This paper analyzes the idea of designing computer hardware inspired by the knowledge of how the brain works. This endeavor has lurked around the twists and turns of the computer history since its beginning, and it is still an open challenge today. We briefly review the main steps of this long lasting challenge. Despite obvious progress and changes in the computer technology and in the knowledge of neural mechanisms, along this history there is an impressive similarity in the arguments put forward in support of potential advantages of neural hardware over traditional microprocessor architectures. In fact, almost no results of all that effort reached maturity. We argue that these arguments are theoretically flawed, and therefore the premises for the success of neuromorphic hardware are weak.