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A Gentle Introduction to the Mathematics Subject Classification Scheme

This is intended to be a guide to the Mathematics Subject Classification (MSC) scheme generally used to classify newly-released mathematics resources. This page is intended for a person with approximately the training of an undergraduate mathematics student; links lead to pages assuming greater familiarity with the sub-disciplines.

The MSC does not include classifications for elementary material. Since there are some materials at this site which border on the elementary (e.g. plane geometry and elementary calculus), I have made the best fit I could, but this implies a slight extension of the MSC system. There is a short page of topics which do not fit neatly into the MSC, including a couple of topics (rings; spheres) on which I have written detailed summary articles. (Also included there are links to a few subjects which do fall within a MSC heading but whose position there may be hard for a newcomer to find.)

The major divisions of the MSC hardly provide an equal division of the current mathematics spectrum. Of course what really would be an "equal division" is open to interpretation. The welcome page for this collection shows an image of the areas of mathematics which shows the relative numbers of recent papers in each area (arranged so as to illustrate the affinities among related areas).

[The MSC is updated at the turn of a decade; the changes for the year 2000 are mentioned only briefly below. The total will be 63 major headings, not 61.]

Warning: this is a single-author document and thus reflects the opinions and deficiencies of that author [Dave Rusin]. (I'm particularly unsatisfied with the descriptions of applications to the physical sciences and other allied disciplines.) Helpful suggestions are always solicited.

What is mathematics, anyway?

Any attempt to distinguish the parts of mathematics must begin with a decision about what constitutes mathematics in the first place! I have tried to 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. We draw the line only at experimental sciences, philosophy, and computer applications. Personal perspectives vary widely, of course!

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 and equations.

How many parts of mathematics -- Two? Seven? Sixty-one?

One way to divide the mathematics literature is to decide which books and articles are designed to reveal the structure of mathematics itself, and which are intended to apply mathematics to closely allied areas.

The first group divides roughly into just a few broad overlapping areas:

Of course, the division of the subject areas into these broad headings is a little fuzzy: combinatorics is only weakly associated to the rest of "algebra"; algebraic groups are arguably a part of analysis or topology instead of algebra, differential geometry is in practice closer to analysis than geometry, and so on.

The second broad part of the mathematics literature includes those areas which could be considered either independent disciplines or central parts of mathematics, as well as those areas which clearly use mathematics but are interested in non-mathematical ideas too. It is important to note that the MSC, as well as the collection of files at this site, covers only the mathematical aspects of these subjects; we provide only cursory links to observational and experimental data, mathematically routine applications, computer paradigms, and so on.

The division between mathematics and its applications is of course vague. In the Math Reviews database, for example, papers in these areas are perhaps over-represented in terms of the number included, yet under-represented in the number carefully reviewed.

Finally note that every branch of mathematics has its own history, collections of important works -- reference, research, biographical, or expository -- and in many cases a suite of important algorithms. The classification allows these topics to be included within each major heading at a secondary level, although there is always some material which cannot otherwise be classified.

The MSC scheme now breaks down these general areas into 61 numbered subject classifications (with widely varying characteristics). We adhere to the polite fiction that these areas are more distinct than the subfields of some of the larger areas; more detail is available in the pages for the various areas.

Logic and set theory

These areas consider the framework in which mathematics itself is carried out. To the extent that this considers the nature of proof and of mathematical reality, it borders on philosophy. But standard mathematical perspectives are used in most topics covered in the MSC.

Algebraic areas

The algebraic areas of mathematics developed from abstracting key observations about our counting, arithmetic, algebraic manipulations, and symmetry. Typically these fields define their objects of study by just a few axioms, then consider examples, structure, and application of these objects. Other fairly algebraic areas include Algebraic topology (55), Information and communication (94), and perhaps Numerical analysis (65).

Geometric Areas

One of the oldest areas of mathematical discovery, geometry has undergone several rebirths over the centuries. At one extreme, geometry includes the very precise study of rigid structures first seen in Euclid's Elements; at the other extreme, general topology focuses on the very fundamental kinships among shapes. (There is also a more subtle notion of "geometry" implied in Algebraic Geometry (14), which is frankly quite algebraic; see above.) Other fairly geometric areas are K-theory (19), Lie groups (22), Several complex variables (32), Calculus of variations (49), Global analysis (58).

Analytic areas

Analysis looks carefully at the results obtained in calculus and related areas. One might characterize algebra and geometry as the search for elegant conclusions from small sets of axioms; in analysis on the other hand the measure of success is more frequently the ability to hone a tool which could be applied throughout science. Thus in particular, most of the calculations are done with the real numbers or complex numbers being implicitly understood.

Analysis includes many of the MSC primary headings, a large portion of the mathematics literature, and much of the most easily applied mathematics. Perhaps, then, it is appropriate to subdivide this topic (although schemes for this subdivision are not very standard):

Calculus and Real Analysis

Complex variables

Differential and integral equations

Functional analysis

Numerical analysis and optimization

We turn now to the parts of mathematics most concerned with developing mathematical tools applicable outside of mathematics.

Probability and Statistics

These areas consider the use of numerical information to quantify observations about events. The tools and development are clearly mathematical; these areas overlap with analysis in particular. On the other hand, the use of the ideas developed here is primarily in non-mathematical areas.

Computer science and Information theory

By design of the MSC, literature concerning specific computations and algorithms is classified with the area of mathematics to which the computations are applied. But mathematics can return the favor and study the process by which computers carry out their information handling.

For mathematical analysis of computation see Numerical Analysis.

Applications to the sciences

Historically, it has been the needs of the physical sciences which have driven the development of many parts of mathematics, particularly analysis. The applications are sometimes difficult to classify mathematically, since tools from several areas of mathematics may be applied. I can comment on these applications not by discussing the nature of their discipline but rather their interaction with mathematics.

I must confess that I have only a cursory acquaintance with most of these fields. -- djr.


Some parts of the mathematics literature seem neither to develop nor apply mathematics; rather, they discuss the nature of mathematics or mathematicians, or recount some very basic mathematical concepts. As a rule, this material is poorly tracked in indices and databases of mathematics, but some of it has a genuine place in the MSC scheme.

More detailed descriptions of these areas of mathematics, including the subdivisions of them, may be obtained through the main index pages. [Only some of them are complete at this time, sorry. -- djr]

You can reach this page through http://www.math-atlas.org/welcome.html
Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org