There is a great and often underappreciated danger in logic thinking itself.

That is the danger of circular reasoning. The easiest example goes:

If A is true then so is B.

If B is true, than so is A.

Therefore A and B are true.

Because it seems so easy to spot, it is vastly underestimated.

But what if hundreds of  theorems and variables are involved?

It is not only possible, it is more than likely, that circular reasoning is involved somehow.

Say your intuition goes against the conclusion you have drawn.

You begin to test your logic, going this way and that way through all the theories and variables,

training your own brain to believe in them.

Proofs of the existence of gods have been done that way.

There is a point where you should look at the concepts you think in, even into the exact meaning of the words you use.

Which things do you see as to trivial to even think about?

That things fall down, for example?

Newton did not find that so trivial.

In mathematics, that is what an axiom system is for. It is a safe standing, a sort of table you can use to build your conclusions on without the danger of circular thinking. That is why axiom systems are so crucial in mathematics.

So, how is an axiom system constructed?

At the very beginning, before the axioms are even formulated, there are the definitions.

They contain the ideas you want to think about.

The hardest steps for explaining "fall down" for Newton must have been finding the physical words for "energy", "impulse" or "gravity".

Newton took words that were around anywhere, but their meaning was and is wholly different outside his world of physics.

So, as a mathematician, that is what a axiom system is for: Giving you a set of words and rules to think in. But beware: Your logical thinking is only true as long as the axioms hold.

If you hear a lot of mathematicians say: This is obviously so, no explanation needed, then look very close:

That is the place to look for a nice little overlooked axiom.

You have to take it, build it into the axiom system, probably overthrowing the whole system, and find new words to think in, until you have a sound and well - build axiom system again.

And then you probably have to proof all the theorems and certainties of the old system anew.

I think, the feelings most mathematicians have for tinkerers on axioms systems is very alike to the feeling of one who has build a 100 floors high playing card house - and then someone comes and shakes the table.