Buddhism and Christianity both enjoin self-denial. But Buddhism is more radical in that it connects self-denial with denial of the very existence of the self, whereas Christianity in its orthodox versions presupposes the existence of the self: Christian self-purification falls short of self-elimination. Nevertheless, there are points of comparison between the 'No Self' doctrine of Buddhism and the Christian doctrine of the self.
In his Combattimento Spirituale (1589), Lorenzo Scupoli writes:
You my mind, are not mine: you were given me by God. Neither are
the powers active within me -- will, with its energy -- mine. Nor
does my feeling, my ability to enjoy life and all my surroundings
belong to me. My body with all its functions and requirements,
which determine our physical well-being, is not mine either....And
I myself belong not to me, but to God. (Unseen Warfare, St. Vladimir's Seminary Press, 1995, p. 172)
Apart from the references to God, this meditation of Scupoli, of which the above is merely an excerpt, bears a striking resemblance to the one contained in the Anattalakkhana Sutta. Buddha there examines each of the khandas, body, feeling, perception, etc., and concludes with respect to each of them that "This is not mine. This is not my self. This is not what I am." In Scupoli we encounter virtually the same litany: body, feeling, mind . . . of all of which it is true that "This is not mine, etc." Of course, nothing depends on the exact taxonomy of personality-constituents. The point is that however one classifies the constituents of personality, no one of them, nor any combination of them, is veridically identifiable as one's very self. I say 'veridically,' since we do as a matter of fact (falsely) identify ourselves with all manner of item both within us (feelings, memories, etc.) and without us (property, progeny, etc.) But these false self-identifications are part of what our ignorance/sinfullness consists in.
Thus Scupoli (who I take to be a representative Christian, and who is of interest here only as such) and Buddha agree with respect to the (negative) thesis that nothing in one's experience is veridically identifiable as one's very self. Thus nothing that we ordinarily take to be ourselves (our bodies, our thoughts, feelings, memories, etc.) can in truth be one's self. But there is also similarity in their reasoning. One way Buddha reasons is as follows. If x (body, feeling, perception, etc.) were my very self, or were something that belonged to me, then I would have complete control over x. But it is evident that no x is such that I have complete control over x. Therefore, no x is either my self or anything that belongs to me. This could be called the 'complete control' argument. Scupoli has something similar:
Let us remember that we can boast only of something which is a
direct result of our own will and is done by us independently of
anything else. But look how our actions proceed. How do they begin?
Certain circumstances come together and lead to one action or
another; or a thought comes to our mind to do something, and we do
it. But the concurrence of circumstances does not come from us;
nor, obviously, is the thought to do something our own; somebody
suggests it. Thus, in such cases, the origin or birth of the
thought to do something cannot or should not be an object of
self-praise. Yet how many actions are of this kind? If we examine
them conscientiously, we shall find that they almost all start in
this way. So we have nothing to boast of. (174)
This passage suggests the following argument: One cannot justifiably take pride in anything (an action, a physical or mental attribute, etc.) unless one originates it 'independently of anything else.' But nothing is such that one originates it in sublime independence of all else. Therefore, one cannot justifiably take pride in anything.
But does this amount to an argument against the self? It does, given the Buddhist assumption, crucial to the reasoning in the Anattalakkhana Sutta, that a self is an entity that has complete control over itself. Such a self could justifiably take pride in its actions and attributes. For it would be their fons et origo. So if one cannot justifiably take pride in any of one's actions or attributes, then one is not a self. Pride is one of the seven deadly sins precisely because the proud person arrogates to himself a status he lacks, namely, the status of being a self in the sense in which this term is employed in the Anattalakkhana Sutta.
In sum, both Buddha and Scupoli are claiming that no one of us is a self for the reason than no one of us is in complete control of any of his actions or attributes. No one of the things which one normally takes to be oneself or to belong to oneself (e.g., one's body, habits, brave decisions, brilliant insights, etc.) is such that one has originated it autonomously and independently.
The main difference between Buddha and Scupoli, of course, is that the latter maintains that God gives us what we do not have under our control. Thus for Scupoli, what we do not have from ourselves, we have from another, and so have. But for Buddha, what we do not have from ourselves, we do not have at all.