You're not going to get a mathematical definition, because we're talking about real-world, complex items with many movable parts. You can't have a perfect, objective definition for anything if you look at things that way. But that doesn't mean we can't have a real-life definition on what RPGs are, that would be agreed by most people. Especially when most people will agree on the extremes:
Fallout, Wizardry - pure RPG. Everything is determined by the rules system. There is no reflexes involved, it is a purely "intellectual" game, in which you try to work the rules in your favor, estimating the risks and the opportunities the rules give you. Story, choices, quests, exploration are strong points.
Quake, Ghosts'n Goblins - not RPG. Purely determined by action and skill. Dodge stuff hitting at you. Personal skill and reaction time, rather than strategic decisions, is all that matters. Sure there's some game progression given by equipment, but only there to scale up challenge and add a little variety in combat. No exploration except for a few secrets, no quests, no choices, no interaction.
Those are the extremes, black and white. Obviously, Morrowind is closer than Oblivion to the "RPG" than to the "not RPGs". I am not saying this in a derogatory way, just saying that being agile, skilled, or having good reflexes is not what RPGs were about in PnP: rule systems that simulated reality in a numeric way, and the ability to roleplay a character that interacts with a gameworld in a meaningful way.