I think the whole point about having beliefs is that they
do constitute a part of reality, whether they concern a
scientific or a
religious cosmology, or a
political universe. To speak of the believer/non-believer instance in terms of
assumptions is a little weak to my mind.
It has always seemed to me that what we strive for is a way to have everything we can't explain, or can't bear, or can't stop, mean something--with the possibility that we might live with it, or even change it. Only belief can do that.
As an undergraduate at the University of Toronto, I used to spend many pleasant hours talking with a variety of religionists on the streets.
We had the questions in common (I think anyone who thinks does), even the partial answer--belief--but I always felt they had put their hat in a place that didn't allow them to grow.
While Your beliefs are your concern, just please don't let them creep into our secular argument may simplify the secular argument for the secular partner in the argument, for the other it is as impossible not to let his belief creep in, as it is not to breathe.
After all, belief is nothing more than looking.