I'm gonna show my ignorance here, but mathematics isn't science?
Nope. Science deals with concrete, measurable, repeatable
facts of the world (for instance, the acceleration rate of a car, or the right combination of substances to achieve a certain chemical result). Mathematics deals with abstract
concepts (numbers, sets, infinity etc). Science employs Mathematics sometimes, and that's it.
A good brain exercise is asking yourself: "Are numbers real?". If you say "of course they are", then follows the question "Well, have you ever seen/touched/smelled/licked one"? To which the answer obviously can only be "No". We've all seen graphic representations of numbers, but not the numbers themselves. Because they have another type of existence: A
metaphysical existence (in contrast with
physical existence -- which is the
rightful domain of Science) and I'm not gonna shock the present company if I tell you that metaphysical existence is the same type of existence that Metaphysics attributes to our ol' pal God (and, you know, it makes sense... I mean, would the Creator be contained, physically, within the Creation? To my mind it would be like trying to make a big bowl fit in a smaller bowl: impossible. Jesus, who according to tradition is God incarnate, came down here only for a limited amount of time. And all major religions stress the fact that we're separate from the Father).
This distinction already signals that there are areas where Science becomes useless: would you ask for romantic advice from a scientist,
in his capacity as a scientist? I certainly wouldn't. What good could come from me knowing that love is probably an interaction of chemical particles and electricity?
Now, back to the question about numbers, if you say "Of course numbers are not real", then follows the question, "Well, then how all these bridges, overpasses and buildings remain erect (let alone the fact that they ever got to be built at all)?" They were built with calculations which employed numbers, so these numbers gotta have *some sort* of existence, otherwise it wouldn't have worked in the real world.
Already you see the subject is more complicated (and way, way, way more interesting) than the usual "faith is dumb, science is smart" that people like Dawkins hock for a living. I hope this explains why I find this subject endlessly fascinating.
I'm gonna come back to your other points in a bit, if that's ok.