Text 2. Fields of Mathematics

The major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinity.

Space

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change. Tensor calculus studies symmetry and the behavior of vectors under rotation.

Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas.

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference—with model selection and estimation; the estimated models and consequential predictions should be tested on new data.

Exercises

1. Practice the pronunciation of the following words:

algebra;

analysis;

astronomical;

calculus;

chaos;

commerce;

concern;

consequential;

deterministic;

dimensional;

encompass;

exhibit;

familiar;

geometry;

integers;

internal;

linear;

major;

mechanics;

property;

quantity;

quantum;

ramification;

randomized;

rational;

statisticians;

statistics;

subdivision;

symmetry;

theme;

to dedicate;

to generalize;

to specify;

trigonometry;

uncertainty;

unpredictable;

variable;

verify.

2. Give Russian equivalents to the following words:

a powerful tool to investigate something; central concept; chaos theory; consequential predictions; dynamic systems; empirical mathematics; general relativity; higher-dimensional geometry; logic; major disciplines; originate with something; polynomial equations; quantity; quantum mechanics; random sampling; set theory; space; study of uncertainty; to be contained with something; to do calculations; to exhibit deterministic behavior; to generalize ideas.