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Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era. While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1. In his doctoral dissertation he gave the first satisfactory proof of the fundamental theorem of algebra. Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying.

Of more importance for algebra itself than Gauss’s proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems. A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many, but not all, of the properties of the real numbers. Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann G?nther 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 G?del, 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 in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at G?ttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilbert’s address at G?ttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the ?Hilbert problems? has been solved, mathematicians all over the world await the details of the story with impatience.

Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (Computer). Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbage’s imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics,

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