Through-axiomatisation
Early in the 20th century, many parts of mathematics began to be treated by delineating useful sets of axioms and then studying their consequences. Thus, for example, the studies of "hypercomplex numbers", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory (in this case, with the specific meaning of associative algebras over the field of complex numbers.) In this context, the quotient ring concept is one of the most powerful unifiers.
This was a general change of methodology, since the needs of applications had up until then meant that much of mathematics was taught by means of algorithms (or processes close to being algorithmic). Arithmetic is still taught that way. It was a parallel to the development of mathematical logic as a stand-alone branch of mathematics. By the 1930s symbolic logic itself was adequately included within mathematics.
In most cases, mathematical objects under study can be defined (albeit non-canonically) as sets or, more informally, as sets with additional structure such as an addition operation. Set theory now serves as a lingua franca for the development of mathematical themes.
Read more about this topic: Unifying Theories In Mathematics