Discussion points.
Proving one's beliefs rationally or logically is a contradiction, as belief is by definition in what lies beyond the probative power of reason and logic. Logic and reason can point to the coherence of one's belief system, but can never unequivocally "prove" it.
On the other hand, science, which is dynamic and ever-changing, is not value free as it's part of a specific paradigm named scientism, always being limited in scope due to its inherent weaknesses with such items as induction and empiricism, amongst other things, in many cases cannot prove something with a level of indisputable certainly, a case in point being evolution.