1 How to Taste Mathematics

Modern culture nowadays faces a real danger as mathematics is being driven outside of the realm of components constituting that culture. Viewing that matter as a real danger is fully justified. Mathematical structures reflect all possible patterns of mental activity, which is creative and unlimited. This merit proves itself especially in mathematical breakthroughs. In one phase of the sequence of its activities, mind formulates some simple or rather involved problem. Soon the problem transforms into a sort of obsession that pushes the mind into a long search for a solution. In case a solution is found, mind is satisfied only temporarily, i.e. until mind creates a new problem. This is the fundamental machinery of doing mathematics. Mathematics is not a pragmatic science in any sense. Hence, the rarity of great mathematicians in our epoch, if not for a reason, but due to the fact that mathematics has been driven to the economic sector. Music is moving towards the same fate. Nobody can pretend having a strict standard for appealing to our deep aesthetic motivations, as in music and other arts. Classical methods of teaching mathematics, which are still adopted, are among the factors generating the aforementioned situation. We have to modify these methods so as to achieve a deeper understanding of mathematics. Usually, we express our incomprehension of mathematics and other subjects by describing it as being too involved, rendering it unimportant, and finally neglecting it altogether. The world of mathematics, just like that of music, yearns for being understood, even partially