While most religious belief systems affirm the existence of certain
things, the belief system of (natural) science tends to deny the
existence of certain "not-reproducible" things.
I don't think this is true. Science - in the limited scope you meant here -
only deals with things that can be demonstrated by reproducible experiments. It can't deal with what is, by design or nature, not subject to such requirements. Are there invisible fairies in my garden? We cannot know, because the claim is unverifiable by design: the fairies are invisible, and so we cannot see them. Can we communicate with the dead, if we have faith? We can't know, bucause the claim is unverifiable by nature: if we were to make an experiment, we would already lack the necessary faith.
It doesn't follow that those things don't exist. It follows that science cannot reach to conclusions regarding them. Collaterally, there is a "belief", which can not experimented in a "scientific", reproductible way, that things that have inbuilt unverificability clauses, such as invisible fairies or any phenomenon dependent on faith, are either not important, or so rare that they cannot be accounted for in any meaningful way.
On the other hand, scientists seem to believe in the Big Bang, the existence of dinossaurs, or dark matter, all of which are unsusceptible of repeatable experimentation. So, while repeatable experimentation is certainly an important part of science, it cannot be all that is in science.
Let's look at two examples to make this question more concrete. There
are many examples in fiction of interactions of no longer living
persons with living persons. But suppose this would happen to you in
"real life". Wouldn't you try to explain it away (or at least keep it
for yourself)? Perhaps you manage to settle for a "I don't know"
There are many real life accounts of interaction with the dead. But, in short, the problem is that either whatever information the dead bring to us can be checked by the living, or it cannot be checked. In the first case, the dead can usually be shaved off by Ockham's razor - they are unnecessary to explain how such information was obtained. In the second case, we cannot know if such information is actual information, or just a fantasy. The dead could, of course, clear up the issue by informing me the numbers of the incoming lotto - but then it seems they have a strict (and convenient) moral code that forbids them from doing that. And when they otherwise talk about the future, they tend to adopt a contorted (Nostradamic) style that makes it impossible to understand what they are predicting.
Now let's look at the different belief systems of (natural) science
and religion. Is there any need to fit this into a scientific belief
system? Isn't (natural) science concerned with "reproducible" things,
so that science doesn't even need to bother whether certain
"non-reproducible" things are "real" or not?
It cannot be so indifferent to such "non-reproducible" things, as they will contaminate the reasoning and result in non-verifiable hypotheses. And so, they must be purged.
Can the same position also be used for religious belief systems?
Probably not, because this touches the kind of questions that
religious belief systems are concerned with. But what about the
opposite case, for example heliocentricism? That's a scientific theory
(or fact) after all, so it shouldn't worry religious belief systems
too much. But apparently it did.
It did, but that due to certain particularities of the Abrahamic faiths, which imply the inerrancy (at a least to some level) of given, dated, sacred texts - sacred texts that make several assertions about existing or past states of the universe. It is worse for literalists, such as a few fundamentalist Protestant sects, and less of a problem for non-literalists, such as the Catholic ortodoxy, or rabinic Judaism, that can always fall back into an alegorical interpretation (yes, the Bible says God created the world in six days, but who says that a billion years is not just a day for God?)
From Philosopher3's answer:
Religion has a conclusion (i.e. deity, after life, creation, etc.) and
then finds evidence to support those conclusions.
Science on the other hand has evidence (i.e. an object falls at a contestant rate regardless of weight) and then draws a conclusion from
It is not like that. Science starts with conclusions (theories) and only then seeks for evidence. The difference is, science seeks for evidence that contradicts its "conclusions" (ie, theories), and only sticks to conclusions that resist such trial. Religion does not seek evidence at all, and tends to reject (or misinterpret) evidence that contradicts its conclusions.