Inner self. In the conventional sense, that is.
Why? Because rather than judging and grasping on what is outside,
we should shine the light (awareness) within to look in
ourselves.
If we always shine a spotlight on others, and see how dirty they
are, we will forever miss the 'dirt' that is within our very own
minds.
That is like being able to distinguish everything out there, but
know nothing about ourselves. This is not the way to
liberation.
Zen Master Dogen Zenji says,
To study Buddha Dharma [or Buddhism], is to study the self. And to
study the self is to forget the self [or to let go of the self].
And to let go of the self, is to be enlightened by all things [the
10,000 dharmas]. To be enlight-ened by the 10,000 dharmas is to
find perfect freedom for yourself and for others, and this trace-
less enlightenment goes on and on and on, not leaving any trace at
all.
To be able to realise no-self, first of all, we must learn what
this self is about.
Now in the ultimate sense, there is no inner or outer. No self. So
in terms of insight practice, we are taught to just observe
everything without judgement. And then eventually we will be able
to penetrate the subject-object duality -- in seeing, only the
seen, no seer. In hearing, only the heard, no hearer. In thinking,
only the thought, no thinker. Everything that is self-arising is
self-luminous and self-liberating, inseparable from
conditions.
At that time you will not ask if there is relationship between
'inner' and 'outer', because everything is seen in Wholeness. There
is no inner to relate to outer in the first place.