ABSTRACT
A formal conception of geometry entailed a radical shift in the conception of geometry as a science. Geometry is the science of space, or more narrowly, it is the science of points, straight lines, angles, circles, planes, and so on. Euclidean geometry is the geometry that most of people have studied in high school. Klein provided a different model of hyperbolic geometry, in which the straight lines of hyperbolic geometry are considered as the open-ended line segments within a disk. The consistency of hyperbolic geometry could be proved with new rigor, and all of mathematics could be seen as one large axiomatic system. This new framework was the product of a new interdisciplinary coalition between philosophers and mathematicians, and the process of developing this interdisciplinary coalition is an interesting area to study. The standard accounts of the development of geometry connect geometry to logic and to foundational issues in the philosophy of mathematics.
