Secondary Calculus: a new area of mathematics

The history of Babylon's tower told in the Bible explains the fate of the greatest scientific project of that time which failed after a direct divine intervention. More precisely, when God realized that the construction of the tower was too much advanced, He mixed the languages of the builders and consequently the work could not be completed. But such a story has not ended yet. Indeed some stubborn people who was curious about the structure of the Universe

**invented a new common language**which now goes under the name of "Mathematics". Such a language helped us a lot in understanding many details of the divine creation. But meanwhile God was not idle and now an then He applied the old trick of

**mixing languages within Mathematics itself**. Such a process can be explained more scientifically by invoking the laws of Thermodynamics, namely the second one, which inevitably affects the scientific progress in general and shows itself in the most bright way in the mathematical branches of Natural Sciences. The modern mixed state of the mathematical language and the deriving form it chaos is perfectly shown by the

**current A.M.S subject classification**. Not to speak of the almost total impotence before the non-linear problems and the screaming mathematical ugliness of the current theory of

*quanta*.

**Diffeotopy is a new area of Mathematics**generated by the attempts to find some path in the multitude of thousands concrete facts growing around the non-linear PDEs and in all those areas of Mathematics itself, Mechanics, Physics, etc., where they are applied and are indispensable. Diffeotopy began to take a concrete shape 20-25 years ago, and now it has

**its own precise methodology, paradigm and philosophy**. In a schematic way diffeotopy can be defined as the study of the

**natural structures of Differential Calculus**- primary and secondary ones - and their

**applications to problems of Mathematics itself and mathematical models from Natural Sciences**. In this sense Differential Geometry and Topology, Mechanics of Continua Media, Field Theory, Control Theory, etc., constitute objects for diffeotopy - obviously not to speak of the non-linear PDEs themselves. Undoubtedly such a list is not complete and as time goes by it will be enlarged. For instance, it can be sharply foreseen that in due time Algebraic Geometry will take part of it - but understood as Differential Geometry over algebras of algebraic functions. And this will happen only after sheaves, schemes and other palliative means have been selected for extinction by the natural Darwinian process for concepts.

From a materialistic viewpoint diffeotopy represents a natural synthesis of many sectors of modern Mathematics among which one can find PDEs themselves, Commutative and Homological Algebra, Algebraic and Differential Topology, Differential Geometry, Lie Group and Algebras, and other things. Therefore the pathway to diffeotopy is not easy. But on the other hand a steady and safe progress in the key areas of Natural Sciences is not possible without a suitable development of diffeotopy.