I don't necessarily distinguish between "religious faith" and "mathematical faith" or "scientific faith"; they all have to do with assuming certain things to be true without proof--faith statements.
And yet there seem to be key differences between the epistemic grounds for each kind of "faith". e.g. As polymath pointed out, what you call, "scientific faith", is based on observation, and what you call "religious faith" is not.
Moreover, your conflation of those two kinds of "faith" does not seem to be epistemologically justified but rather appears to be as arbitrary on your part as if you were to whimsically refuse to distinguish between Macintosh and Granny Smith apples on the logically weak and insipid grounds that "both kinds of apples are apples and therefore should not be distinguished one from the other".
So first of all, not all facts are based on observation. Some facts are based on formal systems of reasoning, like mathematical or geometrical theorems.
Agreed, and good point. In formal philosophy, the distinction you are drawing has for the past 2,300 years been known as the distinction between a posteriori and a priori knowledge.
But even for those facts which ARE based on observation, one must have faith in one's observations--that is, one must have faith that they are not just a brain in a vat being stimulated by a mad scientist.
No, that's simply not true. Contrary to popular belief, it is neither epistemologically nor logically necessary to indulge oneself in distasteful metaphysical speculations (I consider all metaphysics mere speculation and therefore distasteful) by positing an ontological grounds or basis for observation. The reason it is not necessary to posit an ontological grounds for observation is because of methodological naturalism. Methodological naturalism provides a logical and epistemic framework or justification for scientific inquiry involving observation without resorting to detestable speculation.
...faith in the axioms (faith statements) that underlie formal systems of reasoning.
If that were actually true (and I do not believe it is), then how would you explain away those good folks who treat axioms as prescriptive rather then descriptive statements? Such folks certainly do not appear to have any faith at all that axioms correspond to an ontological state of affairs.
Although we disagree on several points, it is obvious to me that you've put some thought into your beliefs in this matter. That makes your remarks interesting. Thank you for that.