Mathematics can be defined as the logically rigorous study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Mathematics is used throughout the world in fields such as science, engineering, medicine, and economics. These fields both inspire and make use of new discoveries in mathematics. New mathematics is also created for its own sake, without any particular application in view.
The word "mathematics" comes from the Greek μάθημα (máthēma) meaning science, knowledge, or learning, and μαθηματικός (mathēmatikós), meaning fond of learning. It is often abbreviated math in the U.S. and Canada and maths in Britain and many Commonwealth countries
History of mathematics
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.
Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.
Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proof."
Science (from Latin scientia - knowledge) refers to the system of acquiring knowledge – based on empiricism, experimentation, and methodological naturalism. The term science also refers to the organized body of knowledge humans have gained by such research.
Most scientists maintain that scientific investigation must adhere to the scientific method, a process for evaluating empirical knowledge which explains observable events in nature as a result of natural causes, rejecting supernatural notions. Less formally, the word science often describes any systematic field of study or the knowledge gained from it. Particular specialized studies that make use of empirical methods are often referred to as sciences as well. This article concentrates on the more specific definition.
Fields of science are commonly classified along two major lines:
Natural sciences, the study of the natural phenomena including biology;
Social sciences, the systematic study of human behavior and society.
Mathematics has both similarities and differences compared to other fields of science, and is sometimes included within a third, separate classification, called formal science. Mathematics is similar to other sciences because it is a rigorous, structured study (of topics such as quantity, structure, space, and change). It is different because of its method of arriving at its results. Mathematics as a whole is vital to the sciences — indeed major advances in mathematics have often led to major advances in other sciences. Certain aspects of mathematics are indispensable for the formation of hypotheses, theories and laws in discovering and describing how things work (natural sciences) and how people think and act (social sciences).
Science as defined above is sometimes termed pure science to differentiate it from applied science, the application of research to human needs.