Samkhya System
The Samkhya philosophy holds that reality is constituted of two principles one female and the other male i.e. Prakriti, Purusha respectively. Prakriti and Purusha are completely independent and absolute. According to this system, Purusha is mere consciousness, hence it cannot be modified or changed. Prakriti on the other hand is constituted of three attributes, thought, movement and the change or transformation of these attributes brings about the change in all objects.
Yoga
Yoga literally means the union of the two principal entities. The origin of yoga is found in the Yogasutra of Patanjali believed to have been written in the second century BC. By purifying and controlling changes in the mental mechanism, yoga systematically brings about the release of purusha from prakriti.
Nyaya
Nyaya is considered as a technique of logical thinking. According to Nyaya, valid knowledge is defined as the real knowledge, that is, one knows about the object as it exists. For example, it is when one knows a snake as a snake or a cup as a cup.
Vaisheshika
Vaisheshika system is considered as the realistic and objective philosophy of universe. The reality according to this philosophy has many bases or categories which are substance, attribute, action, genus, distinct quality and inherence. Vaisheshika thinkers believe that all objects of the universe are composed of five elements–earth, water, air, fire and ether.
Mimamsa
Mimamsa philosophy is basically the analysis of interpretation, application and the use of the text of the Samhita and Brahmana portions of the Veda. According to Mimamsa philosophy Vedas are eternal and possess all knowledge, and religion means the fulfilment of duties prescribed by the Vedas.
Vedanta
Vedanta implies the philosophy of the Upanishad, the concluding portion of the Vedas. Shankaracharya wrote the commentaries on the Upanishads, Brahmasutras and the Bhagavad Gita. Shankaracharya’s discourse or his philosophical views came to be known as Advaita Vedanta.