Regarding the issue you present, in the past I would have answered that spirituality has nothing to do with science, as they are two completely different things: in one hand you have religious beliefs and in the other hand you have observation and study of Nature. That was quite clear and obvious to me. But now, from a new perspective and my personal research in these recent years, I think that “empirical science” is just another kind of belief and spirituality is not really about religion: it is a science! Let me explain my point.
As you may probably know, modern science (quantum physics in particular) has explored the old realms of metaphysics and the role of consciousness in the study of the universe, linking the ancient spiritual traditions and the philosophical schools to the most advanced theories in our comprehension of matter, energy and consciousness. So, from this point of view, spirituality, perceived as the realm beyond matter and our five senses, is indeed a science (you may call it the science of God, of the Universe or of ourselves), because it deals with the most profound questions about who we are, where do we come from and what’s the meaning of our existence. For instance, I think that gnosticism was an attempt to separate true spiritual knowledge from the religious dogma (and it’s funny to see in the present times how evolutionism, which was born to overcome the influence of religious dogma in science, has become very dogmatic in itself, despite some contrary evidence and the lack of proof).
In this way of thinking, the empirical science would be only a “2nd class science”, as it tries to describe and understand the surface of things, somewhat similar to the famous duality in Plato’s myth of the cave (the world of images and the world of ideas, which is “the real world”). So, to cut a long story short, I would say that, if you leave behind the old preconceptions about what is science and what is not, spirituality is in fact connected to a deep, ultimate knowledge of ourselves and there is no contradiction in being spiritual and keeping a scientific approach (if we consider that science is about seeking the truth).