Facts on File, 2011. - 217 pages.
For millennia, Euclidean geometry, the geometry of the ancient Greeks, set the standard for rigor in mathematics—it was the only branch of mathematics that had been developed axiomatically. Euclidean geometry, most philosophers and mathematicians agreed, was the language of mathematics. However, early in the 19th century, mathematicians developed geometries very different from Euclid’s simply by choosing axioms different from those used by Euclid. These new geometries were inteally consistent in the sense that mathematicians could find no theorems arising within these geometries that could be proved both true and false. Some of these alteative geometries even provided insight into certain aspects of physical space. Mathematicians and philosophers, who had longed believed that Euclid’s geometry embodied mathematical truth, came to see Euclidean geometry as one possible axiomatic system among many. Nevertheless, the discipline of geometry, suitably broadened, retained its place at the center of mathematical thought until the last decades of the 19th century.
Nevertheless, geometry proved to be ill-suited to the demands placed upon it. Analysis, for example, the branch of mathematics that grew out of calculus, had originally been expressed in the
language of geometry, but the attempt to express analytical ideas in geometric language led to logical difficulties. Conceptually, geometry was not rich enough to serve as a basis for the new
mathematics that was developed during the latter half of the 19th century. Consequently, mathematicians began to seek another mathematical language in which to express their insights. They found it in the language of sets.
For millennia, Euclidean geometry, the geometry of the ancient Greeks, set the standard for rigor in mathematics—it was the only branch of mathematics that had been developed axiomatically. Euclidean geometry, most philosophers and mathematicians agreed, was the language of mathematics. However, early in the 19th century, mathematicians developed geometries very different from Euclid’s simply by choosing axioms different from those used by Euclid. These new geometries were inteally consistent in the sense that mathematicians could find no theorems arising within these geometries that could be proved both true and false. Some of these alteative geometries even provided insight into certain aspects of physical space. Mathematicians and philosophers, who had longed believed that Euclid’s geometry embodied mathematical truth, came to see Euclidean geometry as one possible axiomatic system among many. Nevertheless, the discipline of geometry, suitably broadened, retained its place at the center of mathematical thought until the last decades of the 19th century.
Nevertheless, geometry proved to be ill-suited to the demands placed upon it. Analysis, for example, the branch of mathematics that grew out of calculus, had originally been expressed in the
language of geometry, but the attempt to express analytical ideas in geometric language led to logical difficulties. Conceptually, geometry was not rich enough to serve as a basis for the new
mathematics that was developed during the latter half of the 19th century. Consequently, mathematicians began to seek another mathematical language in which to express their insights. They found it in the language of sets.