The Nature and Growth of Ideas in Mathematics

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Clay Tokens and the Origin of Numeracy

Tokens were minute arithmetical clay items (cylinders, tetrahedrons, cones, spheres, etc.) that were used before the invention and adoption of writing. Clay tokens, therefore, were three-dimensional shapes that dated back to around 8000 BCE and varied in shapes and sizes. The two commonly used types of tokens were plain and complex. The plain ones existed earlier than the complex variant. Unlike the complex variants that were decorated, plain tokens were simple shapes without decorations.

In those days, the widespread use of tokens was mainly for trade as a medium of exchange since they were a precise representation of the items being traded. Their different shapes represented a variety of items like a bag of wheat or sheep. When the tokens being exchanged were many, they were placed in a clay jar or ball and given to the seller, who opened the ball to verify the number of tokens. Due to social and economic advancement, the complex tokens were modified with impressions and incisions with varying significance. People also made impressions (before the clay dried) on the outer part according to the number of tokens inside. This was meant to serve as a reference in case of a dispute since the ball could not be broken to show the tokens inside. This practice marked the beginning of writing on clay tablets. The complex tokens were used to keep records of produced goods as their complex, and standard nature made them more useful than twigs and pebbles. These tokens were then kept in clay envelopes or on strings for security. Several tokens represented cuneiform signs. Scribes made several incisions on the tokens to show the number of products it represented as well as diagrams symbolizing the item being exchanged (such as a symbol of a goat or sheep). This marked the onset of numeracy and writing. The tokens were later modified as laws were created that demanded the inscription of names of the recipient on the token.

Positional Principle

The positional principle refers to a place-value system where the value attached to a digit or symbol is dependent on its position. For example, the value of 3 in 300 is higher than that of 3 in 30. The commonly used form of the positional system is the decimal-positional number system that uses ten symbols from zero to nine in the generation of all other numbers. This system was devised by Indians and improved upon by Arabs. The other two types of positional number systems are found in computer science. These are the binary and hexadecimal systems. The binary system has only two symbols (0 and 1), while the hexadecimal system is made up of sixteen symbols (0 to 9 followed by letters A to F). In the positional principle, each digit that is part of a number has two values. The first value is intrinsic and is the value of the isolated digit. The second value is the local value (also called place value), which the digit possesses due to its positioning.

The Hindu numeral system is a good example of a purely positional system that makes use of a zero. When balanced against non-positional systems, positional systems are reasonably scarce in the history of mathematics. However, the advantage of embracing a positional system is in its precision and ease of application. The modern Hindu-Arabic system clearly depicts this ease and precision. Instead of using several digits to depict one position, the Hindu-Arabic system uses only one symbol for each position. The major disadvantage of the Hindu-Arabic system is that the numerals are merely abstract symbols that bear no connection with the things they represent. An example of a non-positional system is the Roman numerical system that uses seven symbols to signify digits.

Opportunities and Challenges Presented by Sources of Ancient Mathematics

Historians use various sources like books, scrolls, images, and artifacts in their quest to understand what happened in the past. In this regard, historians studying the mathematical past are not excluded. These materials present a historian with an opportunity to take a look at the lives of people in the past and their contributions to mathematical discourse. Can you imagine studying the history of mathematics without any material facts that can aid the study? History cannot depend on hearsay. These materials, upon interpretation and careful dating, can provide information about the lives of those under study. Historians can reconstruct mathematical history from materials found by archaeologists.

However, the major challenge when dealing with these sources lies in their interpretation. Interpreting inscriptions on a clay token, for example, is not easy since none of the people doing the interpretation understand the contexts in which they were used. Archaeologists and historians can merely speculate about the use of the object. Though such speculation is based on rigorous multidisciplinary procedures, it is still done from an outsider’s perspective by people who have no experience with the context being investigated. The historian must ascertain all the facts about the source. Subsequently, he should use deductive logic to find out additional information about the source.

There are few available sources on mathematics as some people have tampered with most of these sources. This makes it difficult to trust the accuracy of the information inferred from the sources and their validity. The samples of the available papyri containing mathematical information on ancient Egypt, for instance, are limited. These available sources are not sufficient to enable a historian to construct a clear history of ancient Egyptian mathematics.

