It is extremely simplistic to speak of "Eastern" religions as if they all are much the same. Actually, they aren't. For example, in some regards Hinduism is closer to Christian theology than to Buddhist teachings.
Case in point: not-self. Buddhists call this anatman.
The doctrine of anatman (or anatta in Pali) is one of the central teachings of Buddhism. According to this doctrine, there is no "self" in the sense of a permanent, integral, autonomous being within an individual existence. What we think of as our self, our personality and ego, are temporary creations of the skandhas.
Hinduism also uses this term. But in this religion the word means something very different.
Anātman (Sanskrit: अनात्मन्) in Sanskrit means not spiritual, corporeal, unreal, something different from spirit or soul, not self, another. In Advaita Vedanta this word is used to indicate Samsara i.e. the world - existence or the world phenomena, which is the unreal projected by ignorance on the real, that is, on Brahman who is the Absolute.
In other words, Buddhism doesn't consider there is any unchanging, permanent, foundational aspect to reality. Rather, everything is interconnected, "empty."
Not empty in the sense of nothingness, a void. Empty in the sense that every seemingly separate entity lacks a distinct individuality, a quality unique to itself, an eternal aspect, a being-ness all to its own.
However, Hinduism and other like-minded faiths (including, as mentioned above, Christianity) posit the existence of Atman, or soul. Atman supposedly is an everlasting drop of the spiritual ocean, Brahman, our real self.
Here's how David Barash describes the contrast between Buddhism and HInduism in his fascinating book, "Buddhist Biology."
Some of his [Buddha's] impatience was evoked by the ancient Brahmanical texts known as the Upanishads, which predated Buddhism by a thousand years or more, and against which the then-emerging tradition of Buddhism had to define itself -- not unlike Christianity struggling to distinguish itself from its ancestral Judaism.
The Upanishads promote a very different conception of the self, or atman, as an unchanging substance (not altogether different from the Christian sense of a soul) that is eternal and uniquely associated with each individual.
For Hindus, atman is not only indestructible but also beyond description as well as beyond suffering.
...This atman, in classical HIndu thought, is connected to the underlying ground substance of all reality, namely brahman. For the Upanishads, it isn't too much of a stretch to suggest that atman = brahman. Buddhists, as we have seen, view things differently as reflected in the alternative designation, anatman or "not-self."
...Let us stipulate yet again that Buddhists are not fools. They realize that each of us experiences a degree of continuous memory as well as some continuity of body. They are not claiming that there is literally "no self," but rather that the various aggregates of personhood are not one's self. Buddhists certainly know the difference between the "you" who is reading these words and the "I" who has written them (or the other way around).
But when we introspect -- and also when we consider the objective biological facts -- neither you nor I can find a solid, unchanging "me" aside from the feelings, sense data, ideas, and so forth that are apparent to our selves at any given moment.
The great philosopher David Hume -- who, after all, was renowned as a hard-headed empiricist, not as a mystic or in any other way a devotee of Eastern thought -- described in A Treatise on Human Nature the following widely shared experience, one that could as well have come from a Buddhist master:
When I enter most intimately into what I call myself, I always stumble on some particular perception or other, of heat or cold, light or shade, love or hatred, pain or pleasure. I can never catch myself at any time without a perception, and never can observe any thing but the perception.
I've been having an interesting comment conversation over on this blog post, which has quite a few comments (scroll in the comment section down to the January 2014 interchanges between "devpaul" and me).
At first I thought that devpaul's commitment to one wife, one car, one bicycle, one job, one religious belief, and such throughout one's life was based on Indian culture. He is critical of my contention that it is normal to change all of these things, and so much more, during one's lifetime.
Or, if not exactly critical, devpaul contends that it really isn't possible to make spiritual progress without an enduring commitment to a single religious perspective, sticking with it through thick and thin, good times and hard times.
Like marriage, I guess. Of which I have had two. And he, one (an arranged marriage).
There are undoubtedly cultural factors influencing how devpaul and I look upon the world. I'm pretty sure there also are philosophical differences impacting our worldviews. Though I'm not a Buddhist, I resonate with the "not-self" perspective.
As Barash argues in his book, it is pleasingly scientific, being in tune with modern neuroscience, biology, and psychology. Not to mention common sense, as Hume pointed out. Where is the "me" separate from my experiences?
So I have no problem embracing all the changes I've experienced in my life, including changing my religious perspective from believer to churchless. After all, I don't consider there is any "me" or "I" other than those experiences.
Thus there is no self that I can be untrue to, or let down. If I change my mind, I realize there was no enduring me which possessed a mind. I'm just my mind, my body, my experiences, my sensations, my thoughts, my feelings.
In fact, I could have left out all the "my's" in that last sentence. Force of habit to write it the way I did.
I like this passage in "Buddhist Biology."
Ambrose Bierce, incidentally, modified Descartes's dictum to cogito cogito ergo cogito sum -- "I think I think, therefore I think I am" -- arguing that this was as close to certainty as philosophy was likely to get.