Quest for true mathematics

Another thing to be investigated: the perversion of mathematics through abstract algebra in the 20th-century (field theory and abstract ring theory and algebraic geometry based on them).  Our moto; back to Galois, Gauss, Dedekind, Riemann, Weierstrass, Poincaré ! Bring back the study of Euclid and Archimedes.  Several complex variables and 'elementary' number theory. The more concrete analytic study of (partial and ordinary) differential equations rooted in geometric intuition and the reality of 'real' physics (Kepler, Newton, Leibniz, Cauchy, Lagrange, Laplace, Hamilton, etc.).  And study the philosophical light brought by formal logic and recursion theory before it became perverted by logical positivism and ZFC set theory and model theory.

Mathematics though difficult and demanding dedication yet ultimately must be a self-illumination of the intelligible structure of the soul. Once such an illumination is obtained (which involves the development and purification of intuition) then the result should be permanent. If this is not the case then we are in the presence of a 'sick' and 'deviant' mathematics that, besides lacking philosophical (and even logical) value, does spiritual and psychology harm: this appears to be the case of much of abstract algebra and abstract analysis together with ZFC set theory and model theory. Perhaps there is more true mathematics or philosophy in hard classical physics problems than generally realized.

The fountains of authentic mathematics remains: mathematical analysis in ${\mathbb{R}}^n$ including PDEs (and its its extensions and abstractions in complex analysis, measure theory and function spaces) and finite mathematics.

Update: the history of abstract algebra is of utmost interest. We must distinguish between natural and good abstraction and faulty abstraction. Thus Grassmann's linear, exterior and multilinear algebra is a brilliant example of good and natural abstraction.  So too is the theory of groups, specially the theory of finite groups (cf. the famous textbook of Burnside). Likewise Boolean algebras and other structures considered by Whitehead in his Universal Algebra.   But there are other structures in which further abstraction is not a good idea. Thus finite fields, algebraic number fields and quotients of polynomial rings over a common field - these are fine as they are. A more abstract perspective (such as Noetherian rings, etc.) is misleading and wrong.  In fact so much of the theory of finite fields and algebraic number theory can be carried out in a wonderful way in the context of elementary number theory.  The same goes for category theory. The general theory has no interest compared to Abelian categories but this is just a faulty abstraction of categories of modules.   So too is general topos theory basically sterile.  Only toposes of sheaves are really interesting - though there are also some interesting alternative structures.