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If the question is about Hinduism at present, then I think it is faulty. Of the six Astika schools—Sankya, Yoga, Nyaya, Vaisheshika, Mimangsha, Vedanta, each which differed a little bit in their philosophy—are all dead except the Vedanta. Hence, Upanishad means Vedanta now. Later, various philosophies emerged from Vedanta, namely Advaita, Bishista advaita, ...


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There is a different way of classification within Asktika i.e. those who accept the authority of vedas. Astikas can be classified into 2 groups: Vedanta / YogaVedanta, all the schools (including vedanta) mentioned in the question above fall under this category. VedaAgama / Agama: ShaivaAgamas(Shiva), ViashnAgamas(Vishnu), ShaktAgamas(Shakti) (maybe more) ...


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It is my understanding that Karma Kanda and Upasana Kanda are portions of Vedas where there is identity with specific Ishta Devata. In Vedanta the identity shifts to nature of self in this there are various schools of philosophy where in duality , monism etc are discussed as the nature of Self. So there cannot be a connection between shan-Matha and Vedanta


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