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# 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 63 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:

• Search for a topic by keywords.
• Browse a list of subject headings.
• Click on territory in a visual "MathMap".
• Help for using this site.

Perhaps, before we begin, you would like to make sure this is the tour you want!

## What is mathematics, anyway?

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:

• That which mathematicians do.
• The study of well-defined things.
• The study of statements of the form "P implies Q".
• The science one could go on practicing should one wake up one morning to find that the world has ceased to exist. (attrib. to Bertrand Russell)
• The science of patterns (Keith Devlin)
• "Mathematics, at the beginning, is sometimes described as the science of Number and Space - better, of Number, Time, Space and Motion." -- Saunders MacLane, in Mathematics: Form and Function

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

• Courant, Richard; Robbins, Herbert: "What Is Mathematics? An elementary approach to ideas and methods." Oxford University Press, New York, 1941. 521 pp. MR3,144b Reprinted 1979 ISBN 0-19-502517-2
• Davis, Philip J.; Hersh, Reuben: "The mathematical experience" Birkhäuser, Boston, Mass., 1980. 440 pp. ISBN 3-7643-3018-X MR82i:00020 (Hersh also has a new book, "What is mathematics, really?")
• Stewart, Ian: "From here to infinity" The Clarendon Press, Oxford University Press, New York, 1996. 310 pp. ISBN 0-19-283202-6 MR97c:00004
• Snapper, Ernst: "What is mathematics?" Amer. Math. Monthly 86 (1979), no. 7, 551--557. MR80k:03013
• Grant, Hardy: "What is modern about 'modern' mathematics?", Math. Intelligencer 17 (1995), no. 3, 62--66. MR96h:00008
• Dieudonné, Jean Alexandre: "A panorama of pure mathematics. As seen by N. Bourbaki. (Translated from the French by I.G. Macdonald.) Pure and Applied Mathematics, 97. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. 289 pp. ISBN: 0-12-215560-2 MR83e:00003 (French original: Gauthier-Villars, Paris, 1977. 302 pp. ISBN: 2-04-010012-1)
• Mac Lane, Sauders: "Mathematics, form and function", Springer-Verlag, New York-Berlin, 1986. 476 pp. ISBN 0-387-96217-4 MR87g:00041
• Gårding, Lars: "Encounter with mathematics", Springer-Verlag, New York-Heidelberg, 1977. 270 pp. MR57#2796
• Ash, Robert B., "A primer of abstract mathematics" Classroom Resource Materials Series. Mathematical Association of America, Washington, DC, 1998. 181 pp. ISBN 0-88385-708-1
• A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrentcev, eds: "Mathematics: Its Content, Methods and Meaning", 3 vols. (Russian; English translation by S. H. Gould) American Mathematical Society 1962-3; repr. MIT Press 1969; Dover reprint