My principal research interest is foundations of mathematics . In order to know what we are talking about, it's important to define our terms. Here is my attempt to define the terms foundations and foundations of mathematics . I am trying to make the meaning of these terms crystal clear to anybody who wants to understand them. The Hierarchy of Concepts 1.
All human knowledge is conceptual and forms an integrated whole.
Usually if not always, this kind of conceptual unity follows from the existence of a specific subject matter, the real-world object of study. Mathematics is a particular field of study. So, for instance, the science of triangles is the study of common aspects of all triangular bodies. H. G. Apostle, Aristotle's Philosophy of Mathematics . Like any large field of study, mathematics has a number of subfields, for instance functional analysis and algebraic geometry.