 # Upper Level Math

Mathematics is a broad field with ancient roots, reaching back over 4000 years through many civilizations.  Yet it remains relevant in modern times, and has recently experienced a growth explosion, as it is estimated that the great majority of mathematics we have today was just discovered in the last century.  Mathematics is a unique field of study, considered to be simultaneously in the sciences, humanities, and arts.

Calculus and linear algebra constitute the foundation for most of modern, upper level mathematics.  Beyond these core courses, the study of mathematics is roughly divided into the overlapping areas of pure math and applied math.

## What is Applied Math?

Applied math refers to mathematics as a tool used in science, that is, to solve “real-world” problems.  Although mathematics is primarily non-empirical and does not require any application to the physical world, yet it remains the language of choice to describe and study patterns observed in God’s creation.  Indeed, as Galileo Galilei said, “Mathematics is the language in which God wrote the universe.”  Although applied mathematics historically referred to applications in physics, all natural and social sciences have become increasingly “mathematized” in the late 20th century, notably since the advent of computing.  Even newer fields such as ecology, epidemiology, pharmacology, psychology, and management science, now make copious use of mathematical and statistical tools.

Central to applied mathematics is the process of mathematical modeling, by which a mathematician translates an observed real-world phenomenon into a set of mathematical equations.  This process requires a deep knowledge of the system being studied, as well as specialized mathematical tools.  Statistics is the primary tool for analyzing and interpreting experimental data collected from a system, and differential equations is the tool of choice to describe the underlying mechanisms that govern an observed behavior in time and space.

Although some mathematical models are studied using traditional analytical methods, the advent of computing has promoted the development of numerical methods, based on algorithms implemented by a computer.  Many industries are progressively becoming more computational, relying on mathematical models to assist or replace traditional methods of generating, collecting and/or analyzing data.  Employers in these industries seek to hire graduates who have strong mathematical and computational skills, and are familiar with the methods of applying these skills to solving real-life problems.

## What is Pure Math?

Although much of mathematics has its roots in humanity’s attempts to solve “real-world” problems, mathematics is much more than just a “toolbox for science”.  Unlike the other sciences, mathematics is not confined to the three-dimensional, visible, physical world.  According to mathematician Keith Devlin, mathematics is the field that “makes the invisible visible”, allowing us to appreciate patterns, derive truth, and appreciate beauty in God’s invisible creation, beyond the reach of empirical science.

Mathematics allows intuitive concepts to be abstracted and generalized, providing a framework for the human mind to progress beyond the world of empirical science by logic and deductive reasoning.  Mathematicians organize their ideas into systems of axioms, definitions, theorems, and proofs, in a process similar to how ideas are organized and analyzed in philosophy and theology.  The study of these axiomatic systems cultivates the mind’s ability to organize and synthesize thoughts, and to solve problems sequentially and creatively.

Mathematical analysis axiomatizes, extends, and generalizes concepts encountered in calculus, such as infinitesimals, infinite sequences and series, limits, differentiation, and integration.  Modern algebra studies groups, rings, ideals, and other abstract patterns arising in solutions of polynomial equations of higher degree.  Geometry and topology study space, dimension, and transformation, extending and abstracting the concepts of Euclidean geometry.  These fields of pure mathematics constitute the foundation of modern-day mathematics, and of the Wheaton College mathematics curriculum.