If the cap fits wear it.
I have spent a lifetime discussing with religious types why they believe. The important issue is not what people believe but why. The key issue is what constitutes proof. If you can pin people down on that you get to the core of things. This is not only about religion as extreme nationalism and politics share common attributes.
Basis of Proof - A Logical Hierarchy
1) Repeatable, demonstrable experiments. E.g Acid + Base = Salt + Water
2) Historical external evidence e.g geological
3) Logic e.g maths 2 + 3 = 5 Although this is absolutely true by definition we are dealing with closed systems which may not reflect external reality.
4) Recorded events. E.g Film. With the modern ability to edit film recordings the value of this evidence is becoming less.
5) Multiple present day eye witnesses, especially if independent.
6) Written records. As Churchill said 'History will be very kind to me for I shall write it.
Lower categories have some value but not if they are contradicted by higher forms of evidence.
For example if it was written down that Napoleon ate ham and eggs on the Isle of Elba this would tend to be believed as an unremarkable and uncontradicted statement. On the other hand an historical record of something miraculous would not be believed if it is contradicted by normal experience. Much history written by the Romans we believe because it's not contradicted by anything else.
Likewise mathematical proof is subordinate to observation. Mathematical formulae are only of use insofar as they describe the real world.
Criteria which have no value
1) The person making the assertion is very important or has a PhD. Has a Crown, fancy uniform, many medals, dog collar etc. Variant. Person believing is a scientist. Well scientists, engineers, business people etc have to be rational in their day job but is the scientist being scientific in his own personal beliefs? Don't look at the person look at the idea.
2) Millions of people believe it. This is a demonstration of the power of brain washing particularly of children who tend to believe everything their parents tell them.
3) Faith. Valuing an idea which is not supported by any evidence turns logic upsidedown. Valuing an idea which is not supported by any evidence is not a virtue but stupid.
4) The idea is nice/poetic/makes me feel uniquely valued and I want it to be true. Nb Gods chosen people, master race etc. e.g Man is created equal - we hold this truth to be self-evident.
5) It can't be disproved so it might be true. (Variant, just because something cannot be proven does not mean it is not true). There are things which can be shown to be true, things which can be shown to be untrue, some things have a probability of being true and a huge number of ideas which can't be proved or disproved. Generally ideas which can't be tested have no value as they do not influence everyday existence. In practice nobody believes the vast numbers of ideas which could be true. So we are left with people picking and choosing which ideas they like but which they cannot differentiate from a vast number of absurd ideas. If you choose to believe in a religion, why not all religions and why not fairies?
6) Relativism. All existence is perceived through sentient experience. So if it is true in your conscious mind, it is as true to you as any external reality. Similar in many ways to item 5. Technically this is a correct argument but could be used to justify anything whatsoever however fantastic, criminal or insane.
Why believing matters.
Suppose you knew with absolute certainty that a God existed. Why would you bother to worry what other people thought? Do I worry about whether people believe in the existence of Africa or not? The reason people with irrational ideas care about who believes, is that the belief cannot be justified by the logical hierarchy so there is no way of peacefully persuading people. The idea only exists insofar as people believe in it. There is no external reality.