TOUR OF THE SUBFIELDS OF MATHEMATICS
This is the starting page for a brief tour of the broad subfields of mathematics. It is our intention that this tour provide enough description of the terrain to help you select the heading of the Mathematics Subject Classification appropriate for a specific inquiry. (There are 61 main headings and thousands of subheadings; the areas with index pages of their own offer a tour of their subareas and links to adjacent territory.)
Click here to start the tour, or if you prefer, simply load all at once the shorter (41K) "Layman's Guide to the Mathematics Subject Areas", which contains most of the same comments but lacks the pretty pictures.
Other types of navigation tools at the Mathematical Atlas:
Perhaps, before we begin, you would like to make sure this is the tour you want!
We keep the broad definition here, that mathematics includes all the related areas which touch on quantitative, geometric, and logical themes. This includes Statistics, Computer Science, Logic, Applied Mathematics, and other fields which are frequently considered distinct from mathematics, as well as fields which study the study of mathematics (!) -- History of Mathematics, Mathematics Education, and so on. We draw the line only at experimental sciences, philosophy, and computer applications. Personal perspectives vary widely, of course.
A fairly standard definition is the one in the Columbia Encyclopedia (5th ed.): "Mathematics: deductive study of numbers, geometry, and various abstract constructs, or structures. The latter often arise from analytical models in the empirical sciences, but may emerge from purely mathematical considerations."
Some definitions of mathematics heard from others:
Contrary to common perception, mathematics does not consist of "crunching numbers" or "solving equations". As we shall see there are branches of mathematics concerned with setting up equations, or analyzing their solutions, and there are parts of mathematics devoted to creating methods for doing computations. But there are also parts of mathematics which have nothing at all to do with numbers or equations.
We'll give one viewpoint of what's in modern mathematics as we start the tour.
For further reading (and other opinions), see