To explicate certain phenomena, e.g., the possibility of deduction without definition, we hypothesize that an individual is able to understand and appreciate reasoning with a due feeling of its necessity when the concept image of each concept involved in the reasoning has reached a certain level of development; we then speak of "deep intuition". This conception is presented (with a variety of examples) in the framework of D. Tall's theory of three worlds of mathematics ("conceptual-embodied", "proceptual-symbolic", and "formal-axiomatic").