Indian Philosophy or Hindu Philosophy is generally classified into 6 orthodox schools (astika) and 3 heterodox (nastika) schools.
The basic difference between the two branches of Hindu Philosophy schools is said to be based on the recognition of Vedas. Orthodox schools recognize the authority of Vedas while heterodox schools don’t believe in the authority of Vedas.
 
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Six Orthodox Schools (Classical Schools) of Indian Philosophy
The 6 classical schools are Sankhya, Yoga, Nyaya, Vaisheshik, Purva Mimansa and Uttar Mimansa (Vedanta). Almost all Indian schools of thought accepted the theory of karma and rebirth, and the ideal of moksha is conceived as liberation from the cycle of births and deaths. Moksha/liberation is considered as the highest goal of human struggle.
Sankhya Philosophy
Sankhya is the philosphy put forward by Kapila. It is generally believed that Sankhya Philosophy is dualistic and not monistic because it has two entities, purush (spirit) and prakriti (nature) in it. Samkhya emphasizes the attainment of knowledge of self by means of concentration and meditation. Sankhya holds that it is the self-knowledge that leads to liberation and not any exterior influence or agent.
Purush vs Prakriti: In the beginning the philosophy was materialistic as it talked only about Prakrithi, but later the element of purush was also added to it. While Purusha is posited as the only sentient being, ever existent, and immaterial, Prakriti is said to be the material basis of this universe, composed of three basic elements (Gunas) – namely Tamas, Rajas, and Sattva.
Yoga Philosophy
Yoga presents a method of physical and mental discipline. The Yoga presents a practical path for the realization of the self whereas the Samkhya emphasizes the attainment of knowledge of self by means of concentration and meditation. Releasing Purush from Parkriti by means of physical and mental discipline is the concept behind Yoga.
Founder of Yoga is Pathanjai. Yoga does not require belief in God, although such a belief is accepted as help in initial stage of mental concentration and control of mind.
Nyaya Philosophy
Nyaya Philosophy states that nothing is acceptable unless it is in accordance with reason and experience (scientific approach). Founder of this philosophy is Gautam and the principles are mentioned in Nyaya Sutras. Nyaya says that the world is real and the philosophy does not follow a monist view.
Nyaya philosophy relies on several pramanas i.e. means of obtaining true knowledge as its epistemology. According to it, the pradhan pramana or principal means of obtaining knowledge is pratyaksha pramana i.e. the knowledge obtained through the 5 senses. There are also other pramanas like anumana (inference) and shabda pramana (scriptures, Vedas).
Vaisheshik Philosophy
The classical Indian philosophy Vaisheshik, was the physics of ancient times. It propounded the atomic theory of its founder Kannada. At one time Vaisheshik was regarded as part of the Nyaya philosophy, since physics is part of science. But since physics is the most fundamental of all sciences, Vaisheshik was later separated from Nyaya and put forth as a separate philosophy. To make it short, Vaisheshik is a realistic and objective philosophy of universe.
Purva Mimamsa (Mimamsa)
The word Mimamsa means to analyze and understand thoroughly. Purva Mimamsa examines the teachings of the Veda in the light of karma-kanda rituals, ie karma-mimamsa system is called purva-mimamsa. Purva mimansa (or briefly mimansa) lays emphasis on the performance of the yagya for attaining various spiritual and worldly benefits. Hence this philosophy relies on the Brahmana (and samhita) part of the Vedas.
Uttara Mimamsa (Vedanta)
Vedanta says that the world is unreal, Maya. Vedanta is monistic, in other words, it says that there is only one reality, Brahman. Vedanta lays emphasis on self knowledge, hence relies on the Upanishads which are part of the Vedas.
 
 
Heterodox Schools of Indian Philosophy
Schools that do not accept the authority of vedas are by definition unorthodox (nastika) systems. The following schools belongs to heterodox schools of Indian Philosophy.
Carvaka
It is characterised as materialistic and aesthetic school of thought. Accepted direct perception as the surest method to prove the truth of anything. Insists on joyful living. Also known as Lokayata, Carvaka is a materialistic school of thought. Its founder was Carvaka, author of the Barhaspatya Sutras
Buddhist Philosophy
It is a system of beliefs based on the teachings of Siddhartha Gautma. Buddhism is a non-theistic philosophy whose tenets are not especially concerned with the existence or non-existence of God.
Buddhists philosophy of life to get ‘Nirvana’ from suffering is based on the following eight principles
 
Ajivika Philosophy
The Ajivikas may simply have been a more loosely-organized group of wandering ascetics (shramanas or sannyasins)

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