Recent advances in the field of quantum cognition (Pothos and Busemeyer 2013; Wang et al. 2013) suggest a puzzling connection between fundamental physics and the mind. Many researchers see quantum ideas and formalisms merely as useful pragmatic tools, and do not look for deeper underlying explanations for why they work. However, others are tempted to seek for an intelligible explanation for why quantum ideas work to model cognition. This paper first draws attention to how the physicist David Bohm already in 1951 suggested that thought and quantum processes are analogous, adding that this could be explained if some neural processes underlying thought involved non-negligible quantum effects. The paper next points out that the idea that there is a connection between fundamental physics and the mind is not unique to quantum theory, but was there already when Newtonian physics was assumed to be fundamental physics, advocated most notably by Kant. Kant emphasized the unique intelligibility of a Newtonian notion of experience, and this historical background prompts us to ask in the final part of the paper whether we can really make sense of any quantum-like experience (whether experience of the empirical phenomena in the “external world” or the “inner world” of psychological phenomena). It is proposed that intelligibility is a relative notion and that, regardless of initial difficulties, quantum approaches to cognition and consciousness are likely to provide valuable new ways of understanding the mind.