01: History and biography
Formal studies in the history of mathematics developed much more slowly than studies in mathematics itself. There is a particular difficulty in that those who are well trained in historical analysis are typically insufficiently versed in mathematics to be able to appreciate the subject at hand. This has left mathematicians to write their history, by and large, although they are usually untrained for that task.
Nonetheless, the stories of the people involved in developing mathematics are often inspiring or informative. A sense of what motivated mathematicians to pursue their key ideas helps put the importance of those results into a better context.
Thus the study of the history of mathematics and its proponents includes several well-developed parts. The development of comparatively simple mathematics (through the calculus, for example) is now well documented, principally as part of the study of the development of scientific ideas in distinct human cultures through the 18th century. The development of mathematics in the last couple of centuries is instead more frequently studied thematically -- that is, the worldwide development of algebra, or statistics, say -- or through the lives of individual mathematicians.
The broad outlines of the development of mathematics are well known. Practical problems (e.g. the need for resurveying fields after the annual flooding of the Nile) led to techniques for solving elementary numerical or geometric problems in a number of ancient cultures, notably in Egypt, China, India, and Central America. The tradition of abstract mathematics -- with formal definitions and careful proofs -- is usually traced to Greek cultures of the Pythagorean era (ca 500 BC), with the Elements (of Geometry) of Euclid being the longest-used textbook ever in mathematics. Comparatively few works were produced in the Mediterranean area after the late Roman period. Arab scholars continued some of the themes and developed in particular a much better algebraic framework through the Renaissance. Thereafter, European interest in pure and applied mathematics brought it to center stage, a position it maintained until the 20th century. By 1700 or so, the now-familiar patterns of mathematical work were established: scholarly work conducted largely in universities or upon commission of a governing power; publication in books and journals; an emphasis on proofs; the development of useful notation; and so on.
By this period, the role of the individual mathematician became dominant; that is, new mathematics is usually attributed to a person rather than a culture. The roster of key luminaries is of course subjective, but among the most frequently studied mathematicians are Descartes, Fermat, Newton, Leibniz (17th century), Euler, Lagrange (18th century), Gauss, Cauchy, Riemann, Poincaré (19th century), Hilbert, Gödel, von Neumann, and our contemporaries of the 20th century.
No attempt will be made to summarize the development of abstract and applied mathematics during this last few centuries; see the resources below. Some historical comments are included in the index pages for the separate disciplines.
Common publications which provide information about mathematicians include mathematical obituaries, bibliographies, collected works, and festschriften.
A unique feature of late 20th century mathematics is the realization that it constitutes a subculture of its own. There have been a number of studies of the mathematics community both internally and by nonmathematicians. This includes both quantitative studies and qualitative analyses.
For the history of a specific discipline, see the corresponding web page. Historical or biographical material is given classification XX-03 there.
This image slightly hand-edited for clarity.
There is only one division (01A) but it is subdivided: 01A: History of mathematics and mathematicians.
The amount of published material in Section 01 is fairly small but subsection 01A70 (biographies) is the single most used 5-digit code in the MSC.
Browse all (old) classifications for this area at the AMS.
Among comprehensive treatments of the History of Mathematics we mention
More introductory is "Mathematics and its history", John Stillwell (Springer-Verlag, New York-Berlin, 1989, ISBN 0-387-96981-0).
A unique overview of modern mathematics from a unique, um, participant in it is by Bourbaki, Nicolas, "Elements of the history of mathematics", Springer-Verlag, Berlin, 1994. 301 pp. ISBN 3-540-19376-6[English] or 2-225-80320-X[French], MR86h:01003
Two recommended works by Kenneth O. May regarding the conduct of research in this area:
A unique historical resource is D. J. Struik's "A source book in mathematics, 1200--1800" (Princeton University Press, Princeton, N.J., 1986, ISBN 0-691-02397-2).
A comprehensive source for biographical information on 1000+ mathematicians is the "Biographical dictionary of mathematicians", in 4 volumes (Charles Scribner's Sons, New York, 1991, ISBN 0-684-19288-8).
Perhaps more comprehensive for more recent mathematicians is Poggendorff, J. C.: Biographisch-literarisches Handwörterbuch der exakten Naturwissenschaften. Akademie-Verlag, Berlin. This is an ongoing publication, now up to 7 volumes (well over 5000 total pages); the last volume began with A in 1965 and just reached Z thirty years later.
There is a classic collection of biographies which perhaps should be enjoyed as mythos, rather than accepted as fact: Eric Temple Bell, Men of Mathematics (Simon and Schuster).
There are electronic mailing lists in the History of Mathematics maintained by the Mathematical Association of America and the Historia Matematica list (a professional, closed forum).
A starting point for biographical information on living mathematicians is perhaps the Combined Membership List of the AMS, MAA, SIAM, and other organizations of North American mathematicians, together with the cognate directories for European and Russian mathematicians.
Just one entry for "sociology of mathematics": the Erdös Number Project Home Page.