The not-self (anatta) doctrine of Buddhism takes two forms: a special form and a generalised form. In the the special form, not-self means not belonging to me, not mine, not I, and it concerns false identifications of self with, for example, the five aggregates. In the generalised form, it states that all phenomena are not-self, meaning all phenomena are empty of inherent existence, that there is no elusive entity behind phenomena.
Now, there is a curious class of statements known to philosophers as a priori statements. Kant (who invented the term) has defined this as the class of statements that do not rely upon empirical verification. In other words, these are "logical" propositions that cannot be contradicted without violating logic. "All triangles have three sides" is an example of an a priori statement. More precisely, this is an example of an analytic a priori statement. Unfortunately, most analytic statements are quite boring. The other type is called synthetic a priori statement and these tend to be more interesting. For example, "11 + 9 = 20" is such a statement (though it could be seen as a border case because it follows directly from the definition of natural numbers). Anyway, take for example: C/2r= pi. The circumference of a circle divided by its diameter (radius *2) equals pi. There you have an example of a non-trivial synthetic a priori proposition.
You might begin to wonder why I am mentioning this. Well, the class of synthetic a priori statements appears to describe atta/atman, or perhaps better: the essence of things. For example, C/2r=pi describes the essence of the circle. For all we know, it is eternal, universal, and non-changing (within the confines of Euclidean geometry). Furthermore, pi plays an important role in various mathematical and natural laws ranging from physics to statistics. Conclusion: there are certain atman-like properties that we can know about certain phenomena. This brings up a number of interesting questions.
1. Are phenomena themselves manifestations of atta/atman as far as these laws are concerned?
2. Are there classes of mathematical laws that are subject to some kind of impermanence?
3. Are there physical laws that are subject to some kind of impermanence?
4. Does our mind/thoughts somehow experience atta/atman to the extent to which we can understand these laws?
5. What part of ourselves understands these laws?
6. An argument favoured by theologians (rejected by Buddhists): is it our atman/brahman nature that discovers and understands these laws?