Does our grasp on science, technology and logic describe everything in the world, or are there things that are still beyond our understanding, and maybe even beyond our awareness?

I am an engineer by training. At one point, I was a very good engineer and science and technology was a world I felt fully at home in. I regularly came top in my end of year exams at University and graduated with a First. There was no problem I could not find some way to solve, eventually. The world was technical and I was its heir apparent.
Omnipotent? Perhaps not, but omni-potential was a concept that I could really wallow in.
And then I went on a course run by a very strange character, a sort of Zen-mystic prima-donna, and though I initially categorised him in the low place I reserved for all those people who preached a message which had no foundation in provable scientific fact, I was out of the office for seven days and felt I might as well go with the flow just for the hell of it.
The outcome of that simple decision turned my world inside out. With like minded colleagues, I experienced first hand effects and events that defied rational explanation. We repeated things ourselves, and though we worked hard to disprove them, or find alternative explanations, we were frankly at a loss to do so. And I began to realise for the first time that the scientific and rational basis that formed the boundaries of my world was actually only a model that described a small part of it. Instead of using science and technology as a ladder to extend my thinking I had encouraged it to become a cage which imprisoned it.
Looking back on it I find it difficult to credit how I could have been so naïve. It is fairly obvious that scientific discovery abounds continually as we identify new phenomena and develop new models to describe them, or as we refine existing models to better reflect a deeper understanding of what were previously seen to be anomalies. Perhaps my 'engineering' mindset just thought of such things as 'range extensions' and 'enhancements'.
What our ancestors used to see as 'magical', were not things that were outside reality, but simply outside of their limited ability to explain reality. Why should I fool myself into believing that such discovery has now stopped when all our recent experience points in exactly the opposite direction?
My scientific models are trapped within my paradigms and experience of the world, which in turn are limited by my five very physical senses. Simply because I am not equipped (whether by design or by evolution) with the ability to identify or recognise certain phenomena does not necessarily mean that those phenomena do not exist. Does the fly that lands on my VDU screen see it as anything more than a vertical table?
The universe (at least the part of it that we know about) has been in existence for 15 Billion years. We have only been really cognitive of aspects beyond volume, mass and time for the last hundred. And even then, our detail knowledge is limited to a small planet orbiting one star amongst 10 Billion stars that comprise a galaxy amongst 10 Billion galaxies. In our small part of the universe, matter came together to form molecules, which came together to form amino acids, which came together to form the basis for simple life, and single cell organisms, which came together to form more complex symbiotic arrangements that eventually evolved into higher primates - the basis for you and me. This has been without doubt an amazing sequence. Am I confident in all the vastness of time and space that such complex sequences are only feasible for the physical phenomena that I have emerged from? Or could it be that, as fantastic as 'life' is, it has still only evolved to the point where I am equipped to see solely where I came from? Dust!
My own particular decision on this matter, subsequently reinforced by my experiences in wrestling with the next two questions, was to accept that the spiritual realm was a real possibility, and that we are even less well equipped to disprove its existence than we are to demonstrate it beyond doubt.

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