Without significant written historical accounts of past mathematical practices and the thoughts of mathematicians, it is very difficult for modern historians to interpret sources objectively. They are usually impeded by ethnocentrism and other biases. Such biases can influence the results of their interpretation and, ultimately, the way they construct ancient history.

Plato’s Illustration of Greek Mathematics in the Meno

Plato played a significant part in inspiring his students and admirers to pursue mathematics and philosophy. His academy handled mathematics, including plane and solid geometry, as part of philosophy. Some of the greatest mathematicians of that era, like Eudoxus and Theaetetus, were products of his academy. At the beginning of the dialog with Socrates, Meno questioned whether virtue could be acquired through teaching. In response, Socrates proposed that before they could establish if indeed virtue could be taught, a clear definition of the concept was necessary. In some part of the conversation, Socrates told Meno to call his slave. Meno conceded that the slave had no education in geometry. In that dialog, Socrates displayed his technique of questioning to enable recollection of information. The boy, who had no knowledge of geometry, was ultimately able to learn a complicated geometry problem of doubling-of-square. By doing this, Socrates intended to prove to Meno that learning was possible when the correct method was used.

Socrates argued for the immortality of the soul. He believed that the soul had eternal knowledge, which is lost at birth. The purpose of that dialog was to prove that (through questioning), a person could remember the information that was forgotten. The important part of that dialog was that Socrates chose to use the halving of a square by its diagonals. That showed that the great thinkers of ancient Greece possessed knowledge on the concept of geometric dissection. It also portrayed the nature of Greek mathematics as founded on geometry. As shown in the dialog, ancient Greek mathematics used deductive logic to prove or dismiss concepts. The dialog, therefore, clearly portrayed the close relationship between mathematics and philosophy in ancient Greece. The method of proof is still used today to verify theories.

Pythagoras’ Contributions to Mathematics

Known in many quarters as the first genuine Greek mathematician, Pythagoras was born in Samos, Greece (569 BC). He was well educated and had interests in mathematics, philosophical pursuit, music, and astronomy. The main influence in Pythagoras’ life choices was Thales, who inspired his interest in arithmetic and astronomy. He started a school in Croton that later became a sect of faithful followers of his teachings and philosophy (Pythagoreans). In the school, the general philosophy was that everything was made up of a number and that God was a number. For instance, they believed that the number ‘ten’ was holy. They viewed ten as a triangular number comprising the sum of one, two, three, and four. This number was devised in honor of Pythagoras.

Pythagoras and his followers were mainly remembered for the introduction of a more thorough mathematical approach than the one that existed before. They built on the earlier concepts of axioms and logic. Before him, geometry was just a set of laws arrived at through experimental measurement. He came up with a complete mathematical system in which geometric essentials were consistent with numbers. Pythagoras was also remembered for the principle of right-angled triangles, commonly called Pythagoras Theorem. He popularised the principle that any right-angled triangle had a hypotenuse whose square was equivalent to the summation of the squares of the two other sides. As an equation, this principle was written as a² + b² = c². Though the concept of right angles had been in use earlier in Babylon and Egypt, it was Pythagoras who made it definitive.

His studies of numbers that were odd and even, triangular, and perfect were a valuable contribution to the comprehension of triangles, areas, and polygons. He also established a relationship between music and mathematics. While playing his favorite musical instrument, the seven-string lyre, he discovered that notes in music were always proportionate to whole numbers. This discovery led Pythagoras to believe that the entire universe was made up of numbers and that the movement of the planets was in accordance with mathematical equations. The Pythagorean Theorem is very useful today for any regular shape.

Evolution of the Number

The role played by numbers in the world of mathematics is very critical. It has been a predominant occupation of great mathematical minds since antiquity. The origin of numbers was linked to the practical day to day need of prehistoric people to value their properties. This was followed by a series of evaluations through many centuries involving several discoveries and analyses of formulae. This essay provides a historical outlook of the evolution of numbers dating back to antiquity.

