How “mathematics” be able to contribute to social sciences?

-For the foundation work of mathematical sociology-

Abstract

Firstly, it is discussed what "mathematics" means for various people including mathematicians and social scientists. Secondly, through some examples, possibility of verification of models by virtue of its clear logic of mathematics is discussed. Finally, by rather critically investigating “An algebraic reinterpretation of "IKI" by Kosaka (1984,1989), effectiveness and its boundary of mathematical models are discussed but we will point out that our discussion might be able to contribute the development of mathematical sociology.