Most of math can be derived from a deductive system called ZFC set theory. The theory is classical logic equipped with the notion of sets and the axiom of choice. So the jump from logic to math requires us to propose sets and a few extra axioms that describe the behavior of sets.
Is math analogous to fine arts?
I would say not.
Do syllogisms pass out of existence if human beings pass out of existence?
What does it mean to say that an idea exists? Questions like this are a semantical nightmare. Thankfully, we don’t need to discuss the existence of logical arguments for logic to be useful.
Can math be measure, weighed, and counted?
I personally doubt that the Pythagorean Theorem weighs 200 pounds after all that exercise and dieting it’s been through.
Should science use math if it cannot be measured or weighed?
Let’s put it this way: If there comes a day when math is not applicable to science, scientific advancement will end. Look at Aristotelian physics for an example of what happens when science fails to describe phenomena quantitatively.