Mathematical Duality: The Ramsey Approach

The meta-mathematic approach to the mathematical duality is introduced. Systems of mathematical theorems/propositions related each to other by the mutually exclusive relations of “duality” and “non-duality” are addressed. Theorems/propositions are considered as the vertices of the graph, and the relations of duality appear as the links connecting the vertices. Thus, the bi-colored, complete graph emerges. The coloring procedure is exemplified with the theorems of projective geometry. It is demonstrated, that that the graph built of the quartet of vertices/theorems of projective geometry, which contains no mono-colored triangle is possible. The monochromatic triangle will necessarily appear in the graph containing five vertices/theorems of projective geometry. The emerging graph is different from the traditional Ramsey graph. The graph representation is easily generalized for any system (finite or infinite) of theorems/propositions in which the relations of duality are established.