The Specialization of Mathematics and the Rise of Formalism
Overview
Mathematics is the study of the relationships among, and operations performed on, both tangible and abstract quantities. In its ancient origins, mathematics was concerned with magnitudes, geometries, and other practical and measurable phenomena. During the nineteenth century, mathematics, and an increasing number ofmathematicians, became enticed with relationships based on pure reason and upon the abstract ideas and deductions properly drawn from those relationships. In addition to advancing mathematical methods related to applications useful to science, engineering, or economics (hence the term "applied mathematics"), the rise of the formalization of symbolic logic and abstract reasoning during the nineteenth century allowed mathematicians to develop the definitions, complex relations, and theorems of pure mathematics. Within both pure and applied mathematics, nineteenth-century mathematicians took on increasingly specialized roles corresponding to the rapid compartmentalization and specialization of mathematics in general.
Background
Well into the nineteenth century mathematicians continued to scramble to invent and refine analytical methods that would be of use in solving the seemingly endless list of questions and problems being raised by the emerging European industrial revolution's demand for increased experimentation in physics, astronomy, and engineering. By the middle of the century, however, attention began to shift toward the operations of mathematical logic, and, as a consequence, there was an increased emphasis on the relationships and rules for evaluating axioms and postulates.
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