The most difficult concept in Advaita Vedānta is also its most ordinary: the witness. Sākṣin (साक्षिन्) means "the one who sees". The whole system turns on what is meant by "sees".
The setup is familiar. Saṅkara's Advaita holds that ultimate reality is Brahman—pure being, pure consciousness, without qualities, without a second. The phenomenal world (vyavahāra) appears real but is, on final analysis, a superimposition (adhyāsa) on Brahman, the way the appearance of a snake is superimposed on a coiled rope in the dim light.
The question that immediately arises: who is doing the superimposing? The world is illusory, but the illusion has to appear to someone. The dream is unreal but there must be a dreamer.
Saṅkara's answer is precise and somewhat startling. The dreamer is not the jīva (the individual self) and not exactly īśvara (God). It is consciousness itself, considered as the bare fact of awareness, prior to any of its contents. He calls this sākṣin.
The sākṣin is not a thing. It cannot be an object of awareness because it is the awareness in which objects appear. It is not even a subject in the ordinary sense, because subjects have particular qualities—mine is different from yours—and the witness, on this analysis, is the same in all knowers. Sākṣyaḥ sarva-bhūtānām—the witness of all beings.
Three things follow.
First, the witness cannot be doubted. To doubt is itself an act of awareness; the witness is presupposed by the doubt. Saṅkara puts it in Upadeśasāhasrī: na hi pratyakśaṃ jyotir-nirāsaḥ śakyaḥ—the light by which you would deny the light cannot itself be denied.
This is the Indian version of Descartes' cogito, except that it does not conclude in a thinking self. It concludes in awareness, which is not a self at all.
Second, the witness is not a process. Western philosophy of mind has tended to treat consciousness as something that happens—a function, an emergent property, a computational state. The Advaitin response is to point out that any "happening" is already an event in awareness. The awareness is not the happening; it is the field within which the happening shows up.
This is why Advaita resists naturalization. The naturalist account always presupposes what it is trying to explain.
Third, and most consequentially: if the witness is the same in all beings, then the apparent multiplicity of selves is misleading. Tat tvam asi—"that thou art"—is not a piece of mystical poetry. It is a logical consequence of the prior analysis. If consciousness is one and indivisible, and you are aware, then what is aware in you and what is aware in me cannot be two different things.
This is the move that classical critics—Madhva, Rāmānuja—found impossible. The Viśiṣṭādvaita response is that even if the witness is one, the individuated selves are real. The Dvaita response is more radical: the very distinction between witness and witnessed is itself basic to the structure of reality.
Advaita's response to both has always been the same. The objections, taken seriously, presuppose the very witness they are trying to deny.
The argument is twenty-five hundred years old. The question is older. The answer—if there is one—is sitting where you are sitting, reading these words.