Mathematics

A new leader emerged: the German David Hilbert (1862–1943). Work on abstract algebras and the foundations of geometry led him to emphasize the importance of axiomatizing mathematical theories (including the axioms of Euclidean geometry that Euclid had not noticed) and to study their foundations metamathematically. But his mathematical knowledge was vast enough for him to propose twenty-three problems for the new century; while a personal choice, it exercised considerable influence upon the community. He presented it at the International Congress of Mathematicians, held in Paris in 1900 as the second of a series that manifested the growing sense of international collaboration in mathematics that still continues.

One of Hilbert's problems concerned the foundations of physics, which he was to study intensively. In physics Albert Einstein (1879–1955) proposed his special theory of relativity in 1905 and a general theory ten years later; according to both, the ether was not needed. Mathematically, the general theory both deployed and advanced tensor calculus, which had developed partly out of Riemann's interpretation of geometry.

Another main topic in physics was quantum mechanics, which drew upon partial differential equations and vector and matrix theory. One of its controversies concerned Werner Heisenberg's principle of the uncertainty of observation: should it be interpreted statistically or not? The occurrence of this debate, which started in the mid-1920s, was helped by the increasing presence of mathematical statistics. Although probability must have had an early origin in mathematical thinking, both it and mathematical statistics had developed very slowly in the nineteenth century—in strange contrast to the mania for collecting data of all kinds. Laplace and Gauss had made important contributions in the 1810s, for example, over the method of least-squares regression, and Pafnuty Chebyshev (1821–1894) was significant from the 1860s in Saint Petersburg (thus raising the status of Russian mathematics). But only from around 1900 did theorizing in statistics develop strongly, and the main figure was Karl Pearson (1857–1936) at University College, London, and his students and followers. Largely to them we owe the definition and theory of basic notions such as standard deviation and correlation coefficient, basic theorems concerning sampling and ranking, and tests of significance.

Elsewhere, Cantor's set theory and abstract algebras were applied to many parts of mathematics and other sciences in the new century. A major beneficiary was topology, the mathematics of location and place. A few cases had emerged in the nineteenth century, such as the "Möbius strip" with only one side and one edge, Riemann's fantastical surfaces, and above all a remarkable classification of deformable manifolds by Poincaré; most of the main developments, however, date from the 1920s. General theories were developed of covering, connecting, orientating, and deforming manifolds and surfaces, along with many other topics. A new theory of dimensions was also proposed because Cantor had refuted the traditional understanding by mapping one-one all the points in a square onto all the points on any of its sides. German mathematicians were prominent; so were Americans in a country that had risen rapidly in mathematical importance from the 1890s.