I read what you wrote, and I admit I am not quite getting it. So let me describe my position and hope that it is a worthy response to what you wrote. I'm not quite sure whether I am disagreeing with you or not.
There is a way in which people try to persuade other people of their belief system. I think that our current scheme (axioms, tautologies, modus ponens etc) we use in math is a very good description of what we feel are really effective ways of persuading other people. (Of course we know that using pure reason doesn't work with most people, but we have this sense that there is a Platonic reality as to how an ideal argument should work.) However, I believe that our ultimate sense of what we think is true and not true comes down to simply "what do I believe." And the methods of logical deduction are no more than one other set of things that I simply "believe."
On a completely different tack, while methods of logical deduction seem to be a good description of how to do mathematics, I don't think it is how mathematicians really think. The example I would cite is Euler's (rather brilliant) discovery that sum 1/n^2 = p^2/6 (which basically involves pretending that the Taylor series of sin(x)/x is a polynomial). His original proof certainly was not rigorous. Rather it was a flash of insight, and certainly thinking out of the box.