What is Mathematics?
As of today there is no clear cut deﬁnition of mathematics. According to Aristotle, mathematics is the science of quantity. This deﬁnition endured until the18th century, when the study of mathematics became rigorous and addressed abstract topics such as group theory and projective geometry, mathematicians and philosophers began to emphasize the deductive character of mathematics. However, three leading deﬁnitions of mathematics exist. They are called logicist, intuitionist and formalist.
The logicists deﬁne mathematics as the science that draws necessary conclusions: The intuitionists’ deﬁnition is, “mathematics is the mental activity which consists in carrying out constructs one after the other.”
The formalists identify mathematics with its symbols and the rules for operating them. Gauss called mathematics the queen of the sciences. Some mathematicians believe that calling mathematics a science downplays the importance of its aesthetic side. Others believe that to ignore its connection to the sciences is to discard the interface between mathematics and its applications in science and technology. Many problems in mathematics arise from problems found in the sciences, business, astronomy, architecture, engineering, medicine etc.
Some areas of mathematics are relevant only in those areas that inspired them and it is often applied to solve problems in those areas. In many other cases, mathematics inspired by one area has been found to be useful in many other areas. Before the invention of mathematical notations, mathematics was written out in words. Modern notations make mathematics easier for the professional mathematicians but difﬁcult for the beginner because mathematical ideas are more abstract and more encrypted than the natural language where people can equate a word with the physical object that correspond to it. Mathematical symbols are abstract and lack any physical analog.
The development of mathematics is attributable to the early Babylonians, Egyptians, Greeks, Romans, Chinese, Indians, Persians and Medieval Europe. The term mathematics was coined from the ancient Greeks word mathema. The ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering and even arts and crafts. During the period of the Renaissance in Italy in the 15th Century, new mathematical developments, interacting with new scientiﬁc discoveries were made at a rapid pace. This has continued till the present day. The 17th Century saw an increase of mathematical and scientiﬁc ideas across Europe.
Johannes Kepler studied planetary motion. He formulated mathematical laws of planetary motion which was made possible by the work of Rene Descartes in Cartesian coordinates. Kepler’s laws brought together the concepts now known as Calculus. Gottfried Leibniz who is regarded as one of the most important mathematicians of the 17th Century developed calculus and much of the calculus notation in use today. The most inﬂuential mathematician of the18th Century was arguably Leonard Euler. He founded the graph theory and named the square root of minus one, (−1) with the symbol “i". He popularised the Greek letter π for the ratio of a circle’s circumference to its diameter. He made contributions in Topology, Graph Theory, Calculus, Combinatorics and Complex Analysis. Another important mathematician in this period was Joseph Lagrange who worked on Number Theory, Differential Calculus, Algebra and Calculus of Variations.
The 19th Century saw a great deal of abstraction in mathematics. Bernard Riemann developed the Riemann geometry which generalises the three types of geometry. He also deﬁned the concept of manifolds which generalises the ideas of curves and surfaces. Other developments in mathematics were made by the following mathematicians.
• William Hamilton developed noncommunative algebra.
• George Boole devised the Boolean algebra in which the only numbers were 0 and 1.
• Abel and Galoi laid the ground work for the development of group theory.
• George Cantor established the foundation for set theory.
A number of national mathematical societies were founded during this period. For example, the London mathematical society in 1865, the societe mathematique de France in 1872, the circolo matematico di Palermo in1884,the Edinburg mathematical society in 1888.
Mathematics became a major profession in the 20th Century with universities turning out thousands of Ph.Ds every year. A list of 23 unsolved problems in mathematics was set out by David Hilbert in 1900. These problems span many areas of mathematics and to this day only 10 have been solved. The development of computers allowed industry to deal with large amounts of data to facilitate mass production. New areas of mathematics were developed to deal with this; for example, computability theory, complex theory, Signal processing, data analysis etc. Computers enabled the handling of mathematical problems that were too time consuming with pen and paper calculations leading to new areas such as numerical analysis and symbolic computation.
In the year 2000 the Clay Mathematics Institute announced the 7 millennium prize problems. In 2003 one of these problems known as the Poincare conjecture was solved by Grigori Perelman (who declined to accept the award of one million dollars as he was critical of the mathematics establishment). With the advent of the computer, mathematics started growing and new applications are rapidly expanding.
