If mathematicians were like philosophers, trying to fully resolve an issue before advancing, then there would be no progress in mathematics. So, wherever there is doubt, they create an axiom and tell their colleagues to invoke it in their work, if and when they like. In this way, they avoid the search for truth, so that to focus solely on mathematical work, numbers and proof: they just need to say which axioms were used, and leave the 'what really is the case' to philosophers. Mathematical axioms, in the way they are used by mathematicians, do not have anything to do with 'truth' in the real sense, they are just there to help them in their games. After all, if something was obviously true and accepted by everyone, we wouldn't have an axiom for it, would we? :)

I think that for the unmoved movers, he meant them as a final cause.

It is the constructivist/finitistic approach to maths MO is supporting. But because there is not universal concensus in mathematical and philosophical communities alike, in maths we have axioms for this stuff, like the axiom of infinity or the axiom of choice.

It causes motion to others by being a final cause, to add.

Is this in contrast to Ancient Science, where belief in deities was common? Or else, when was science, modern or otherwise, ever concerned with deities?