Category Theory As A Rival
Category theory is a unifying theory of mathematics that was initially developed in the second half of the 20th century. In this respect it is an alternative and complement to set theory. A key theme from the "categorical" point of view is that mathematics requires not only certain kinds of objects (Lie groups, Banach spaces, etc.) but also mappings between them that preserve their structure.
In particular, this clarifies exactly what it means for mathematical objects to be considered to be the same. (For example, are all equilateral triangles the same, or does size matter?) Saunders Mac Lane proposed that any concept with enough 'ubiquity' (occurring in various branches of mathematics) deserved isolating and studying in its own right. Category theory is arguably better adapted to that end than any other current approach. The disadvantages of relying on so-called abstract nonsense are a certain blandness and abstraction in the sense of breaking away from the roots in concrete problems. Nevertheless, the methods of category theory have steadily advanced in acceptance, in numerous areas (from D-modules to categorical logic).
Read more about this topic: Unifying Theories In Mathematics
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