Vedanta means the completion of the Veda, which was the sacrificial and ritualistic faith of the horse-worshipping Aryans who conquered the Indus Valley around 1,500 B.C.
Friedrich Max Müller (1823-1900) was the first European scholar to publish complete translations of the Rig Veda and the principal Upanishads in English, as well as the Vedanta Sutra of Adi Shankara (8th c.) which finally systematized Vedanta down to its minutest details, and therefore provided Europe with a much more complete picture of this Hindu philosophy and religion than had been available hereto. His influence on Victorian England was huge, and he was instrumental in founding the study of comparative religion as an academic field, and in developing the idea of the inseparable relation of language and culture.

The brief introduction constructed below was extracted from lectures Müller gave on Vedanta in 1894 and designed to be as brief and pithy as possible. Since this is the dominant system of philosophy in India, and also in yoga and “spiritual” schools worldwide, I present it here in Müller’s own words. As refreshing and liberating as its ideas might seem to a person familiar only with the major monotheistic religions, in my opinion its view of reality is rather primitive and dated. In support of this statement, I have inserted critical comments in brackets throughout.
What Is Vedanta?
In India the prevailing philosophy is still the Vedanta. The most extraordinary feature of this Vedanta philosophy consists in its being an independent system of philosophy yet entirely dependent on the Upanishads — nay, chiefly occupied with proving that all its doctrines, to the very minutest points, are derived from the revealed doctrines of the Upanishads. In these Upanishads not only are all sacrificial duties rejected, but the very gods to whom the ancient prayers of the Veda were addressed are put aside to make room for one Supreme Being, called Brahman.
The same Upanishads had then to explain the true relation between Brahman, Supreme Being, and the soul of man. The soul of man was called Atman, literally the self, also Jivatman, the living self. After the substantial unity of the living or individual self with the Supreme Being or Brahman had been discovered, Brahman was called the Highest Self or Paramatman.
These terms were not new technical terms coined by philosophers. Some of them are very ancient and occur in the oldest Vedic compositions, in the hymns, the Brahmanas, and finally in the Upanishads. The etymology of Brahman and Atman is extremely difficult, and this very difficulty shows that both of these words, from the point of view of historical Sanskrit, belong to a prehistoric layer of Sanskrit (i.e., these terms are older than Sanskrit).
Significance of the Veda
I have often pointed out that the real importance of the Veda will always be the opportunity which it affords us of watching the active process of the fermentation of early thought. The growth of the divine idea is laid bare in the Veda as nowhere else. We see in the Vedic hymns the first revelation of Deity, the first expressions of surprise and suspicion, the first discovery that behind this visible and perishable world there must be something invisible, imperishable, eternal or divine.
[Must there be?]
Nearly all the leading deities of the Veda bear the unmistakable traces of their physical character. Their very names tell us that they were in the beginning names of the great phenomena of nature, of fire, water, rain and storm, of sun and moon, of heaven and earth. We see before our eyes the bright powers of heaven and earth who became the Devas, the Bright Ones.
We see how these individual and dramatic deities ceased to satisfy their early worshippers, and we find the incipient reasoners postulating One God behind all the deities of the earliest times. This was the final outcome of religious thought, beginning with a most natural faith in invisible powers or agents behind the starting drama of nature, and ending with a belief in One Great Power, the unknown.
Philosophy of Vedanta
The fundamental tenet of the Vedanta school consisted in contending that the existence of matter has no essence independent of mental perception, that existence and perceptibility are interchangeable terms, that external appearances and sensations are illusory and would vanish into nothing if the divine energy which alone sustains them were suspended even for a moment. To the Brahmans, to be able to mistrust the evidence of the senses was the very first step in (their) philosophy.
[But why mistrust the senses? Why was this chosen as the foundation of Vedanta “philosophy” — a word whose original Greek meaning is “love of truth”? So Vedanta has postulated a more real world, eternal and unchanging, and has rejected the sensory phenomenal world as “illusory” and unreal, the product of Avidya (ignorance or delusion). And so dualism crept into their thought, about which we shall read more as we progress.]
From the Maitrayaniya Upanishad:
Thoughts alone cause the round of a new birth and a new death. Let a man therefore strive to purify his thoughts. What a man thinks, that he is. This is the old secret. (VI, 34)
Exactly the same idea is expressed by Buddha in the first verse of the Dhammapada (the collected words of the Buddha). Buddha’s hostility toward the Brahmins has been very much exaggerated, and we know by now that most of his doctrines were really those of the Upanishads.
