The majority of mystic practices (from what I can determine) seem to indicate a key concept: quieting the mind - basically ignoring the sensory input and the random thoughts that are constantly flowing through the brain. This silencing of the physical reality allows the presence of a non-physical reality to enter the consciousness.
The problem is that this non-physical reality returns no experience - it is as though nothing has happened and no time has passed.
How can the non-physical, then, provide physical proof of its existence?
This question is the biggest hurdle to convincing the non-believer - although it may turn out that some minds are simply incapable of making the leap into a non-physical [Spiritual] dimension. Subjective "experience" of the event is the current proof. Logic can get the intellect to the door, but the mind must step through its doorway to completely understand what lies on the other side.
And since there is no return of cognitive information, it is difficult to speak of the non-physical without sounding deranged or contradictory; thus the 'silence' of the mystic. If the experience is mentioned at all, it is in the form of poetic verse or Kōan-like riddles that force the mind to hold two contradictory thoughts at the same time (this sets up a static interference and the brain shuts down allowing the other reality to flow in).
It's not so much that science doesn't know a non-physical reality exists, but so far, it has no physical application and is seen as pretty much worthless.
Religion uses the non-physical [Spiritual] environment as a power base, since most people can sense the presence of its reality without being able to explain its fact.
Here is a way to get close to a conception of the non-physical:
Let's say that all you know is 'east' as a direction. All other directions - west, south, north - are viewed as 'non-east'. Since east is the only familiar referent, anything not east is a basic unknown described only as being 'not east'. So it is with the non-physical. There may be many dimensions to it, but we can only say 'non-physical'.
The 'non-physical' lies cloaked in mystery as long as 'physical' remains the experience which fills the consciousness to the exclusion of all else - but it does not go away. It waits patiently for any who care to find it.