r Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange. Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula. Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to a geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. In the 20th century, algebra has also been applied to a general form of geometry known as topology. Another subject that was transformed in the 19th century, notably by Laws of Thought (1854), by the English mathematician George Boole and by Cantor's theory of sets, was the foundations of mathematics (Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantor's theory. One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set ( Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising. They left open the question, however, of whether other paradoxes might arise in these restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, proved in 1931 by the American logician Kurt Gdel, that in any axiom system complicated enough to be interesting to most mathematicians, it is possible to frame propositions whose truth cannot be decided within the system. Current Mathematics At the International Conference of Mathematicians held ...