Man is immortal as soon as he knows himself, that is, as soon as he knows the eternal Self within him. Subject, for the Vedantists, is not a logical but a metaphysical term. It is, in fact, another name for Self, or for whatever name has been given to the eternal element in man and God. As soon as the Self is conceived and changed into something objective, Avidya (“absence of true knowledge”) steps in, the illusory cosmic life begins, the soul seems to be this or that, to live and to die — while as subject, it can be touched by neither life nor death, it stands aloof, it is immortal.
If the Hindu philosopher is clear on any point, it is this: that the subjective soul, the witness or knower, or the Self, can never be known as objective but can only be itself, and thus be conscious of itself. We can only know ourselves by being ourselves. If other people think they know us, they know our phenomenal self, never our subjective self, which can never be anything but a subject — it knows, but it cannot be known.
Problems with Vedanta
Soon, however, a new question arose: Whence come all these upadhis or (limiting) conditions, this body, these senses, this mind and all the rest? And the answer was, from Avidya.
The Vedantist sees the work of Avidya everywhere. He sees it in our not knowing our true nature, and in our believing in the objective world as it appears and disappears. He guards against calling this universal Avidya real, in the sense in which Brahman is real, yet he cannot call it altogether unreal, because it has at all events caused all that seems to be real, though it is itself unreal.
[Here the argument starts to fall apart.]
Its only reality (Avidya’s, that is) consists in the fact that it has to be assumed, and there is no other assumption possible to account for what is called the real world (emphasis added).
[Is there no other assumption possible to account for what is called “the real world”? First the Upanishads posit a more real world, eternal and unchanging, but are then faced with the problem of explaining the sensory phenomenal world — the physical world, the world as it is, as we perceive it, the world we inhabit. So the only “assumption possible” is that the world we see and inhabit is not real, in the sense that the “real” world is real (here they qualify it), all the while missing the fact that this “real” world was born as an idea in the mind of man. Without this assumption, the whole philosophy falls apart.
[Because man does not perceive correctly, he misses the real world and falls under the spell of a world he himself creates, the illusory one. This is dualism par excellence, unfortunately found in what is termed Advaita (“without two,” usually translated as Nondual) Vedanta, and its presence in this philosophy is the reason I call primitive and dated. A yet finer conception of nondualism is possible (as we shall see elsewhere), one that does not posit and distinguish between real and unreal worlds.]

The Triumph of Shankara: Vedanta in a Nutshell
Avidya, however, is not meant for our own individual ignorance, but as an ignorance inherent in human nature — nay, as something like a general cosmic force, as darkness inevitable in the light, which causes the phenomenal world to seem, and to be to us, what it seems and what it is.
Hence Avidya came to be called Maya, original power (also Shakti, “power”), the productive cause of the whole world. This Maya soon assumed the meaning of illusion, deception, fraud — nay, it assumed a kind of mythological personality.
[To me it seems that all the undesirable remnants of this philosophy, such as the “productive power” that leads to deception and fraud, were personified as the female principle (Shakti) and rejected in favor of the male (Brahman).]
However, the word Maya never occurs in the principal Upanishads in the same sense a Avidya, though in some of the later Upanishads it has taken the place of Avidya.
[Maya as a deceptive veil thrown over the world to conceal its true nature, leading to suffering and rebirth, is also a central concept in Buddhism.]
In some places, certain latent powers or shaktis (“powers”) are ascribed to Brahman in order to account for the variety of created things in each period. But this is strongly objected to by Shankara, who holds that the universe, though it has all its reality in and from Brahman, is not to be looked upon as a modification — for Brahman, being perfect, can never be changed or modified, and what is called the created world, in all its variety, is and remains with the Vedantist the result of a primeval and universal turning aside or perversion caused by Avidya.
Brahman, as Shankara says, though ignorantly worshipped, remains unaffected by our inadequate conceptions. He is not tainted by our ignorance, as little as the sun is tainted by the clouds that pass over it. Nay, we may learn in time that as the human eye cannot look upon the sun except when covered by those passing clouds, the human mind cannot possibly conceive God except behind the veil of human language and human thought. The phenomenal Brahman is therefore nothing but the real Brahman, only veiled in time by Avidya.