The first thing to correct is the assumption that Indian philosophy is primarily mystical. Nyaya logicians produced formal analyses of inference and fallacy as technical as anything in the Western tradition. Debates about what makes a belief count as knowledge, and about which sources of knowledge are legitimate, ran for centuries with the same argumentative rigour found in contemporary epistemology.
Six schools are traditionally classified as āstika, meaning they accept the authority of the Vedas — which is a statement about their relationship to a textual tradition, not about whether they argue. Nyaya developed logic and epistemology, holding that knowledge arises from perception, inference, comparison and reliable testimony. Vaisheshika proposed an atomistic account of the physical world. Samkhya distinguished pure consciousness from material nature, and Yoga supplied the practical discipline for realising that distinction. Mimamsa addressed scriptural interpretation and the nature of obligation. Vedanta, the last, became the most influential.
Advaita Vedanta, systematised by Shankara around the eighth century, makes the claim most often encountered outside India. Ātman — the self — and Brahman — ultimate reality — are not two things but one, and the appearance of separation is the product of avidyā, a fundamental ignorance. On this view liberation is not an achievement that adds something but a recognition of what was always the case. The proposal is metaphysically radical: individual selfhood and the world of distinct objects are, at the deepest level of analysis, not ultimately real in the way they appear.
It was vigorously contested within Vedanta itself. Ramanuja's qualified non-dualism held that the self is genuinely distinct from Brahman while wholly dependent on it — preserving a real relationship, and with it the possibility of devotion, which strict non-dualism strains. Madhva argued for outright dualism. These are not sectarian squabbles but arguments about identity, dependence and reality, conducted with detailed attention to what the texts can support and what reason requires.
Two further concepts travel widely and are widely mangled. Karma in these traditions is a principle of moral causation, not cosmic punishment, and the schools differ over how it operates. Moksha, liberation from the cycle of rebirth, is the goal most schools share while disagreeing sharply about its nature and how it is reached. Notably, Buddhist philosophers rejected the ātman entirely — arguing there is no enduring self to be identified with anything — and the resulting exchange between Buddhist and Hindu thinkers produced some of the most sophisticated metaphysics anywhere.