionary theory of matter beginning with primordial matter. In evolution, Prakriti is transformed and differentiated into multiplicity of objects. Evolution is followed by dissolution. In dissolution the physical existence, all the worldly objects mingle back into Prakriti, which now remains as the undifferentiated, primordial substance. This is how the cycles of evolution and dissolution follow each other.
Goal of Samkhya - According to Sankhya, the Purusha is eternal, pure consciousness. Due to ignorance, it identifies itself with the physical body and its constituents - Manas, Ahamkara and Mahat, which are products of Prakriti. Once it becomes free of this false identification and the material bonds, Moksha ensues.
P.S. - Very beautiful philosphophy which has potential to explain most of the questions of life without having a need to assume existence of god. When coupled with Yoga, it becomes all pervading philosophy with slight theistic flavor, as Yoga acknowledges the existence of Non-acting Non-Interfering God, who although is a master of universe (Ishvara), never participates in any of the activities of Universe. So practically, he is Non-existent in most of Yoga, except when it comes to final aim of Moksha.
Yoga - Almost same as Samkhya. If Vedanta and Saamkhya is theory, then Yoga is Practical.. Knowledge, IMHO, can't be attained by any one single approach. This is the opinion of all the Astika schools.
Yog literally means Union.. It extensively deals with psychology and training which mind and body needs to be given for achievement of supreme knowledge. It is defined as controlling the tendencies of Mind. (Yog: Chitta Vritti Nirodh:). Control need not necessarily mean oppression. Some times in some conditions, some tendencies need to be over-expressed.
It is relative, and should be governed by Dharma (righteousness and Duty, more fittingly basic nature, in terms of Yoga). When, tendencies of mind are controlled, the characteristics of mind which is governed by Prakriti of three characteristics slowly change to neutral state of mind. This is called the process of going back towards birth (Prati-Prasava) of these tendencies.
For eg. If a person gets angry too often, trying to get rid of this quality by suppressing anger is illogical. Instead, slowly, the quality is taken towards its birth. Finding out the root cause of why one gets angry, and trying to work on the cause. Slowly, the intensity of anger will reduce and eventually vanish. Thus, one Characteristic of the given mind is controlled. Similarly to all characteristics. When all the characteristics of mind are controlled, person attains a state where the mind and the ego are dissolved.
Hence, a seedless state of mind (Nirbeej Samadhi) is attained, where the existence of individual as mind, intellect, and separate identity vanishes and he becomes Kaivalya or Singularity. This is state of Liberation or Moksha - Liberation from one'e mind, body, intellect and even a sense of separate individual existence. Thus we see, Samkhya, Yoga and Vedanta are mutually complementary to each other and belief in God does not matter.
Mimansa - Atheist school. Emphasizes on performance of rituals precisely and elaborately as essential thing for attainment of Moksha.
Nyaya - Indian Logic.. More mathematical, than spiritual. Presents the methodology of how knowledge should be attained and how to differentiate between true data and false data. Very scientific in approach. Atheist school of Philosophy.. Since gaining of knowledge is focus of the Samkhya, Yoga, and Vedanta, the seeker of knowledge must be aware of Nyaya in order to save himself from getting lost in the forest of knowledge. Nyaya provides a methodology of segregation of obtained data.
Vaisheshika - Indian Atomist school. Very similar to Nyaya school. again deals more with description of the universe whose truth a seeker wishes to decipher. Universe made up of different atoms. Different atoms of different substances have different characteristics. Different combinations of differently flavoured atoms gives rise to differentiality of the world.
Vedanta - It is a theistic school. Dvaita is one school which might be closest to Abrahmic religions. However, it must be emphasized that the characteristics of Supreme god in dvaita school although similar to abrahmic sky daddy in some aspects, differs a lot in many other aspects..
Three branches of Vedanta are popular.
1) Advaita Vedanta (Non-Dualism) - Everything is Brahman, without any duality
2) Vishishtadvaita (Qualified or Special dualism) - Brahman and world is like light and sun. Same but not the same
3) Dvaita (dualism) - Brahman and Universe are different entities with Ishwara = God = Master of Universe.
Thus from Astika branch of India religions we have Two theist schools - Yoga, Vedanta schools. three atheist schools - Nyaya, Mimansa, Vaisheshika one school which was originally atheist, but later a theist version was developed - Saamkhya. So in principle 4 Atheist schools and 2 theist schools. Furthermore, the The Hymn of Creation - Nasadiya Sukta (Rigveda 10:29), clearly states that " No one can say for sure whether He exists or not. Only He can know whether He exists or not".