Mathematics as a Broad Subject
In Trinity College we just have one Department of Pure and Applied Mathematics. Just as Pure Mathematics covers a multitude of areas, such as Algebra, Analysis, Geometry, Topology etc, so Applied Mathematics covers Theoretical Physics, Computing, Numerical Analysis etc. Courses in all of these areas, plus Statistics and even Economics are available to mathematics students, so when we use the word `mathematics' we do so in a very broad sense.
Mathematics has evolved as an abstract subject because of its usefulness as a tool in science and industry. The range of its applications is increasing all the time. New applications demand new mathematics, which often leads to further applications, and so on.
Theoretical Physics has always been a fertile ground for this interaction. Recently physical considerations have led to speculative results in the geometry of four-dimensional manifolds. These have now been proven by `pure' mathematicians.
It is also true that over time mathematics seems to go through phases of divergence and convergence. In the divergent phase many seemingly independent theories are developed, motivated by particular problems. However, despite the fact that the problems have been totally different it often emerges that there is a unity underlying the mathematical theories that have been developed. This is the convergent phase, when the seemingly disparate threads are woven into a pattern.
``Specialisation, long seen as dooming mathematics with a multiplicity of separate disciplines, led to deeper understanding and a new unity in mathematics.'' Bers
The main thing to understand in all of this is that as mathematics pushes back the boundaries of knowledge the subject is constantly in a flux. Theorems don't change, but the relative importance of theorems and even areas do, and so the need is for a broad background.