The Beginning

The initial use of numbers in trading started thousands of years ago. During that time, numerical systems probably consisted of only whole numbers. Bones and several relics have been revealed, with incisions believed to have been used for tallying. That form of the tallying system was the earliest method of numeracy and was used to count the number of days, seasons, or animals. The system had no place value, which limited its application to large numbers. Nevertheless, the tallying system remains the earliest abstract form of numbering.

The Babylonian Era

In the Babylonian era, practical geometry arose that required the application of square roots (Hodgkin 25). The Babylonians adopted their numbering system from Sumer. Basically, their numerical system was sexagesimal. It was from this base of sixty systems that modern mathematicians derived the concepts of sixty minutes in one hour, sixty seconds in one minute, and the concept of three hundred and sixty degrees in a circle. The Babylonians’ rapid mathematical progress was because they had devised a place value system, unlike the Romans and Egyptians.

Adoption of Roman Numbers

Prior to the adoption of the Hindu-Arabic numbering system, Roman numbers were already in use. The Roman system of numeration was a heritage of the Etruscan era. It was founded on a five-digit principle adopted from the five fingers of a human hand. The numbering system was formed due to the demand for a uniform technique of counting that would benefit trade and communication. Before that, it was difficult to count beyond ten as a person only had ten fingers. The system was fundamentally made up of seven symbols (I, V, L, X, C, D, and M). Roman numbering system had its fair share of problems. The main weakness, for instance, was that it lacked a symbol for the number zero. Anyone using this method also had no way of calculating fractions. That made conducting commerce using the Roman system difficult. It also slowed down the universal application or the Roman numerical symbols. Consequently, the system was gradually abandoned for a more dynamic Hindu-Arabic system.

In the modern era, Roman numbers are used in movie credits and titles, tables of contents, and notation symbols in music. They are also used to identify different groups of the periodic table. These applications are mainly for aesthetic reasons rather than functional.

Hindu-Arabic Numbers and the Use of Zero

Many ancient mathematical systems such as Babylon, India, ancient Greece, and Egypt adopted the use of zero. Ancient Egyptian accountants and mathematicians employed nfr to symbolize a zero balance. Indians also employed the use of the word shunya to mean emptiness. Most Indian books on the mathematics used this word to mean the number zero. Greek manuscripts indicated the uncertainty that ancient Greeks possessed regarding the value of zero as a number. There were several numerical systems that adopted the use of zero as a number. Therefore, the adoption and use of zero marked an important stage in the evolution of mathematics.

To understand the evolution of numbers, we must first understand the nature of Hindu-Arabic numbers. The Hindu-Arabic numerical system, which is widely used today, is a mixture of ten digits. These digits (ranging from one to ten) were initiated in Europe by an Italian mathematician called Fibonacci (Stakhov 60). He received his education in the northern part of Africa. While pursuing his studies, he was taught Hindu-Arabic numerals, which he introduced back in Italy. The Indians were using the number zero long before it was introduced in Europe. Nevertheless, the usage of the Hindu-Arabic numerals was initially described in the fifth century by a Roman called Boethius. In one of his books on arithmetic, he described the operation of the abacus by use of tiny inscribed cones with symbols of Hindu-Arabic symbols as an alternative to using pebbles. A cone was referred to as an apex. It was inscribed with symbols of the nine digits now known as numbers. Consequently, the initial symbols of early numbers that appeared in Europe were referred to as apices. Every one of the nine apices was given its own name. The table below contains the names used and the digits they represented.

Apex	Digit
Igin	1
Andras	2
Ormis	3
Arbas	4
Quimas	5
Caltis	6
Zenis	7
Temenisa	8
Celentis	9

The origin of the names indicated in the table above is still not clear, but some of them were derived from Arabic. The numbers were easily adopted and spread throughout Europe.

Conclusion

From the days of mechanistic tallying to the modern-day complex computing, numbers have truly evolved to suit changes in technology and lifestyle. This evolution makes mathematical practice much easier and increases the application of numbers in our everyday lives. Only time and research can tell the future magnitude of this evolution.

Works Cited

Hodgkin, Luke. A History of Mathematics: From Mesopotamia to Modernity, New York, USA: Oxford University Press, 2005. Print.

Stakhov, P. Alekseĭ. The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, Singapore: World Scientific, 2009. Print.

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