There is a sentence "I am what I do" by Martin Buber, but is there such a sentence as "I do what I am"?
Do both sentences mean the same or is there a difference?
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Those sentences are not the same.
"I am what I do" means that my personality is determined/can be described by my actions. "I do what I am" means that my actions are determined/can be described by my personality.
Technical speaking, the difference is what is a cause and what is an effect (philosophical concept of casuality). In the first sentence "I" is an effect of my action — "I do". In the second my actions "I do" is an effect of what "I am".
And is there such a phrase? Don't think that it exists in exactly those words. Some of our prominent philosophers, Freud among them, do postulate this same principle though.