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What is Mathematics?

math on the chalkboard
Math on a chalkboard.
Credit: Handmade Pictures | Shutterstock.com

Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

Since the beginning of recorded history, mathematic discovery has been at the forefront of every civilized society, and in use in even the most primitive of cultures. The needs of math arose based on the wants of society. The more complex a society, the more complex the mathematical needs. Primitive tribes needed little more than the ability to count, but also relied on math to calculate the position of the sun and the physics of hunting.

History of mathematics

Several civilizations — in China, India, Egypt, Central America and Mesopotamia — contributed to mathematics as we know it today. The Sumerians were the first people to develop a counting system. Mathematicians developed arithmetic, which includes basic operations, multiplication, fractions and square roots. The Sumerians’ system passed through the Akkadian Empire to the Babylonians around 300 B.C. Six hundred years later, in America, the Mayans developed elaborate calendar systems and were skilled astronomers. About this time, the concept of zero was developed.

As civilizations developed, mathematicians began to work with geometry, which computes areas and volumes to make angular measurements and has many practical applications. Geometry is used in everything from home construction to fashion and interior design.

Geometry went hand in hand with algebra, invented in the ninth century by a Persian mathematician, Mohammed ibn-Musa al-Khowarizmi. He also developed quick methods for multiplying and diving numbers, which are known as algorithms — a corruption of his name.

Algebra offered civilizations a way to divide inheritances and allocate resources. The study of algebra meant mathematicians were solving linear equations and systems, as well as quadratics, and delving into positive and negative solutions. Mathematicians in ancient times also began to look at number theory. With origins in the construction of shape, number theory looks at figurate numbers, the characterization of numbers, and theorems.

Math and the Greeks

The study of math within early civilizations was the building blocks for the math of the Greeks, who developed the model of abstract mathematics through geometry. Greece, with its incredible architecture and complex system of government, was the model of mathematic achievement until modern times. Greek mathematicians were divided into several schools:

  • The Ionian School, founded by Thales, who is often credited for having given the first deductive proofs and developing five basic theorems in plane geometry.
  • The Pythagorean School, founded by Pythagoras, who studied proportion, plane and solid geometry, and number theory.
  • The Eleatic School, which included Zeno of Elea, famous for his four paradoxes.
  • The Sophist School, which is credited for offering higher education in the advanced Greek cities. Sophists provided instruction on public debate using abstract reasoning.
  • The Platonic School, founded by Plato, who encouraged research in mathematics in a setting much like a modern university.
  • The School of Eudoxus, founded by Eudoxus, who developed the theory of proportion and magnitude and produced many theorems in plane geometry
  • The School of Aristotle, also known as the Lyceum, was founded by Aristotle and followed the Platonic school.

In addition to the Greek mathematicians listed above, a number of Greeks made an indelible mark on the history of mathematics. Archimedes, Apollonius, Diophantus, Pappus, and Euclid all came from this era. To better understand the sequence and how these mathematicians influenced each other, visit this timeline.

During this time, mathematicians began working with trigonometry. Computational in nature, trigonometry requires the measurement of angles and the computation of trigonometric functions, which include sine, cosine, tangent, and their reciprocals. Trigonometry relies on the synthetic geometry developed by Greek mathematicians like Euclid. For example, Ptolemy's theorem gives rules for the chords of the sum and difference of angles, which correspond to the sum and difference formulas for sines and cosines. In past cultures, trigonometry was applied to astronomy and the computation of angles in the celestial sphere.

After the fall of Rome, the development of mathematics was taken on by the Arabs, then the Europeans. Fibonacci was one of the first European mathematicians, and was famous for his theories on arithmetic, algebra, and geometry. The Renaissance led to advances that included decimal fractions, logarithms, and projective geometry. Number theory was greatly expanded upon, and theories like probability and analytic geometry ushered in a new age of mathematics, with calculus at the forefront.

Development of calculus

In the 17th century, Isaac Newton and Gottfried Leibniz independently developed the foundations for calculus. Calculus development went through three periods: anticipation, development and rigorization. In the anticipation stage, mathematicians were attempting to use techniques that involved infinite processes to find areas under curves or maximize certain qualities. In the development stage, Newton and Leibniz brought these techniques together through the derivative and integral. Though their methods were not always logically sound, mathematicians in the 18th century took on the rigorization stage, and were able to justify them and create the final stage of calculus. Today, we define the derivative and integral in terms of limits.

In contrast to calculus, which is a type of continuous mathematics, other mathematicians have taken a more theoretical approach. Discrete mathematics is the branch of math that deals with objects that can assume only distinct, separated value. Discrete objects can be characterized by integers, whereas continuous objects require real numbers. Discrete mathematics is the mathematical language of computer science, as it includes the study of algorithms. Fields of discrete mathematics include combinatorics, graph theory, and the theory of computation.

People often wonder what relevance mathematicians serve today. In a modern world, math such as applied mathematics is not only relevant, it's crucial. Applied mathematics is the branches of mathematics that are involved in the study of the physical, biological, or sociological world. The idea of applied math is to create a group of methods that solve problems in science. Modern areas of applied math include mathematical physics, mathematical biology, control theory, aerospace engineering, and math finance. Not only does applied math solve problems, but it also discovers new problems or develops new engineering disciplines. Applied mathematicians require expertise in many areas of math and science, physical intuition, common sense, and collaboration. The common approach in applied math is to build a mathematical model of a phenomenon, solve the model, and develop recommendations for performance improvement.

While not necessarily an opposite to applied mathematics, pure mathematics is driven by abstract problems, rather than real world problems. Much of what's pursued by pure mathematicians can have their roots in concrete physical problems, but a deeper understanding of these phenomena brings about problems and technicalities. These abstract problems and technicalities are what pure mathematics attempts to solve, and these attempts have led to major discoveries for mankind, including the Universal Turing Machine, theorized by Alan Turing in 1937. The Universal Turing Machine, which began as an abstract idea, later laid the groundwork for the development of the modern computer. Pure mathematics is abstract and based in theory, and is thus not constrained by the limitations of the physical world.

According to one pure mathematician, pure mathematicians prove theorems, and applied mathematicians construct theories. Pure and applied are not mutually exclusive, but they are rooted in different areas of math and problem solving. Though the complex math involved in pure and applied mathematics is beyond the understanding of most average Americans, the solutions developed from the processes have affected and improved the lives of all.

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