# Mathematics

The Merriam-Webster Dictionary defines **Mathematics** as * the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations*.^{[1]}
Mathematician Michael Resnik argues that mathematics is a *science of patterns.*^{[2]}
Such discussions include the recognition of mathematics alternately as a formal language or as a *world* within which discoveries are possible.

### Theoretical or "pure" mathematics

Mathematics encompasses a tremendously broad collection of activities, everything from formal logic to numerical computational methods. The issues of mathematics are primarily aimed at the expansion of the topics of mathematics, without necessarily being directed to specific applications. Mathematicians are interested in posing well formed statements called "propositions." The goal of the mathematician is the establishment of the truth of the proposition via "mathematical proof" or the recognition of its falsehood through a "counter example". The precision of mathematical language is such that there can be no contradictory statements, meaning that a single proof or a single counter example is sufficient for establishing the truth or falsity of a mathematical proposition.

New mathematics is created continuously through the increasingly precise investigations of patterns and structures within mathematics. New mathematics also comes from the needs of physical scientists, who have proposed mathematical objects, which were later integrated into the system of mathematics.

### Applied Mathematics

The precision of mathematics has been employed in the diverse technical fields, including finance, land surveying, business, commerce, navigation, engineering, agriculture, and the physical sciences for all of human
history. This began with the invention (or discovery) of the concept of the "natural" or counting numbers, which led to arithmetic. The need for precision in spatial reckoning in surveying and
architecture led to *geometry* (earth measure). Initially, geometry and arithmetic were considered to be totally separate, but with the invention (or discovery) of "irrational" numbers, these two disciplines were unified. The issue of posing problems for solution, led to algebra. Polynomial equations posed in algebra lead to the extension of the number system to include "complex numbers."
The issues of time and motion from the physical sciences led to differential and integral calculus. Imprecision
in the classical application of calculus led to the most modern type of mathematics, which is called *analysis."*

In the physical sciences "ordinary" and "partial differential equations" are found to be useful representations of physical processes. These include the behavior of potential fields, the flow of heat, or the propagation of waves. From attempts to solve partial differential equations, as well as integral equations, have sprung such diverse topics as "asymptotic series," transform methods, operator theory, and numerical analysis.

### Mathematical topics of interest to geophysicists

Geophysicists make routine use of the mathematical topics seen in mathematical physics. Geophysicists also make use of numerical methods for modelling physical processes, and use sophisticated
mathematics---translated into software algorithms---to process geophysical data. Much of geophysics may be thought of as the process of *solving for the Earth* given measured data and governing equations of physical processes.

The following topics are of interest to geoscientists

- Calculus
- Complex Analysis
- Ordinary Differential Equations
- Partial Differential Equations
- Numerical Analysis