I think the concept of edge comes from probability theory.
Each bet with a predifined reward to risk ratio has a mathematical edge if the implied probability of the bet is lower than the real probability for the outcome.
If you bet on flipping coins with a reward bigger than the risk, you have a mathematical edge, because the probability of a win is 50%. If you divide 1 to the reward to risk ratio + 1, you get the implied probability for that bet. If someone offers you 1.1$ profit for rach 1$ risked at flipping coins, the implied probability for that is 1/(1+1.1)=0.47619 -> 47.618%. If you bet on that you have a mathematical edge as long as you don't over bet (according to the kelly criterion formula) and the law of large numbers is saying that you will make money in the long run.
Casinos have a mathematical edge because they offer bets with a higher implied probability than the real one (think of the 0 and double 0 at roulette).
When talking about mathematical edge it is crucial to factor in the percentage of capital risked. If the risk is higher than what kelly criterion formula recommends, those bets lead to losses in the long run according to the law of large numbers, regardless of the apparent edge.