Honestly the easiest explanation is just that Aumann is a very muddled thinker when it comes to religion. That's hardly a surprising explanation - muddled thinking is extremely common when it comes to religion. And it doesn't mean that such a person can't otherwise be very rational. I know I'm being blunt here, but I see no reason to look beyond superficial appearances here. Just because a statement is religious doesn't mean it's deep.
All this stuff about non overlapping magisteria is honestly just confused nonsense. There's only one truth. Sure you can approach that truth from different directions, but not if those directions directly contradict each other. Spending a great many pages talking about this just obscures that obvious fact.
And it's all good and beautiful to say that "Religion is very different from science. The main part of religion is not about the way that we model the real world". But I've never met a religious person who said that and then didn't immediately turn around and started applying their religion to the real world. Aumann also does that. In fact in this very interview he literally says that copying software is wrong because his religion says so. So he clearly is modelling the real world based on his religion.
I don't think he was saying that copying software is wrong because his religion says so. When he is talking about that example, he says:
"In short, you can be a moral person, but morals are often equivocal. In the eighties, copying software was considered moral by many people. The point I am making is that religion - at least my religion - is a sort of force, a way of making a commitment to conduct yourself in a certain way, which is good for the individual and good for society."
In other words, he is saying that copying software is bad for society...
Just came across this interview with Robert Aumann. On pgs. 20-27 he describes why and how he believes in Orthodox Judaism. I don't really understand what he's saying. Key quote (I think):
Anybody have a clue what he means by all this? Do you think this is a valid way of looking at the world and/or religion? If not, how confident are you in your assertion? If you are very confident, on what basis do you think you have greatly out-thought Robert Aumann?
Please read the source (all 7 pages I referenced, rather than just the above quote), and think about it carefully before you answer. Robert Aumann is an absolutely brilliant man, a confirmed Bayesian, author of Aumann's Agreement Theorem, Nobel Prize winner, and founder / head of Hebrew University's Center for the Study of Rationality. Please don't strawman his arguments or simply dismiss them!