Intuition and rigor are the 'yin and yang' of conducting interesting applied math research (specializing to my field because that's all I can speak to). Initial research is almost entirely driven by intuition. Experimentation, testing, trying new ideas, all founded on the basis of rigor but in the heat of the moment (read: weeks) it's not really what I'm thinking about. Then, during testing and validation, I come back and properly build up the rigor blocks; revealing both errors and potential new directions forward. Cycle over and over, both compose a robust and creative scientific process. Ideally.