ABSTRACT
One aspect of mathematics that distinguishes it from other scientific disciplines is its emphasis on iterative processes of abstraction. In mathematics, the objects under study are not physical objects but rather ideal versions of those objects that are understood and defined in terms of relationships between the elements of those objects. New mathematical objects and structures such as groups and rings stem from abstracting commonalities across families of previously abstracted objects and structures. Abstraction is such a central facet of mathematical thinking and reasoning that domains of mathematics (e.g., algebra and geometry) are characterized by progressions toward arguments entirely comprising legal manipulations of syntactic sentences. The validity of an argument is defined independent of the content of the argument, but as consisting of the application of inferences that are judged to be truth preserving based upon their abstract form.
