You're right, I'm not adding addresses together. If you read my post, I say I'm adding to addresses, which you do to--now hold onto your hat--find the next one in sequence. And that, quite obviously, works because they are integers and map to an address space that matches the complete space, from 0x00000000 to 0xFFFFFFFF (and the equivalent 128-bit space for IPv6) of their bit length. It's not "a convenient coincidence". It's the definition of the thing in question.

It's like saying a pointer to RAM isn't an integer. Of course it is, and you add to them every time you de-reference an array in C. That it has additional semantic meaning doesn't mean it stops being a integer. The pedantry you're peddling doesn't fly.

I think what baq is saying is that IP addresses are torsors http://math.ucr.edu/home/baez/torsors.html . It makes sense to give them a type where you can't add two IP addresses to each other (but you can still subtract to get an integer, and add integers).

Sure, fine, but it'll still be an i128 somewhere, yeah? Like, you can put whatever abstraction you want on it, I still need to directly poke those bits and treat it in all ways as fundamentally the same thing as an integer, defined in those terms in order to use integral operations upon them, at the levels that somebody who's worrying about them has to care about. (Also, look at halomru's post for reasons one might intentionally add and subtract addresses from one another in the first place, which kind of holes this argument.)

That is a cool link though, I'm gonna give it a deeper read.