# Some facts about the history of mathematics.

The evolution of mathematics might be seen as 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 was a breakthrough in human thought. In addition to recognizing how to count *physical* objects, prehistoric peoples also recognized how to count *abstract* quantities, like time — days, seasons, years. Arithmetic (addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to knowledge of geometry.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu 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 relating to 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 has since been greatly 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 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 proofs.”

Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments come to light only in a few locales. The most ancient mathematical texts available are from ancient India circa 1500BC-500 BC and ancient Egypt in the Middle Kingdom period circa 1300-1200 BC, Mesopotamia circa 1800 BC. All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. The Han Dynasty in ancient China contributed *The Nine Chapters on the Mathematical Art* from the 2nd century BC to the 2nd century AD. Ancient Greece and the Hellenistic cultures of Egypt, Mesopotamia and the city of Syracuse increased mathematical knowledge. Jain mathematicians contributed from the 4th century BC to the 2nd century AD, while Hindu mathematicians from the 5th century and Islamic mathematicians from the 9th century made major contributions to mathematics.

One striking feature of the history of ancient and medieval mathematics is that bursts of mathematical development were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an ever increasing pace, and this continues to the present day.