Your use of "belief," e.g. "everything science does requires belief," is not at all the same as what we mean by religious belief. Deciding on which "demonstrations" or explanations of natural phenomena are more persuasive is not at all the same as taking a "leap of faith."

As for "Mathematics has no need of belief," yes, that's probably true. But to be fair, we should note that the acceptance of the truth of axioms is fundamental to math; that acceptance is not the same as religious belief, perhaps (I don't know; maybe it is; I would not like to define "religious belief" too narrowly; anyway it's not the same as what Christians mean when they talk about "coming to believe," e.g. at the end of John 20), but it does involve putting aside the requirement for proof.