Sanatan Dharam has six systems of philosophy, which is called Shad Darshan. The six darshans are - Nyaya, Vaisheshik, Samkhya, Yog, Purva Mimansa and Uttar Mimansa or Vedanta, originated by sages - Aksapada, Kanada, Patanjali, Kapila, Jaimini and Veda Vyas respectively. They generally deal with four topics:
Nyaya: "System, rule; logic, analytic." Its tools of enquiry and rules for argumentation were adopted by all schools of Hinduism. Nyaya Darshan propounded by Akshapad Gautam son of rishi (sage) Deeraghatama in 4th Century, is on the existence of an external world. It holds that human suffering results from mistakes/defects produced by activity under wrong knowledge (notions and ignorance). Moksha (liberation), is gained through right knowledge.
It seeks to prove Parmatma’s existence based on following arguments:
Ultimate goal of Spiritual aspirant is ascertaining truth by applying the 16 padaraths or categories viz. means of knowledge (praman), objects of inquiry (prmeya), doubts (samshay), purpose (prayojan), examples (drushtant), established principles (siddhant), syllogistic premise (avayava), confutation (tarka), decisive ascertainment (vad), arguing constructively and destructively (jalpa), mere destructive argument of the opposition (vitanda) but not consolidating one’s stance, fallacious reason (hetvabhasha) in vitanda, quibbling (chala), futile objections (jati).
The number of means of knowledge (pramans) under Nyaya system are four viz. pratyaksh (perception), anuman (inference), upaman (comparison) and shabda (testimony, may be vedic or by trustworthy authority).
Vaishehika: Naturalism or atomism. Vaisheshika Darshan (370 Sutras, 10 chapters) was founded by Kaṇāda Kashyapa around the 2nd century BC. It postulated that all objects in the physical universe are reducible to paramāṇu (atoms), and one's experiences are derived from the interplay of substances (a function of atoms, and their count and spatial arrangements), quality, activity, commonness, particularity and inherence. And liberation is attained through understanding the nature of existence.
Nyaya and Vaisheshika are viewed as a complementary pair - with Nyaya emphasizing logic and Vaisheshika analyzing the nature of the world.
Vaisheshika Darshan deals with the following aspects:
Sankhya: "Enumeration, reckoning, relating to numbers." A philosophy founded by the sage Kapila (5th century BC), author of the Sankhya Sutras. Sankhya is realistic, dualistic and pluralistic darshan.
The philosophy revolves around the interaction of prakṛti and purusha. Prakṛti remains unmanifested as long as the three gunas - sattva, rajas & tamas - are in equilibrium. This equilibrium of the gunas is disturbed when prakṛti comes into proximity with consciousness or Purusha. The disequilibrium of the gunas triggers an evolution that leads to the manifestation of the world from an unmanifested prakṛti.
It aims to acquire vivek – to discriminate between purush and prakriti through knowledge of the 25 tattvas:
Dominance of sattva causes the evolution of the five organs of perception, five organs of action and the mind. Dominance of tamas triggers the evolution of five subtle elements – sound, touch, sight, taste, and smell from self-consciousness. These five subtle elements are themselves evolvents and cause the creation of the five gross elements - space, air, fire, water and earth. Rajas is cause of action in the evolutes. Purusha is pure consciousness absolute, eternal and subject to no change. It is neither a product of evolution, nor the cause of any evolute.
Prakriti, out of which all things evolve, is the unity of the three gunas: sattva, rajas and tamas. If Sattva increases, Rajas & Tamas decreases and vice versa. Sankhya and Yoga are considered an inseparable pair whose principles permeate all of Hinduism.
Yoga: joining." Ancient tradition of philosophy and practice codified by Patanjali (2nd century BC) in the Yoga Sutras. It is also known as raja yoga - "king of yogas," or ashtanga yoga - "eight-limbed yoga." Its objective is to achieve, at will, the cessation of all fluctuations of consciousness, and the attainment of Self Realization. Yoga is wholly dedicated to putting the high philosophy of Hinduism into practice, to achieve personal transformation through transcendental experience, samadhi. Râja-Yoga is divided into eight steps. Yama and Niyama, as we see, are moral trainings; without these as the basis no practice of Yoga will succeed. As these two become established, the Yogi will begin to realise the fruits of his practice; without these it will never bear fruit. Pratyahara, Dharna, Dhyan and Samadhi. Pratyâhâra, or restraint of the senses from their objects; Dhâranâ, or fixing the mind on a spot; Dhyâna, or meditation; and Samâdhi, or superconsciousness. The mind can exist on a still higher plane, the superconscious.
Purva Mimamsa: "Inquiry" (or Purva, "early," Karma Mimamsa or Karma Kanda), Ritualism. Founded by Jaimini (dating back to 2th century BC), taught the correct performance of Vedic mantras & rituals (Samhitas and Brahmanas) as the means to liberation.
Karma is broadly classified into five types as follows:
Vedanta (or Uttara "later" Mimamsa or Jnanakanda): Jñānakāṇḍa deals with the meditation, reflection and knowledge of Self, Oneness, Brahman (the Upaniṣads said to be 108).
The teaching of Vedånta is that there is one absolute reality - brahman. Man is one with brahman, and the object of life is to realize that truth through right knowledge, intuition and personal experience. The Vedanta Sutras (or Brahma Sutras) were composed by Rishi Badaråyana (400 BC).
There are various commentaries on the Vedanta. However, post Buddhistic commentators are Shankara (Advaita, non-dualistic form), Ramanuja (Vishishtadvait, qualified non-dualistic form), Nimbark (Dvaitavait), Madhava (Dvait, dualistic form) and Vallabh (Shudhadvait).