The lecturer explains that there is a rational explanation for these symbols but the details will be kept from the students. However the lecturer reassures the students that the symbols do behave in a sensible way. As this carries with it the suggestion that the symbols in some way have a life of their own, the stage is set for a course that is nearer to mystery than to mathematics. One possible explanation is that my bias is to the discrete while the analysts' bias is to the continuous. So I particularly admire the way that analysis uses logic to tame the wild natural world of limits and related ideas that arise in a half-grasped intuitive way outside mathematics.
Analysis gives substance to the shadows.
Now the analyst realises that once the shadowy ideas of limit, etc. have been given substance then the most productive way to work is with the intuitive shadows which allow the imagination to range freely through analysis. However it is surely the case that such an approach will encourage the students simply to believe that in mathematics it is all right to work with intuitive shadows irrespective of whether those shadows have been mathematically legitimised. For the great majority of our students, the aim of an analysis course is not to train them as analysts but to give them an appreciation of the following. How does mathematics tackles the tricky problem of taking intuitive shadows which have proved to be useful tools in understanding how the world is, and putting those shadows on a firm foundation. If, on the strength of the above argument, we decide to give the details then we are faced with a new problem.