This article by Karen Armstrong contains the best explanation I have found for how the religious and the secular modes of thinking do not have to be in conflict. They are separate spheres and we do ourselves and the world a great disservice by confusing the two :
We tend to assume that the people of the past were (more or less) like us, but in fact their spiritual lives were rather different. In particular, they evolved two ways of thinking, speaking, and acquiring knowledge, which scholars have called mythos and logos. Both were essential; they were regarded as complementary ways of arriving at truth, and each had its special area of competence. Myth was regarded as primary; it was concerned with what was thought to be timeless and constant in our existence. Myth looked back to the origins of life, to the foundations of culture, and to the deepest levels of the human mind. Myth was not concerned with practical matters, but with meaning. Unless we find some significance in our lives, we mortal men and women fall very easily into despair. The mythos of a society provided people with a context that made sense of their day-to-day lives; it directed their attention to the eternal and the universal. It was also rooted in what we would call the unconscious mind. The various mythological stories, which were not intended to be taken literally, were an ancient form of psychology. When people told stories about heroes who descended into the underworld, struggled through labyrinths, or fought with monsters, they were bringing to light the obscure regions of the subconscious realm, which is not accessible to purely rational investigation, but which has a profound effect upon our experience and behavior. Because of the dearth of myth in our modern society, we have had to evolve the science of psychoanalysis to help us to deal with our inner world.
Myth could not be demonstrated by rational proof; its insights were more intuitive, similar to those of art, music, poetry, or sculpture. Myth only became a reality when it was embodied in cult, rituals, and ceremonies which worked aesthetically upon worshippers, evoking within them a sense of sacred significance and enabling them to apprehend the deeper currents of existence. Myth and cult were so inseparable that it is a matter of scholarly debate which came first: the mythical narrative or the rituals attached to it. Myth was also associated with mysticism, the descent into the psyche by means of structured disciplines of focus and concentration which have been evolved in all cultures as a means of acquiring intuitive insight. Without a cult or mystical practice, the myths of religion would make no sense. They would remain abstract and seem incredible, in rather the same way as a musical score remains opaque to most of us and needs to be interpreted instrumentally before we can appreciate its beauty.
In the premodern world, people had a different view of history. They were less interested than we are in what actually happened, but more concerned with the meaning of an event. Historical incidents were not seen as unique occurrences, set in a far-off time, but were thought to be external manifestations of constant, timeless realities. Hence history would tend to repeat itself, because there was nothing new under the sun. Historical narratives tried to bring out this eternal dimension. Thus, we do not know what really occurred when the ancient Israelites escaped from Egypt and passed through the Sea of Reeds. The story has been deliberately written as a myth, and linked with other stories about rites of passage, immersion in the deep, and gods splitting a sea in two to create a new reality. Jews experience this myth every year in the rituals of the Passover Seder, which brings this strange story into their own lives and helps them to make it their own. One could say that unless an historical event is mythologized in this way, and liberated from the past in an inspiring cult, it cannot be religious. To ask whether the Exodus from Egypt took place exactly as recounted in the Bible or to demand historical and scientific evidence to prove that it is factually true is to mistake the nature and purpose of this story. It is to confuse mythos with logos.
Logos was equally important. Logos was the rational, pragmatic, and scientific thought that enabled men and women to function well in the world. We may have lost the sense of mythos in the West today, but we are very familiar with logos, which is the basis of our society. Unlike myth, logos must relate exactly to facts and correspond to external realities if it is to be effective. It must work efficiently in the mundane world. We use this logical, discursive reasoning when we have to make things happen, get something done, or persuade other people to adopt a particular course of action. Logos is practical. Unlike myth, which looks back to the beginnings and to the foundations, logos forges ahead and tries to find something new: to elaborate on old insights, achieve a greater control over our environment, discover something fresh, and invent something novel.
In the premodern world, both mythos and logos were regarded as indispensable. Each would be impoverished without the other. Yet the two were essentially distinct, and it was held to be dangerous to confuse mythical and rational discourse. They had separate jobs to do. Myth was not reasonable; its narratives were not supposed to be demonstrated empirically. It provided the context of meaning that made our practical activities worthwhile. You were not supposed to make mythos the basis of a pragmatic policy. If you did so, the results could be disastrous, because what worked well in the inner world of the psyche was not readily applicable to the affairs of the external world.
From “The Battle for God”