Formalizing the world in rigorous mathematical terms is no less significant than its fundamental understanding and modeling in terms of ontological constructs. Like black and white, opposite sexes or polarity signs, ontology and mathematics stand complementary to each other, making up the unique and unequaled knowledge domain or knowledge base, which involves two parts: • Ontological (real) mathematics, which defines the real significance for the mathematical entities, so studying the real status of mathematical objects, functions, and relationships in terms of ontological categories and rules. • Mathematical (formal) ontology, which defines the mathematical structures of the real world features, so concerned with a meaningful representation of the universe in terms of mathematical language. The combination of ontology and mathematics and substantial knowledge of sciences is likely the only one true road to reality understanding, modeling and representation. Ontology on its own can’t specify the fabric, design, architecture, and the laws of the universe. Nor theoretical physics with its conceptual tools and models: general relativity, quantum physics, Lagrangians, Hamiltonians, conservation laws, symmetry groups, quantum field theory, string and M theory, twistor theory, loop quantum gravity, the big bang, the standard model, or theory of everything material. Nor mathematics alone with its abstract tools, complex number calculus, differential calculus, differential geometry, analytical continuation, higher algebras, Fourier series and hyperfunctions is the real path to reality (Penrose, 2005).
The Real Forms of the Fractional Supergroup SL(2,C)