Project Focus

In this project we will attempt to scrutinize the applicability of the new paradigm in relation to the process of semantic unfolding characterizing major developments in mathematics; most precisely, in complex geometry, algebra, topology, mathematical logic, differential geometry, algebraic topology, and gauge theory.

By consulting the original sources, we explicate in detail the semantic unfolding process permeating the advancement of mathematical thought in the following characteristic instances:

• Riemann's geometric function theory on the complex domain;

• Galois' theory of field extensions via permutation groups;

• Grothendieck's unification of Riemann's and Galois theories in the topological domain of covering spaces and the fundamental group;

• Poincare's uniformization strategy in complex surface geometry;

• Cayley's geometrization of group theory with emphasis on the non-commutative free group on two generators;

• Klein's uniformization strategy of Riemann surfaces based on potential theory and the free group;

• Goedel's first incompleteness theorem in mathematical logic;

• Cohen's method of forcing and the emergence of topoi as non-standard models of set theory;

• Grassmann's extension theory in the context of lineal geometric analysis;

• Leray's topological theory of sheaves and the compatiple passage from the local to the global;

• Grothendieck's theory of sheaves as coefficient systems for cohomology;

• de Rham's theory of differential forms and cohomology;

• Weyl's program of localizing gauge symmetry to articulate matter;

• Chern's theory of bundles with connection and the emergence of integrality;

• Steenrod's theory of integrable connections and local systems;

• Hilbert's program relating the monodromy action of the fundamental group with the theory of differential equations.

The scrutinization of all these novel mathematical developments from a common semantic perspective allows us to trace their thought-provoking abstraction affinities and attempt a unification of their conceptual roots under the new "constellatory unfolding'' paradigm.

The term semantic is not restricted in the sense that it is currently employed in the field of Mathematical Logic. Rather, we retain the original ancient Greek meaning of the term as pertaining to semasiology and signification.