What is mathematics?

Mathematics is the science of logic, shape, quantity, and arrangement. Math is everywhere around us. It is the foundation of everything we do, from our daily lives to mobile devices, computers and software to architecture (ancient or modern), art and money to engineering and sports.
Mathematical discovery has been at forefront of every civilized society since the beginning of recorded human history. Even the most primitive cultures have used math. Math was required by societies all over the globe due to increasingly complex needs. This necessitated more sophisticated mathematical solutions as described by Raymond L. Wilder, a mathematician (Dover Publications 2013).

Complex societies require more complicated mathematical calculations. Primitive tribes had little more than the ability count. They also used math to calculate sun position and the physics behind hunting. Wilder, 1968, wrote that all records, historical and anthropological, show that counting and eventually numeral systems were used as a means of counting.

Who was the first to use mathematics?

Mathematics as we know it today was influenced by several civilizations, including Egypt, China, India, Egypt, Central America, Mesopotamia, and Central America. Wilder claims that the Sumerians, who lived near the current southern Iraq, were among the first to create a base 60 system for counting.

According to Georges Ifrah's book "The Universal History Of Numbers", (John Wiley & Sons 2000), this was based upon using bones from the fingers to count and then use them as sets. These systems form the basis for arithmetic. It includes basic operations such as addition, multiplication and division, fractions, and square roots. Wilder explained that the Sumerians passed from the Akkadian Empire to Babylon around 300 B.C. Six hundred years later in Central America, six hundred years after the Sumerians' system, Wilder explained that the Akkadian Empire passed through to Babylonians around 300 B.C. In India, the concept zero was created around this time.

Mathematicians started to use geometry as a tool for calculating areas, volumes, angles and other practical purposes. Geometry is used for everything, from home construction to interior design and fashion. Richard J. Gillings, in " Mathematics in the Time of the Phharaohs" (Dover Publications 1982), wrote that the pyramids of Giza, Egypt are a stunning example of an ancient civilization's advanced use of geometry.

A statue of Muhammad ibn Musa al-Khwarizmi, which is located in Khiva (Uzbekistan). (Image credit to Konstik

Algebra and geometry went hand-in-hand. According to Philip K. Hitti (a Princeton history professor and Harvard University historian), the earliest known work on algebra was written by Muhammad ibn Musa al-Khwarizmi, a Persian mathematician. Al-Khwarizmi also invented quick methods to multiply and divide numbers. These are called algorithms, a corruption of his Latin name which was later translated into Algorithmi.

Algebra provided civilizations with a method to allocate resources and divide inheritances. Mathematicians were able to solve linear equations, quadratics, and find positive and negative solutions through algebra. In "Introduction to Analytic Number Theory" (Springer 1976), Tom M. Apostol, an academic at the California Institute of Technology wrote that number theory was also a topic of interest to mathematicians. Number theory, which has its roots in the construction and characterization of shapes, examines figurate numbers, the character of numbers, as well as theorems.

Ancient Greece: Mathematics

Douglas R. Harper, author the "Online Etymology Dictionary", says that mathematics is a word that derives its name from the ancient Greeks. The model of abstract mathematics was developed by the ancient Greeks using geometry and other mathematical studies from other ancient civilizations.

G. Donald Allen, a Texas A&M University professor of mathematics, described how the Greek mathematicians were broken into different schools in his paper "The Origins of Greek Mathematics":

Apart from the Greek mathematicians mentioned above, a variety of ancient Greeks also made an impact on the history and evolution of mathematics. These included Apollonius, who was most well-known for the Archimedes principle around buoyant force; Diophantus who was the first Greek mathematician who recognized fractions as numbers; Diophantus who worked with parabolas; Diophantus who was best known for his hexagon Theorem; and Euclid who first described the golden proportion.

The golden ratio is one the most well-known irrational numbers. It can go on forever and cannot be expressed accurately with infinite space. (Image credit: Shutterstock)

Mathematicians started to use trigonometry during this period. This is a method that studies the relationships between sides and angles of triangles. It also computes trigonometric functions such as sine, cosine and tangent. Trigonometry is based on synthetic geometry, which was developed by Greek mathematicians such as Euclid. Trigonometry was used in past cultures to calculate angles in the celestial circle and astronomy.

According to Wilder, the Islamic empires were responsible for the development of mathematics. Leonardo Fibonacci, a medieval European mathematician, was well-known for his theories about arithmetic and algebra. Advancements in number theory, projective geometry, and logarithms were made possible by the Renaissance. The number theory was vastly expanded, and theories such as probability and analytic geometry ushered in a new era of mathematics with calculus at its forefront.

Calculus development

Isaac Newton, an Englishman, and Gottfried Leibniz, a German mathematician, independently created the foundations of calculus in the 17th century. Carl B. Boyer is a science historian who explains this in "The History of the Calculus" (Dover Publications 1959). Three periods were involved in the development of calculus: anticipation, development, and rigorization.

Mathematicians tried to create infinite processes in the anticipation stage to find areas under curves and maximize certain quality. Newton and Leibniz combined these techniques in the development stage through the derivative (the curve for mathematical function) as well as the integral (the area below the curve). Although their methods weren't always sound, mathematicians of the 18th century were able justify their methods and created the final stage in calculus. We define integral and derivative in terms of limits today.

Calculus is a type continuous mathematics that deals with real numbers, but other mathematicians take a more theoretical approach. Discrete mathematics refers to the mathematical branch that works with objects that have a distinct, separate value. Richard Johnsonbaugh, a mathematician, explained this in "Discrete Mathematics" (Pearson 2017, 2017). Integers can be used to describe discrete objects, instead of real numbers. Because it involves the study of algorithms, discrete mathematics is also the mathematical language of computer sciences. Discrete mathematics includes combinatorics and graph theory, as well as the theory of computation.

Complex math is important in finance, travel and computing, even though it may not seem so to most people. (Image credit: Anton Belitskiy/Getty)

Why mathematics is so important

People often wonder about the relevance of mathematics in their everyday lives. Math such as applied mathematics has become a crucial part of modern life. Applied mathematics includes all branches of mathematics that deal with the physical, biological, and sociological worlds.

Alain Goriely wrote in "Applied Mathematics: A Very Brief Introduction" (Oxford University Press 2018, 2018) that "the goal of applied mathematics was to establish connections between different academic fields." Mathematical physics, mathematical biology and control theory are all modern areas of applied mathematics. Goriely said that applied math not only solves problems but also uncovers new problems and develops new engineering disciplines. The common approach to applied math is to create a mathematical model of the phenomenon and then solve it. Finally, we make recommendations for improvement.

Although pure mathematics can be considered as an alternative to applied mathematics, it is driven more by abstract problems than real-world problems. Many of the topics pursued by pure mathematics have roots in concrete physical phenomena, but deeper understanding of these phenomena leads to technicalities and problems.

Pure mathematics attempts to solve these abstract problems and technicalities. These efforts have resulted in major discoveries for humanity, including the universal Turing Machine, which was created by Alan Turing in 1938. The foundations for modern computers were laid by this machine, which started as an abstract idea. Pure mathematics is abstract and based on theory. It is therefore not limited by the limitations of physical reality.

Goriely says that applied mathematics is to pure mathematics what pop music to classical music is to pure mathematics. Although they can be used in the same way, pure and applied do not have to be mutually exclusive. However, both are rooted in different areas and methods of problem solving and math. Although the math involved in applied and pure mathematics is complex and beyond most people's comprehension, many have benefited from the solutions.

Original publication on Live Science