Classiﬁcation of Mathematics
Mathematics can roughly be divided into pure and applied mathematics. These divisions are however not mutually exclusive. Mathematicians have always been involved in the debate regarding the difference between pure and applied mathematics. According to G. H Hardy, applied mathematics is ugly and dull. He compared pure mathematics to painting and poetry. He believed that applied mathematics sought to express physical truth in a mathematical framework whereas pure mathematics expressed truths that were independent of the physical world. To him, pure mathematics “is the real mathematics which has a permanent aesthetic value and applied mathematics the dull and elementary parts of mathematics that have practical use.” However, he agreed that just as the matrix theory and group theory (which are pure mathematics) have come to have applications in physics, several other areas of mathematics may have practical applications in the future.
Let us examine some of the sub-ﬁelds of pure mathematics:
(i) Analysis: Analysis deals with properties of functions and concepts such as continuity, differentiation, limits and integration. It provides the foundation for calculus. Analysis is further classiﬁed into:
• Real Analysis: Is concerned with studying the behavior and properties of functions, sequences and sets on the real line which is denoted mathematically by R. Concepts that are examined are properties like Limits, continuity, Derivatives and Integration
•Complex Analysis: Studies functions of complex numbers. Some of the well-known mathematicians associated with complex numbers include Euler, Gauss, Reimann, Cauchy and Weiertrass. Complex analysis has important applications, especially in string theory which is involved in quantum ﬁeld theory.
•Functional analysis: Studies inﬁnite dimensional vector spaces and views functions as points in these spaces. Listed below are some branches of physics where functional analysis has found applications:
– The spectral theory of operations is applied in the theories of quantum physics.
– Scattering theory is also applied in quantum physics
– Banach algebras are applied in quantum ﬁeld theory like axiomatic ﬁeld theory and various integrable models of quantum ﬁeld and statistical mechanics.
– Perturbation theory for linear operators is applied in all domains of mathematical physics.
(ii) Abstract Algebra: Abstract algebra studies sets with binary operations deﬁned on them. Abstract algebra sometimes referred to as modern algebra, is the study of algebraic structures such as groups, rings, ﬁelds, modules, vector spaces, lattices and algebras. Some of the problems that played a role in the development of Abstract algebra are related to the theory of algebraic equations.
• attempt at solving systems of linear equations which led to linear algebra;
• ﬁnding formulae for solutions of general polynomial equations of higher degree that led to the discovery of groups; and
• investigation of Diophantine equations that produced notions of rings and ideals.
Some of the applications of abstract algebra are in Physics where groups are used to represent symmetry operations. Group theory also helps to simplify differential equations.
(iii) Number theory: This is the theory of the positive integers. Its ideas are divisibility and congruence.
(iv) Geometry: Geometry studies shapes and space. Projective geometry is concerned with the group of projective transformations that act on the real projective plane. Inverse geometry on the other hand deals with the group of inversive transformations acting on the extended complex plane. The beginning of Geometry can be credited to ancient Mesopotamia and Egypt. In the 7th Century BC the Greek mathematician Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.
(v) Topology: Topology is a recent subject and it is an extension of geometry. Topology involves the properties of spaces or objects that are preserved under smooth operations such as bending or twisting. Topology refers to the structure imposed on a non-empty set that makes it a topological space. Properties such as convergence, connectedness and continuity can be discussed. Topology involves the study of qualitative properties of topological spaces that are invariant under certain kinds of transformations known as continuous maps. The origin of Topology is generally credited to the Swiss mathematician Leonard Euler who in1735 attempted to ﬁnd a solution to the Konigsberg problem. This problem had to do with ﬁnding the shortest route through the town of Konigsberg, that would cross each of its seven bridges exactly once. This problem led to the branch of mathematics known as graph theory. Graph theory has applications in Electrical Engineering, Computer Science, and Operations Research. Topology has three branches: algebraic topology which measures degrees of connectivity; point set topology which investigates concepts like compactness and connectedness; and geometric topology which studies manifolds and their embedding in other manifolds.
Applied mathematics is a branch of mathematics that deals with mathematical models in engineering, science, computer science, business and industry. The activity of applied mathematics is connected with research in pure mathematics. In the past, applied mathematics consisted only of differential equations, approximation theory and applied probability. Presently, applied mathematics includes ﬁelds that are classiﬁed as pure mathematics that are now important in applications. For example, number theory is now applied in cryptography. The use and development of mathematics to solve industrial problems is called industrial mathematics.
Samuel A. Iyase
Professor of Mathematics, Department of Mathematics
Covenant University, Canaan Land, Ota.