By Dr. Martha Stillman
Associate Professor of Mathematics at American Public University
Modern mathematics has spread into some interesting and incredibly useful areas of modern life. Highway engineers use optimization techniques and linear algebra to analyze traffic patterns and minimize travel time for commuters. Airlines, hotels, and Broadway theaters (not to mention some retailers) use complex mathematical models, so complex that only computer programs can sometimes solve them, to set prices on a dynamic basis. Physicists use group theory and tensor analysis to solve the mysteries of sub-atomic particles. There are numerous other examples, from statistics being used by the Census Bureau and research firms to predict market trends, to partial differential equations being used by brokerage houses to formulate models of where the market is going.
Origins of Mathematics
But where did mathematics originate from? How did it get started, and how complicated (or simple) were its beginnings in the ancient world?
While arithmetic in some form (counting) has been with us since people banded together in primitive tribal groups 35,000 years ago, formal mathematics could not begin until writing was invented. This event occurred around 3200 BCE in the Fertile Crescent (specifically Mesopotamia, near the ancient city of Sumer); some authorities think it was also independently invented in ancient Egypt around the same time, as well as in China 1200 BCE. The ancient Egyptians may have made important contributions as well.
For example, the “Moscow mathematical papyrus” (so called because it is held in a Moscow museum and dates to 1850 BCE) contains a problem analyzing the dimensions of a truncated pyramid. While the hieroglyphic explanation would seem like any other Egyptian manuscript to modern eyes, the diagram accompanying it would be readily recognizable in any modern algebra class.
So what is the link between mathematics and writing, and why was it necessary for the development of writing to precede the development of a mathematics that could go beyond counting and simple arithmetic?
Besides the relative permanence of writing and the ability to transmit information from one generation to the next, writing is distinguished from other symbolic representational systems (e.g., cave art, or temple decorations) by the fact that its symbols (letters) are not related to their meaning, but rather represent sounds or other phonemes as the abstract building blocks of language. This development moves writing beyond short-hand pictorial representations, so that the story is told by putting together abstract letters to form words, and not by recounting a story in pictures.
Likewise, mathematics could not get started until the beginnings of a symbolic language to represent basic mathematical concepts had been developed. Arithmetic started roughly at the same time as writing began, around 4000 BCE in the Fertile Crescent (in what is now modern Iran), and at first was based only on counting techniques. For example, arithmetic at that time might be used to represent that 4 apples plus 5 apples yields 9 apples.
Mathematics as we understand it today, where math symbols take on abstract meanings beyond simple enumeration and arithmetic, did not really have its beginnings until the time of the ancient Greeks (eighth century BCE). The Pythagorean theorem (sixth century BCE), which says that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, is generally thought to be the most ancient mathematical formulation to go beyond simple arithmetic and geometry. The Greeks are also credited with being the first to develop deductive logic, a type of reasoning fundamental to mathematics, whereby one can prove a theorem or statement to always be true.
Again, while the text would be incomprehensible to anyone today who did not read ancient Greek, the diagrams in the fragment pictured below are readily recognizable to any college (or high school) algebra student. Here is a fragment of Euclid’s “Elements” found at Oxyrhynchus (in Egypt) and dating to roughly 100 CE.
Ancient Greek Contributions to Mathematics
So what are some of the contributions of ancient Greek mathematicians to the math we use today?
A key Greek contributor, Pythagoras (570 BCE to 495 BCE) is the developer of the Pythagorean Theorem, and made important contributions to religious philosophy and general philosophy as well as mathematics. Indeed, some scholars argue that the ancient Greeks considered mathematics to be a specialized form of natural philosophy and not a separate branch of study at all.
Another ancient Greek philosopher, Thales, used geometry to solve real world problems such as the height of buildings (and the pyramids), and the distance between ships and the shoreline. Plato (428/427 BCE — 348/347 BCE) made important contributions by clarifying the distinction between assumptions and data, as did Euclid (c. 300 BCE), who strengthened the mathematical rigor of proofs by introducing the explicit concepts of definition, axiom, theorem, and proof.
Archimedes (c.287—212 BC) defined the surface area and volume of a sphere and worked with infinite series, as well as contributing to the study of physics with the principle of buoyancy and the creation of the Archimedes screw, which could transfer water from one location to another (including raising it against the pull of gravity).
Chinese and Indian Contributions to Mathematics
Important contributions to ancient mathematics were also made by the Chinese, the Indians, and the Muslims, although the Muslims operated mostly in the eighth century CE and later, so that they are considered to be more modern than ancient times. The Chinese and Indian cultures, however, flourished roughly contemporaneously with the ancient Greeks.
Chinese mathematics, in particular, is so different in its approach and formulations, that scholars generally agree it was developed independently. The Chinese are credited with developing a decimal positional notation system (so that powers of 10, 100, 1000 and so on are distinguished from each other), a variety of geometrical theorems, and mathematical proofs for the Pythagorean Theorem and Gaussian elimination, a technique used in modern day Linear Algebra to manipulate matrices using row reduction.
Unlike ancient Greek math, Chinese mathematics continued to develop well beyond ancient times, so that, for example, in the 13th century CE, Zu Chongzhi calculated the value of pi to seven decimal places and Chu Shih-chieh formulated a method for solving higher order algebraic equations).
The Indian culture for mathematics flourished somewhat later (eighth century BCE to second century CE), and included calculations for the square root of two, a statement (but not proof) of the Pythagorean Theorem, and astronomical treatises from the fourth and fifth centuries CE involving various trigonometric proofs.
Clearly, the mathematics that we use today is the cumulative knowledge of a great many people, stretching back in time more than three thousand years and across a number of different cultures located across the globe.
About the Author
Dr. Stillman holds degrees which straddle two worlds. While in her twenties she earned bachelor’s and master’s degrees in mathematics and physics, and a post-master’s in computer science. She then had a twenty-plus year career in banking and marketing, followed by the decision to go back to school to earn a master’s and doctorate in religion. In 2006 she was awarded a Ph.D. in religion. Today, she pursues a variety of professional activities, among which is teaching mathematics at American Public University.
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