Six arguments against the view that conscious experience derives from a material basis are presented, none of which is entirely new taken in isolation but whose conjunction is compelling. These arguments arise from epistemology, phenomenology, neuropsychology, and philosophy of quantum mechanics. It turns out that any attempt at proving that conscious experience is ontologically secondary to material objects both fails and brings out its methodological and existential primacy. No alternative metaphysical view is espoused (not even a variety of Spinoza’s attractive double-aspect theory). Instead, an alternative stance, inspired from F. Varela’s neurophenomenology is advocated. This unfamiliar stance involves (i) a complete redefinition of the boundary between unquestioned assumptions and relevant questions ; (ii) a descent towards the common ground of the statements of phenomenology and objective natural science: a practice motivated by the quest of an expanding circle of intersubjective agreement.