“Hamsa” in Sanskrit is an important motif in Advaita Vedanta, or nondualism. Ham-sa when inverted reads as sa-ham or so-hum, which in Sanskrit means the oneness of human and the divine.
During pranayama, the inhalation is believed sound like ham, while the exhalation is believed to sound like sa. Thus, a hamsa came to epitomize the prana, the breath of life. It also means "I am That, That I am," referring to the individual's oneness with pure consciousness. The mantra "hamsa" is always with us, in the breath. It is for us to notice it and tune into it: "I am You, You are me." Ham-sa mantra is the mantra of Oneness and surrender.
Hamsa also means "swan" in Sanskrit. In Hindu mythology, it is said that the swan or hamsa can separate milk from water and drink only milk -- just as in Vedanta, we practice Viveka, or discrimination between the Real and unreal, the permanent and the always-changing, the Self and the not-self.
Photo: Detail of Ramakrishna Mission headquarters at Belur Math, Kolkata - taken on my 2017 India trip
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"Realizing your innermost Self, as the Witness of the intellect, and its disturbances and ever maintaining the thought, 'That I am,' shed your identification with the not-Self."
-Vivekachudamani verse 269
"The Self within is to be known as the Witness of the intellect and its thoughts. Having known and understood Its nature, one should move towards it. To step towards the Self is to assert, "I am That" ---(Hamsa); "He alone am I" -- (Soham). Start living as a mere Witness of all the pulsations of the body, mind and intellect (BMI). By this practice, the idea 'I am the Self' becomes rooted in our understanding. At present we have this understanding rooted only in the BMI. This should be renounced and the feeling 'I am the be Self' should be cultivated."
-Swami Chinmayananda's commentary on above