How can aluminium oxide be called an acid? 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.
The most basic definition of "acid" is that it is a proton donor (or one which accepts a lone pair)

All of this stuff is done in an aqueous medium, so we can assume that all aqueous ions and molecules are present. With this assumption (in this case, we are assuming that $\ce{OH-}$ is available to react), we get the following equation:

$$\ce{Al2O3 + OH- -&gt; 2AlO2- + H+}$$

Similarly, we get:

$$\ce{Al2O3 + 6H+ -&gt; 2Al^3+ + H2O}$$

where it is acting like a proton acceptor (base).