Science is a wonderful thing if one does not have to earn one's living at it. One should earn one's living by work of which one is sure one is capable. Only when we do not have to be accountable to anybody can we find joy in scientific endeavor. -Albert Einstein

It has been known since the time of Euclid^{w} that all of geometry can be derived from a handful of objects (points, lines...), a few actions on those objects, and a small number of axioms^{w}. Every field of science likewise can be reduced to a small set of objects, actions, and rules. Math itself is not a single field but rather a constellation of related fields. One way in which new fields are created is by the process of generalization.

A generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole.^{[1]}

The purpose of this article is threefold:

To give a broad general overview of the various fields and subfields of mathematics.

To show how each field can be derived from first principles.

To provide links to articles and webpages with more in depth information.

Foreword:

Mathematical notation^{w} can be extremely intimidating. Wikipedia is full of articles with page after page of indecipherable text. At first glance this article might appear to be the same. I want to assure the reader that every effort has been made to simplify everything as much as possible.

The following has been assembled from countless small pieces gathered from throughout the world wide web. I cant guarantee that there are no errors in it. Please report any errors or omissions on this articles talk page.

If you wish you can see all the sections on a single page here: Intermediate mathematics/All. But that page is very large and will take a full minute to finish loading.