I dispute that investigation in the context of "supernatural"
factors is intrinsically impossible. But perhaps we disagree
on the definition of "supernatural".
My take: space, time, matter, and energy exist, and constitute
the "natural system". This phrase already has other uses, so
please bear with me. Elements of this natural system interact
naturally, and, in general, the behaviour of the system can be
explained completely by understanding the principles of these
interactions; that is to say, the laws of nature. These principles
include mathematical relationships. They do not include
hidden intelligence. They seem to include random events
at the scale of atoms, which is awkward if we want to have
definite conclusions, but is manageable if events actually
are random.
"Supernatural" then means any behaviour of the natural
system that deviates from natural law. This may come from
outside the natural system, or from entities within the natural
system that can produce an un-natural influence.
For instance, a human body appears to be natural, but
religion usually treats human identity as including a "soul"
that has supernatural properties, such as preserving
personal identity after you die. Indeed, evidently your
identity resides in this supernatural object, and acts that
you perform with your physical body are somehow
initiated by a supernatural interaction between your soul
and your body.
Testing this is tricky. Attempts to distinguish a body with
a soul and a body without one have been unsatisfactory.
Also, your identity does seem to reside mainly in your
physical body. I think it is fair to call it unproven that a
human being has a soul of the type that I suggested.
Transubstantiation should be another supernatural event.
Bread on a plate and wine on a glass are converted into
human tissue and blood - before they enter a living human
body. (To convert bread and wine into human tissue after
they enter a human body is not supernatural.) However,
they do not in any evident way become flesh and blood,
except that a priest tells us that they have. Examination
of this material is furiously discouraged because it's sacred.
Breadcrumbs are treated with great reverence. And people
who are allergic to bread or to wine are not excused from
the ceremony. After all, the material is transubstantiated!
But if the original material is different, then it doesn't work!
I do not believe that transubstantiation happens. I think it
is a long term misunderstanding. It certainly is not proven,
partly because doing it in a laboratory isn't allowed.
Materialism 2, The Supernatural, 0.
It is a pity, because it would be a really good way to impress
unbelievers. A god could do it, presumably, while the material
was in someone's hand. If that /was/ allowed to scientists,
then I expect it would be extremely reliable. Just what scientists
like.