It's often been said that Gottfried Wilhelm Leibniz was one of the last people to master and contribute to all of the fields of knowledge in existence during his time (eg http://plato.stanford.edu/entries/leibniz/ ).
Speaking of just mathematics, we're WELL past the point where it's possible to cover the entire interior of the "mathematical circle". As an illustration, MathSciNet (http://www.ams.org/mathscinet/) is a mathematical review database that consists of prettymuch every mathematical book and peer-reviewed article in mathematics since the 1940's (and some well before 1940). As of this writing, there are just under three million publications stored in MathSciNet. And these articles are so specialized that a researcher in one area would have to exert a significant amount of effort to understand articles in a completely different area of research.
So, how do we go from the center of the mathematical circle to the edge? By starting with the basics and moving through a path of what's considered "important" (ie the broad survey provided by undergraduate mathematics courses: the calculus sequence, real analysis, abstract algebra, combinatorics, topology, etc) along the way.
It's often been said that Gottfried Wilhelm Leibniz was one of the last people to master and contribute to all of the fields of knowledge in existence during his time (eg http://plato.stanford.edu/entries/leibniz/ ).
Speaking of just mathematics, we're WELL past the point where it's possible to cover the entire interior of the "mathematical circle". As an illustration, MathSciNet (http://www.ams.org/mathscinet/) is a mathematical review database that consists of prettymuch every mathematical book and peer-reviewed article in mathematics since the 1940's (and some well before 1940). As of this writing, there are just under three million publications stored in MathSciNet. And these articles are so specialized that a researcher in one area would have to exert a significant amount of effort to understand articles in a completely different area of research.
So, how do we go from the center of the mathematical circle to the edge? By starting with the basics and moving through a path of what's considered "important" (ie the broad survey provided by undergraduate mathematics courses: the calculus sequence, real analysis, abstract algebra, combinatorics, topology, etc) along the way.