Mathematics

The amount of mathematical activity has usually increased steadily or even exponentially, and the growth from the mid-twentieth century has been particularly great. For example, the German reviewing journal Zentralblatt Math published at the beginning of the twenty-first century a six-hundred-page quarto volume every two weeks, using a classification of mathematics into sixty-three numbered sections. To suggest the rate of increase, the other reviewing journal, the U.S. periodical Mathematical Reviews, published 3,800 octavo pages in 1980, 7,500 pages a decade later, and 9,800 pages in 2003. It would be impossible to summarize this mountain of work, even up to 1970; instead, some main points are noted relating to the previous sections and to the companion articles on algebras and on logic.

Not only has the amount increased; the variety of theories has also greatly expanded. All the topics and branches mentioned above continue to develop (and also many more that were not noted), and new topics emerge and fresh applications are found. For example, beginning with the 1940s mathematics became widely utilized in the life sciences and medicine and has expanded greatly in economics and other social sciences relative to previous practice.

Much of that work lies in statistics, which after its very slow arrival has developed a huge community of practitioners in its own right. Often it functions rather separately from mathematics, with its own departments in universities.

Another enormous change has been the advent of computing, again particularly since World War II and indeed much stimulated by war work as on cryptography and the calculation of parameters in large technological artifacts. Mathematics plays a role both in the design, function, and programming of computers themselves and in the formulation of many mathematical theories. An important case is in numerical mathematics, where approximations are required and efficient algorithms sought to effect them. This kind of mathematics has been practiced continuously from ancient times, especially in connection with all sorts of applications. Quite often algorithms were found to be too slow or mathematically cumbersome to be practicable; but now computer power makes many of them feasible in "number crunching" (to quote a popular oversimplification of such techniques).

A feature of many mathematical theories is linearity, in which equations or expressions of the form
(A) ax by cz … and so on finitely or even infinitely
make sense, in a very wide range of interpretations of the letters, not necessarily within an algebra itself (for example, Fourier series shows it). But a dilemma arises for many applications, for the world is not a linear place, and in recent decades nonlinear theories have gained higher status, partly again helped by computing. The much-publicized theory of fractals falls into this category.

From the Greeks onward, mathematicians have often been fascinated by major unsolved problems and by the means of solving them. In the late 1970s a proof was produced of the four-color theorem, stating that any map drawn upon a surface can be colored with four colors such that bounding regions do not share the same color. The proof was controversial, for a computer was used to check thousands of special cases, a task too large for people. Another example is "Fermat's last theorem," that the sum of the nth powers of two positive integers is never equal to the nth power of another integer when n 2. The name is a misnomer, in that Fermat only claimed a proof but did not reveal it; the modern version (1994) uses modern techniques far beyond his ken.

This article has focused upon the main world cultures, but every society has produced mathematics. The "fringe" developments are studied using approaches collectively known as ethnomathematics. While the cultures involved developed versions of arithmetic and geometry and also some other branches, several of them also followed their own concerns; some examples, among many, are intricate African drawings made in one unbroken line, Celtic knitting patterns, and sophisticated rows of knotted strings called quipus used in Mexico to maintain accounts.

A thread running from antiquity in all cultures, fringe or central, is recreational mathematics. Unfortunately, the variety is far too great even for summary here. Often it consists of exercises, perhaps posed for educational use, or perhaps just for fun; an early collection is attributed to Alcuin in the ninth century, for use in the Carolingian Empire. Solutions sometimes involve intuitive probability, or combinatorics to work out all options; with games such as chess and bridge, however, the analysis is much more sophisticated. Several puzzles appear in slightly variant forms in different cultures, suggesting transmission. Some are puzzles in logic or reasoning rather than mathematics as such, and it is striking that for some games the notion of decidability was recognized (that is, is there a strategy that guarantees victory?) long before it was studied metamathematically in the foundations of mainstream mathematics.

Lastly, since the early 1970s interest in the history of mathematics has increased considerably. There are now several journals in the field along with a variety of books and editions, collectively covering all main cultures and periods. One main motive for people to take up historical research was their dislike of the normal unmotivated way in which mathematics was (and is) taught and learned; thus, the links between history and mathematics education are strong. For, despite many appearances to the contrary, mathematics is a human activity.