Is Science compatible with spirituality?

The difficulty with trying to answer this question is that "spirituality" is such a vague and nebulous concept for which anyone who considers him or herself "spiritual" gives it an individual, personal meaning.

Spirituality is primarily an emotional response, and although religious people combine it with a belief in a supernatural deity, it does not seem to need such a belief in others.

Science seeks to explain what it finds observable and open to test - at a simple, everyday level if it were not for science we would not be able to discuss it like this - in fact we would not even have electricity supplies let alone telecommunications. It does not try to state all it suggests is true - it works by Hypothesis and Theory before establishing Law - this lack of certainty may be what drives some religious fundamentalists to regard their faiths' founders statements of belief, entirely True, and Science axiomatically but selectively False.

Humanity has for as long as can be established, an innate need for some sort of religious or spiritual support. Now, until historically quite recently that support helped give some sort of reason for existence and an attempt to explain how things exist, although if you examine a strongly-religious view on these you arrive at the rather uncomfortable thesis that the "reason" is merely some supernatural being's whim! 

Religion is no longer necessary to explain "how" because Science is attempting to unravel what really happens on a cosmic or local scale; but neither give any satisfactory "why". Indeed, questioning divine whim on the basis that there might be a deity to have a whim, leads to questioning why that deity might even "exist" in the first place. A riddle beyond all of theology, philosophy and science. Really, the only value in religion these days is in its second major role since (probably) Palaeolithic times, that of lessening the fear of death and comforting the bereaved, but then only among its believers. (Its third of three main roles was bolstering a social and moral code, but you do not need to be religious to be moral and well-behaved; and at the other end of the scale, some of the worst-behaved use their religion to excuse their actions.)

So where does that leave Spirituality? I think one convenient definition or explanation is an emotional need to place one's existence in the greater scheme of things - religion can do that but only if you believe in a god. I believe it to be entirely internal, and it may find expression in as many ways as there are people who believe themselves spiritual beings.

If so spirituality may manifest itself in any of many ways. For example:
   Being moved by a beautiful sunset, majestic wild animal or the processes and sights of astronomy, geology or biology. 
   In the deep satisfaction gained by creating a work of art of craft requiring great skill, patience etc to achieve.
   In the emotional reward of being moved to help others in distress.
   In finding peace in solitude or with like-minded companions, anywhere in the countryside - from a quiet hay-meadow to a great mountain-side.  
   In friendship, companionship, love even, in its widest sense!

Note that I include the three main Natural Sciences. One might even find one's own inner "spirit" not just in admiring the aesthetic beauty of the sunset, animal or mountain, but also in understanding the science in it.  

I'll finish with a tale that illustrates that perfectly; one I find deeply moving even though my interests in the natural sciences are as an amateur.

One night some years ago I happened to be admiring a star-rich Winter night-sky with some acquaintances, on the Mendip Hills in South-West England. The air was clear, with no moon and a goodly distance from street-lights shielded anyway by the hills. My companions said a year or so previously, they had been gazing at a similar scene with a young Canadian visitor. She stared at the sky for a long time then broke down in tears. Somewhat embarrassed, her hosts asked what was the matter. She explained that she was a professional astronomer but worked with a radio-telescope near a city with common fog and persistent light-pollution that usually blanked out the sky at night. So she spent her working day performing meticulous measurements and analyses of stellar radio signals, investigating mathematically their beautiful physics. This though, she had said, was the first time that she had seen the Milky Way - our and her subject stars' own galaxy - with her own eyes.  

Was this astronomer religious? I have no idea. Spiritual? I would count her response to that experience as "spiritual" for her.