Hinduism
In Hinduism, equanimity is just another term that attempts to describe the nature of Brahman (not to be confused with Brahmin). In Vedanta the term Brahman points to Absolute Reality. In a true sense, Brahman cannot be described as any description or attribute introduces the idea of boundedness, hence it must be recognized that these terms are only meant to serve as pointers to the intellectual concept of Brahman. In Vedanta the term Brahman points to the Absolute, also referred to as the only Reality.
Vedanta states that Brahman alone is Real and the world is unreal. By the term 'real' what is being pointed to is that which is unchanging in all circumstances and independent of Spacetime or the Spacetime manifold. The physical world and mental world hence do not qualify as being "real". The idea of equanimity refers to being in pure awareness. Being in pure awareness requires dissolution of mind. The term mind is also known as Ego or Identity. When there is no distraction or attachment to thoughts, there is equanimity. As per Vedanta, 'Equanimity' is our true nature. When the sense of individual discrete identity is dissolved, one transcends the apparent duality and see themselves as one with everything.
In lack of better terms, 'Equanimity' can be used to refer to any of the following terms such as Beingness, Undisturbed, Unattached. It should be recognized that 'Equanimity' does not refer to a state of mind, rather it describes our real nature. Sense of attachment or doership is always individual and operates at the level of Individual Identity or Ego. Gita says by renouncing our limited identity, we can reveal our true nature, which is 'Brahman'. When we are aware of our true nature, the individual ego does not operate anymore, hence the outcome is equanimity. When one is fully aware, one does not attach to the world, rather sees the world. The world is apparent and unfolds in front of our awareness, but due to lack of clarity, we identify with the body and the mind and become finite and limited. The only unchanging reality is pure awareness. It refers to being witness and not having a sense of individual doer which creates attachment and makes one behave otherwise.
According to the Bhagavad Gītā, every one can eventually achieve equanimity through spiritual practice leading to self realization.
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