Abstraction and generalization have often been useful elements of the mathematician's toolkit. They gained in prominence in the nineteenth century and became perhaps the most distinguishing features of mathematics during the period under consideration. For example, Galois's theory of polynomial equations with real-number coefficients was developed by Ernst Steinitz in 1910 over arbitrary fields. The definition of distance between points in Euclidean space was extended by Maurice Fréchet in 1906 to arbitrary sets, giving rise to the important concept of a metric space. This also enabled mathematicians to extend the notion of continuity of a function to such a space. About a decade later Felix Hausdorff generalized these ideas further by noting that distance is not needed to define continuity: "neighborhoods" would do. This was done in the context of his definition of a topological space. The basic notion of function also underwent a grand generalization, embodied in the theory of distributions (or generalized functions) of Sergei L'vovich Sobolev and Laurent Schwartz, introduced in the 1930s and 1940s, respectively. These, of course, were not generalizations for generalizations' sake. They shed light on what is essential, thus extending the range of applicability of concepts and results.
The driving force behind this movement toward high levels of abstraction and generalization was the axiomatic method. While in the past it was used primarily to clarify and codify specific mathematical systems (for example, Euclidean or projective geometry, and the natural numbers), it now became a unifying and abstracting device. Thus, mathematical entities defined by systems of axioms admitting different (nonisomorphic) interpretations (models), such as groups, topological spaces, and Banach algebras, now became fundamental objects of study. The greatest triumph of the axiomatic method has perhaps been in algebra (groups, rings, fields, vector spaces), but the method has also been essential in such diverse areas as analysis (Banach and Hilbert spaces, normed rings), topology (topological, metric, and Hausdorff spaces), set theory (Zermelo-Fraenkel and von Neumann axioms), and even probability (axiomatized by Andrei Nikolaevich Kolmogorov in 1933). Of course, what is abstract to one generation may be concrete to another. In 1945 Samuel Eilenberg and Saunders MacLane introduced category theory, which is an abstraction of an abstraction: instead of studying a group or a topological space, one now studied the class of all groups or of all topological spaces. This idea turned out to be very fruitful in the second half of the twentieth century.
Problems and theory are two opposing pillars of the mathematical enterprise. But they are very much interconnected: attempts to solve problems often give rise to important theories (in fact, this is how theories usually arise), and these, in turn, aid in the solution of problems. Because the supply of mathematical problems is immense, one must choose fruitful ones for investigation (of course, what is fruitful is usually known only in retrospect). The mathematics of the twentieth century began auspiciously with an address by David Hilbert entitled "Mathematical Problems," delivered in 1900 before the second International Congress of Mathematicians in Paris. Although Hilbert favored abstraction and axiomatics and developed important mathematical theories, he was very conscious of the vital role of problems in the subject, noting that "as long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development." He proposed 23 problems for the consideration of twentieth-century mathematicians, and he was confident that they would all be amenable to solution. As he put it: "There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." Indeed, many of the problems were solved during the century, leading to important concepts and theories—though some have resisted solution these past 100 years!
Another pair of opposing pillars of the mathematical enterprise is pure versus applied mathematics—mathematics for its own sake versus mathematics as a tool for understanding the world. There is no doubt that pure mathematics has often turned out to be useful. For example, the two foremost theories in physics of the first half of the twentieth century, general relativity and quantum mechanics, used ideas from abstract mathematics, namely Riemannian geometry and Hilbert-space theory, respectively. But some prominent mathematicians (notably Courant) lamented the fact that mathematics was becoming too abstract (witness the Bourbaki phenomenon, discussed later in this chapter) and thus sterile, not—as in past centuries—maintaining its intimate connection with the physical world, from which it often drew inspiration. As John von Neumann put it: "At a great distance from its empirical source, or after much 'abstract' inbreeding, a mathematical subject is in danger of degeneration." Although there were examples of mathematical developments prompted by nonmathematical considerations, such as the founding of game theory, Minkowskian geometry, linear programming, and aspects of differential geometry, there was undoubtedly some validity to Courant's and others' concerns. The balance, however, was restored with great vigor in the last decades of the twentieth century.
The nineteenth century saw the creation of new fields; these were expanded and extended, and some were axiomatized, in the first half of the twentieth century. Among the numerous new advances were abstract algebra, the Lebesgue integral, ergodic theory, metamathematics, class field theory, Fourier analysis on groups, and the theory of complex functions of several variables. A most important field—topology—was (essentially) newly founded during this period. It joined algebra and analysis as one of three pillars underpinning much of modern mathematics. A central feature of the period was the cross-fertilization of distinct areas, resulting in new departures—new fields, for the most part. Important examples are algebraic topology, algebraic geometry, algebraic logic, topological groups, Banach algebras, functional analysis (analysis and algebra), Lie groups (algebra, analysis, and topology), and differential geometry. Note the predominance of algebra in this list. Some have spoken of the "algebraization of mathematics" as a major trend of this period.
Yet another new development was the emergence of the foundations of mathematics as an important field of mathematical (rather than philosophical) study. The impetus came from the paradoxes of set theory, the use of the Axiom of Choice, and the emergence in the nineteenth century of various geometries and algebras, resulting in the loosening of the close connection between mathematics and the "real world." The questions that were asked had to do with the consistency and completeness of major branches of mathematics defined axiomatically, and, more broadly, with the nature of mathematical objects and the methods of mathematical reasoning. The results were the founding of three "philosophies" of mathematics—logicism, formalism, and intuitionism—and Kurt Gödel's Incompleteness Theorems. The latter showed that large parts of mathematics (including elementary arithmetic and Euclidean geometry) are incomplete, and their consistency can not be established (within the given system). These were most profound results, placing (at least theoretical) limits on the axiomatic method, indispensable in much of modern mathematics. Another great achievement in mathematical logic, soon to be of profound practical impact, was the work of Alan Turing, Emil Post, and Alonzo Church in the 1930s, giving (independently) precise expression to the vague notions of algorithm and computability, thus providing the necessary mathematical underpinnings for the advent of the computer.
Finally, a comment on professionalization in mathematics. The subject grew enormously in the first half of the twentieth century, so much so that its output during that period was (likely) larger than that of all previous centuries combined! The age of the mathematical universalist, such as David Hilbert or Jules Henri Poincaré, who could master substantially the whole subject, came to an end at mid-century. Strong new mathematical centers began to flourish (e.g., in the United States, the former Soviet Union, Japan, China, India, Poland), with the center of gravity of mathematical activity gradually moving away from Western Europe. Many universities were established in various countries, doing both teaching and research in mathematics. The research was channeled into formal institutions: hundreds of mathematical periodicals were founded (there were only a handful in the nineteenth century), three of which were "reviewing" journals, acquainting mathematicians with research done worldwide. Numerous mathematical societies came into being in various countries, including the International Mathematical Union. National meetings were held regularly, and the International Congress of Mathematicians, which began in 1897, took place every four years. At each Congress "Fields Medals" (mathematical counterparts of the Nobel Prizes) were awarded to outstanding mathematicians. Mathematics was thriving!