(Spoiler: the mysteries are the old reliables, existence and life.)
Too often discussions of science or religion or both degenerate into
"Is so";
"Is not";
"Is so";
"Is not";
"Is";
"Isn't"...
modes of discussion. I have no hope of ending that, but I thought these notes might be useful for those who view these discussions from a little distance and get a feeling that the arguments are not about the reality of religion or of science. (At least it offers support for the view that no, these discussions are not getting down to the basic issues.)
The great mystery is why there is something instead of nothing. It's a mystery because there is no way of getting outside of existence to investigate it. You are free to believe it's an accident or that it's intentional and no one can prove you wrong but you can't prove you're right either.
There certainly seems to be some focusing of energy, some wisp of intention, behind bothering to let existence exist. If there is, it would seem to imply a certain generosity of spirit.
~~
Anyway, one way or another, existence does exist and this is where science comes in. Science is based on observations. Being able to point to something anyone can see has the enormous advantage of providing evidence to test each assertion. This has made science, as a whole, immensely reliable for the last few centuries. The disadvantage of science compared to faith-based certainty is that there is always more to find out -- there is never a possibility of completeness.
The particular interest of science is cause and effect. It turns out that all the parts of the universe are interrelated, so pretty much anywhere you dip in you can find plenty of topics for scientific study. In fact, the entire universe appears to be a single thing. To exist, each part depends on the rest of the universe being just the way it is.
No part of the universe can be said to exist independently.
~~
Until you consider life. To be sure, each life form is totally dependent for its existence on the rest of the universe being arranged just right. But there remains that slight wiff of independent existence, of self-motivation, of a focus of energy whose purpose is to maintain that organism.
It's as if the great mystery of existence is echoed in the little mystery of our own individual existence.
Again, as with existence, one can believe what one wants to. If you believe life is just an accident, a logical outcome of the unfolding of the physical universe, no one can prove you wrong. On the other hand, if you believe that some sort of spirit emerges when life appears, then you are free to picture (to image) a whole series of steps channelling the spirit behind existence into the forms which appear in existence and a reconnection from the individual spirit back to the all-encompassing spirit.
~~
And this is where religion comes in. The word "religion" comes from the Latin ligare = "link, bind, connect together". Thus, religion = "reconnection", reconnecting the separate self to the the whole of reality.
When cities first started, religion included everything except the farmers, and soldiers. The king was either a god or the direct representative of one and the priests were responsible for all laws, taxes, administration, arts, sciences, history, etc. Ever since, it's been a long story of separating out these functions and leaving religion with only it's basic function of reconnection.
Of course, religion has been used -- just like nationality, race, language, ideology and so on -- as a marker for groups to declare their separation from others. It's only about religion when that is the most salient characteristic available for creating a privileged caste, but it seems to be the nub of much of the debate about religion and science.
~~
Both science and religion seem especially conducive to the delusion that you've got it all covered, even though both lead to the conclusion that you do not.
In the end, either religion or science is valuable to the extent that it is helpful to someone for finding their way in the world and their place in the universe.