Aluminium oxide is amphoteric. It is easy to see that it is a Brønsted-Lowry base through the following reaction:
$$\ce{Al_2O_3 + 6HCl \rightarrow 2AlCl_3 + 3H_2O}$$
The aluminum oxide splits and the oxygen accepts a proton, forming water.
But what about the reaction with a base? In my textbook, they say:
$$\ce{Al_2O_3 + NaOH \rightarrow 2NaAlO_2 + H_2O}$$
Now, the textbook claims that Aluminium oxide is an acid merely because it reacts with a base to form a salt and water, as is characteristic of a neutralization reaction.
But I'm not satisfied with this definition. I mean, acids aren't defined as 'things that neutralize bases', we have well-established definitions for them.
I tried to figure out for myself how this could be. Clearly, the Brønsted-Lowry theory cannot be applied here since the compound in question has no protons to donate. Therefore, the only alternative is the Lewis concept. I cannot see how that is applicable in this case.