Lazs, it's not a matter of science admitting the possibility of the supernatural. Science simply IGNORES the supernatural because it does not apply - by DEFINITION. For this discussion, there's not even a judgment call over right or wrong, just that the supernatural excludes science because it is not concerned with predictable, observable, measurable results. Again, as soon as you can predict, observe, test, and measure something, by definition it is no longer supernatural.
Where we get hung up, mostly, is not whether there are supernatural explanations to phenomena that are true, but whether we introduce those explanations into the realm of science and even go so far as to include supernatural explanations in our science curriculum.
The supernatural excludes science. Science excludes the supernatural. This by definition, not by bias or arbitrariness.
Let's say we go ahead and REDEFINE science to include the supernatural, and accept this redefined science as school curriculum and a basis for higher research:
Now, I want a more effective treatment for cancer. Or, I want a faster graphics card. We call on this redefined science to help. Do we establish a research group to pray for the treatment and expect it to appear in our pharmacies one day? Do create a laboratory to light incense near computers and rub them with special oils and expect them to get faster? Do we publish journals on how special dances with ceremonial headdresses increase the throughput of fiber optic networks? Those are now valid scientific approaches.
At some point, if we wish to advance our understanding of the natural universe and make medical breakthroughs and have faster computers and networks, we need to have a discipline that IGNORES the supernatural and seeks to find predictable, observable, measurable, testable explanations wherever we can. Now that we've redefined science to be something else, we need to give this discipline a new name. But whatever we call it, it is necessary AND it BY DEFINITION ignores supernatural explanations.
I suppose we can argue that it's not necessary to have this disciple that ignores the supernatural and that having this discipline creates a soulless society. I think there are plenty of examples of cultures who have made that choice (Afghanistan under the Taliban) and I'm quite glad that the western world and particularly the U.S. (so far) has not.
What I don't quite understand is how the strongly pious can't appreciate the huge rewards they've reaped by living in a society where some do have the discipline to ignore the supernatural in pursuit of a scientific understanding of the world. I, for one, am glad that science has figured out how to create drought and disease resistant grains so I don't have to fear my daughter being sacrificed on the altar to ensure a good wheat crop this year.