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The purpose of the dream example is to make us see that the waker's experience of duality is not any different. While the difference between the waker and the dreamer is accepted in terms of qualities (viśeṣa)s the basic non-difference is shown in detail in the kārikā.
In the Jyotir-Brāhmaṇa of the Bṛhadāraṇyakopaniṣad, the invariable ātmā in dream and waking is presented as the light of awareness (jyotis-svarūpaḥ). The svarūpa of the ātmā is not the dreamer, dream, or dreamt; nor the waker, waking experience, or waker's objects. But the knower, known and knowledge vikalpa is also not separate from the ātmā and therefore the division is mithyā. It is obvious that ātmā is always nirvikalpa, in spite of the apparent division. That is what is said in the Kenopaniṣad, “in every form of knowledge, ātmā is understood by the discriminative as the invariable.” Therefore, the knowledge that I am thought-free (nirvikalpa) is in spite of the experience of vikalpa. This is entirely different from a state wherein there is absence of thoughts.
In aṣṭāṅga-yoga, the aṅgī, the main thing to be achieved, is nirvikalpa-samādhi, a state wherein there is the absence of subject-object relationship. Even though it is a desirable accomplishment, the state itself is jaḍa (inert) inasmuch as there is no thought (vṛtti) that can destroy ignorance. In samādhi (a mental state of absorption) and also when there are thoughts, what obtains as invariable is the svarūpa of ātmā, which is nirvikalpa. Again, the notion that when there is no more thought, then there is enlightenment, implies a duality such as ātmā and thought. When thought is, ātmā is not. When ātmā is, thought is not. Both become equally real because one exists in the absence of the other. But that is not true. If one exists whether or not the other exists, both the objects enjoy the same order of reality, like the table and chair. If one exists only in the absence of the other, they both belong to the same order of reality, like illness and health. Both are equally real.
Does thought deny ātmā? Is there a thinker without ātmā? In fact, thought is ātmā. But ātmā is not just a thought. Ātmā is satyam, being present in all situations—while situations are mithyā, dependent as they are for their existence on ātmā. There is no mithyā without adhiṣthāna-ananyatvam, that which is non-separate from its cause.
The wave being not independent of water, you don't have to remove the wave in order to see the water. So too, if the thinker, the thought and what is thought of are dependent upon the ātmā, which is satyam, you don't have to remove any of them to recognise the ātmā. The recognition is that all three are ātmā while ātmā is not any of them.
'Teaching Tradition of Advaita Vedanta' pp24-26, Swami Dayananda
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