Rapid growth in the nature and applications of mathematics means that the Newtonian core – calculus, analysis, and differential equations – is now just one part of a more diverse mathematical landscape. Yet most scientists have explored only this original territory, because that is all that was included in their curriculum in high school, college, and graduate school. With the exception of statistics, an old science widely used across all disciplines that has become largely mathematical during the 20th century, the narrow Newtonian legacy of analysis is the principal connection between practicing scientists and broad mathematical foundations of their disciplines.
The dramatic changes in the mathematical sciences of the last quarter century are largely invisible to those outside the small community of research mathematicians. Today’s mathematical sciences, like yesterday’s Gaul, can be divided into three parts of roughly comparable size: statistical science, core mathematics, and applied mathematics. Each of these three major areas is led (in the United States) by a few thousand active researches and receives approximately $50 million in federal research support annually. Although the boundaries between these parts overlap considerably, each province has an identifiable character paradigm established by Newton: data, deduction, and observation.
Core mathematics investigates properties of number and space, ideas rooted in antiquity. Its tools are abstraction and deduction; its edifices include functions, equations, operators, and infinite-dimensional space. Within core mathematics are found the traditional subjects of number theory, algebra, geometry, analysis, and topology. After a half-century of explosive specialized growth, core mathematics is experiencing a renaissance of renewed integrity based on the unexpected but welcome discovery of deep links among its various components.
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