The development of information sciences, in their widest sense, has pointed out the crucial importance of the complementary role that innovative ideas and the power mathematical and technical tools play in this process showing how new emerging fields establish themselves as valid. This is also evident in the methodological attitude of a scientist like Eduardo Caianiello who, as a physicist, thought that the tradition of physics could provide some breakthroughs to the development of the new disciplines related to the mental and the investigation of intelligence. In the present paper some topics will be briefly discussed trying to point out the importance of the connection between intuition and rigor and, in particular, the process underlying the passage from a new intuitive idea to its formal realization. This process which has been outlined by Rudolph Carnap in his analysis of the notions of explicandum and explicatum is a form of what Gregory Bateson has superbly described as the tension existing between imagination and rigor, the couple of terms we chose as title of this volume of essays in honor of Eduardo Caianiello. The chosen topics all concern Eduardo’s teachings and the multiplicity of his scientific interests, although in different ways and in various degrees, as will be clear in the following.