Ontological Reductions in Mathematics. Part III: On Reconstruction of Some Parts of Mathematics

This is the third part of the study concerning the problem of ontological reductions in mathematics. In this part, the problem of reconstruction of (parts of) mathematics in theories weaker than full ZFC is discussed. The tools from reverse mathematics are used, and the results are discussed from the point of view of various versions of realism (Godel's realism, Quine's quasi-empiricism and Balaguer's Full-Blooded Platonism). Some problems concerning the possibility of discussing these problem outside the conceptual system of set theory are also addressed.