We hope that there is much potentia

We hope that there is much potential in the suggested didactic ideas and we are now engaged in a teaching experiment in which they are systematically put to test. Notwithstanding our belief that the method may work, there is a question that keeps bothering us: even if our students succeed in acquiring a versatile and adaptable assortment of perspectives, how durable and robust will their flexible knowledge be in the long run? The chances are that pseudostructural thinking may sometimes be compared to secondary illiteracy. After all, the mechanical mode has so much to offer to intellectual sluggards: it exempts the problem-solver from the need for constant alertness, from the strain which inevitably accompanies going back and forth from one perspective to another. As testified by Souriau, mathematicians themselves yield to the charm of mechanized symbolic manipulations: Does [the algebraist] follow [the original meaning of symbols] through every stage of the operations he performs? Undoubtedly not: he immediately loses sight of them. His only concern is to put in order and combine, according to known rules, the signs which he has before him; and he accepts with confidence the results thus obtained.