Mathematics comprises both a set of pragmatic procedures (applied mathematics) and a set of rigorous deductive systems (pure mathematics). Pure mathematics derives its ultimate inspiration from practical problems, but its domain is purely mental. Beginning with axioms, which are rules for manipulating undefined terms, mathematicians proceed through the purely mental process of deductive inference to derive new mental constructs called theorems. While this abstractness seems to make pure mathematics utterly remote from experience, it does facilitate the application of these constructs to new areas. Also, through abstraction, pure mathematics strives to achieve absolute certainty. Alas, such certainty has proved elusive. Gödel’s Theorem (1931) established that no formal system rich enough to include the natural numbers could be both consistent and complete. Either deductive reasoning cannot yield all that is true in mathematics, or it will yield contradictory results. Since pure mathematics is a mental construct, such a conclusion raises fascinating questions about the nature of the human mind.

Applied mathematics has been wildly successful in describing the natural world and in proving its usefulness in nearly every area of human endeavor. Since the time of Sir Isaac Newton (1642–1727), the physical sciences (especially physics) have sought mathematical description rather than ultimate causes. The effectiveness of a mathematical procedure as applied to science is typically evaluated not by the rigor of its justification, but by its ability to make predictions about the outcomes of experiments.

While the mathematization of the underlying laws of the life and social sciences has met with spotty success, there can be no doubt that the branch of mathematics called “statistics” forms the basis of nearly all research. Any study that is based on more than anecdotal evidence relies on statistical design and analysis. It follows that a rudimentary understanding of statistics is essential for anyone making decisions based on the results of research, including, for example, decisions about the efficacy and safety of new medications.

Computers have naturally had an enormous influence on mathematics. To begin with, by removing the computational burden from humans, computers affect the way we think about math education. Americans have always had difficulty deciding whether the goal of math education should be the mastery of procedures or the understanding of abstract concepts. The ubiquity of computers and calculators at the very least diminishes the need for arithmetical proficiency, while increasing the need to understand algorithms and numerical accuracy.

Computers have also opened the doors to fascinating new areas of mathematics, including chaos theory The self-similarity of some fractal images (based on chaotic systems) suggests links between the visual arts and designs seen in nature. Chaos theory also links the qualitative and quantitative descriptions of phenomena with wide application to domains previously beyond mathematical analysis.