Thank you for your post. The notion of appealing to intuition is a very serious one. Intuition does not exist, at least not in the sense that it is commonly u

Thank you for your post. The notion of appealing to intuition is a very serious one. Intuition does not exist, at least not in the sense that it is commonly understood. There exists simply familiarity and permutation, but most explanations of intuition appeal to some form of ex nihilio proposition generation. What we refer to as intuition is the subconscious process of synthesizing previous experiences and transformations (permutations) of component parts. It is precisely for the rigor of explicit predicates and rules governing the transitions from one statement of fact to another that distinguish the reliability of math from the fallibility of informal thought. As you so deftly described, the ability to entertain varying predicates and explore the possible routes without violating the rules is a stimulating task that is unfortunately absent from most lower math education. I whole-heartedly agree that the capacity to identify your assumptions is a valuable skill that rigid & formal thinking help to cultivate.