Abstract We describe in a very concise way the multidisciplinary program of artificial mathematical intelligence and the need of a new and improved paradigm of formal computation based on conceptual generation, called conceptual computation, with the potential of being a global paradigm-shifting technique in theoretical physics and computational mathematics. Furthermore, we describe some of the most outstanding results regarding fundamental cognitive abilities needed for the materialization of conceptual computation such as analogical reasoning, metaphorical thinking and conceptual blending. Finally, we present a concrete example of the conceptual computation of the notion of binary continuous operation as a formal blend (i.e . many-sorted first order categorical colimit (pushout)) of the notions of continuous function between topological spaces and perfect square topological space.