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	The Divisions of Mathematics

In order to find one's way around the collection of mathematical ideas, it is useful to organize them and classify them in some way into parts.

Among the ways to divide the field of mathematics is by field of application. There are many books and courses in schools labeled "Engineering mathematics", "Financial mathematics", "Mathematics for social scientists", and so on. While it is perhaps easier for the reader to have the material pre-filtered according to application, this hides the fact that the underlying mathematics is really quite similar --- radioactive decay is essentially the same as inflationary depreciation of investments, for example. At this site we emphasize the mathematics itself rather than the intended application, so this method of dividing material is inappropriate for us.

Another way to divide the portions of mathematics is by level of complexity. Elementary topics include arithmetic and measurement; intermediate topics include simple algebra and plane geometry. From there we may pass to somewhat more complex topics built upon these: trigonometry, "advanced" algebra, analytic geometry, and calculus.

This website is limited to topics more advanced than these; little mention will be made of topics which are typically not considered (except in their most elementary aspects) until a student has progressed through some University studies. Our intended audience at the site is the person who has already studied some mathematics courses beyond these at the university level, although in this tour we try to be more inclusive.

That said, we proceed to divide mathematics along thematic lines.

How many parts of mathematics -- Two? Eight? Sixty-three?

The image at right shows a "map" of the subfields of mathematics. These are the major classification groupings used at this site and by most research mathematics projects. The sizes and positions of the "bubbles" are computed to reflect the sizes and relatedness of the various disciplines. On our tour, we'll highlight some of the main groupings of these areas (the different color groups).

One first step in dividing the mathematics literature is to decide which books and articles intend to reveal the structure of mathematics itself, and those which intend to apply mathematics to closely allied areas. This division between mathematics and its applications is of course vague. Indeed, we'll see that the two groups cut across each other on the MathMap.

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

• Foundations considers questions in logic or set theory -- the very language of mathematics.
• Algebra is principally concerned with symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations; one might think of this as the outgrowth of arithmetic and algebra classes in primary and secondary school.
• Geometry is concerned with shapes and sets, and the properties of them which are preserved under various kinds of motions. Naturally this is related to elementary geometry and analytic geometry.
• Analysis studies functions, the real number line, and the ideas of continuity and limit; this is perhaps the natural successor to courses in graphing, trigonometry, and calculus. (This is a very large area; we subdivide it later into five areas which we may label Calculus and Real Analysis, Complex Analysis, Differential Equations, Theory of Functions, and Numerical Analysis and Optimization.)

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 involve non-mathematical ideas too. It is important to note that 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.

• Probability and Statistics, for example, has a dual nature -- mathematical and experimental. This classification scheme focuses on the former -- the study of the validity of the measurements one might make.
• Computational sciences have obviously flourished in the last half-century, and consider algorithms and information handling. Here we are concerned with what might be computed, not with compilers, architectures, and so on.
• Significant mathematics must be developed to formulate ideas in the physical sciences, engineering, and other branches of science. Again it is the theoretical underpinnings which concern us here rather than the experiment or tangible construction.

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 MSC classification allows these topics to be included within each major heading at a secondary level. However, these themes are sometimes best woven together into areas of study which are not so much research into mathematics as research into the enterprise of mathematics -- "epi-mathematics", perhaps.

The Mathematics Subject Classification (MSC) scheme breaks down these general areas into 63 numbered subject classifications with widely varying characteristics. (This is the classification system used by the research mathematical societies.) 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.

Continue the tour by clicking on any of the major branches of mathematics described above. You might want to begin with a tour of foundations.

But is this division "real"?

In a word, "no". It's false to assume that mathematics consists of discrete subfields, it's false to assume that there is an objective way to gather those subfields into main divisions, and it's false to assume that there is an accurate two-dimensional positioning of the parts. For example, a division into "Pure" and "Applied" Mathematics is traditional, but the boundaries are unclear and cross-fertilization is common. Within the first part it is also traditional to identify Algebra, Geometry, and Analysis as the three largest areas, but again this division is somewhat artificial as we have noted.

Yet the picture we have described above is consistent with the images painted in other sources. Some other systems for classifying mathematics are presented for browsing in the set of subject headings used at this site. Each system is different and yet it is generally possible to match parts of one classification scheme with parts of another.

The National Science Foundation, for example, organizes its mathematics programs into

• Algebra and Number Theory
• Topology and Foundations
• Geometric Analysis
• Analysis
• Statistics and Probability
• Computational Mathematics
• Applied Mathematics

To get another perspective, consider the division into branches of mathematics taken from The Concise Columbia Electronic Encyclopedia, Third Edition or Wikipedia.

For another comparison, here are lists of divisions of comparable length used by the American Mathematical Society to track the employment of new Ph.D.'s. The "Field of Thesis" record uses the following divisions in 1998 and 1999 (shown here with number of new Ph.D.'s in the US in each field that year); here the fields are sorted to mimic the divisions used above:

1998 Logic/Discrete Math/Combinatorics/Computer Science (109) Algebra and Number Theory (160) Geometry and Topology (143) Differential, Integral, and Difference Equations (98) Real or Complex Analysis (39) Functional Analysis (41) Harmonic Analysis and Topological Groups (44) Numerical Analysis, Approximations (61) Probability and Statistics (291) Applied Mathematics (122) Linear, Nonlinear Optimization and Control (27) Other/Unknown (23)

1999 Logic/Discrete Math/Combinatorics/Computer Science (90) Algebra and Number Theory (169) Geometry and Topology (132) Differential, Integral, and Difference Equations (98) Real, Complex, Functional, and Harmonic Analysis (105) Numerical Analysis, Approximations (79) Probability (51) Statistics (269) Applied Mathematics (100) Linear, Nonlinear Optimization and Control (23) Mathematics Education (14) Other/Unknown (3)

So we must accept that different tour guides would partition the landscape in different ways; yet it seems clear that there are some commonly-recognized divisions within the discipline.