Pythagoras (lower left with a manuscript), in Raphael's fresco "The School of Athens," in the Vatican. Wikipedia
From Ability to Mathematical Proficiency
The ability to estimate quantities, known as numerosity is inherent not only in humans but also in various animal species like monkeys, rats, birds and even bees. The question that arises is how this natural ability evolved into the development of numerical systems, calculations and the language of mathematics.
In his book titled "Why Is the World Mathematical?" (Laterza) renowned astronomer John Barrow explains that some human societies have languages that only encompass concepts such, as 'one' 'two' and 'many.'Furthermore it is interesting to note that in European languages the adjectives used for 'first' and 'second' don't have direct connections to the words for 'one' and 'two' whereas the terms 'third' 'fourth' and so on clearly correspond to 'three,' 'four' and beyond. These abstract concepts (such as one, two many few) initially emerged out of needs.
When examining findings that precede even the cuneiform tablets we can gain insights into how our ancestors counted and what their requirements were. Mathematician Bruno D'Amore in collaboration with Silvia Sbaraglia highlights in their essay "La matematica e la sua storia" (Daedalus) that one of the counting methods may have been based on carvings.
For instance a baboon bone dating back 37,000 years discovered in Swaziland, Africa bears 29 notches. Some speculate that this bone could have been a hunting tool used for keeping track of kills while others propose its connection to the duration of the lunar cycle which lasts 29 days.
Similarly intriguing is an engraved bone handle with a quartz tip found near Lake Edward in Ishango, Africa. The notches, on this handle are arranged in a sequence resembling numbers.
Discovered during the transition from the Paleolithic to eras the purpose of this artifact remains shrouded in mystery just like the level of mathematical knowledge possessed by its creators when it comes to prime numbers.
Early Methods of Counting; From Carvings to Thumbels Numbering
Besides using carvings for counting purposes there was also a method called "Thumbels" numbering, which involved using pebbles or similar objects. At the ruins of Nuzi, a city in Iraq archaeologists found a clay container that listed 48 heads of livestock; 21 sheep, 6 lambs, 8 rams, 6 goats, etc. each accompanied by a corresponding clay ball. This simple technique is still employed in traditional societies today and allowed shepherds from three millennia ago to keep track of their animals by adding one ball for each animal.
The Origins of Early Calculation Systems
It is believed that our ancestors likely used "base 2" or "base 5" calculation systems. For instance Australian Aborigines still adhere to a base 2 system where they only have words for "one". Two." Another example comes from Moravia in the Czech Republic where a wolf bone with notches dating back, over 30,000 years was discovered. These notches—grouped in sets of five—represent one of the known records displaying principles of computation based on hand like digits.
In civilizations like Mesopotamia, India, China, Egypt, Greece and Rome people commonly used body parts such as hands, fingers, shoulders, heads and feet as tools for calculations. This practice continued during the Middle Ages despite the introduction of the abacus. The Liber abaci ("The Book of Calculus") by Leonardo Fibonacci played a role in spreading Arabic numerals in Europe during the 13th century. On the side of the Atlantic Ocean the Maya and Aztecs developed a 20 based system to create accurate calendars and sophisticated time measuring systems.
The evolution of numbers also had its roots in different regions. The Sumerians and Babylonians used a base 60 (numbering system influenced by their cultural transition towards urban civilization in Mesopotamia. This transition coincided with developments, in China and the Indus Valley. As trade networks expanded and grain storage became important advanced computational skills were required.
The Sumerians, around 5,300 to 4,500 years ago were pioneers in inventing the abacus (also known in the Far East). Using number symbols. These symbols were used for tasks like creating lists of goods army supply management and making purchase and sale agreements. D'Amore explains that the Sumerians also introduced the positional system, which was later refined by the Indians.
The concept behind this system was that each symbol had a value that could be added to its symbols by summing them up. Today we still use a method for forming numbers; however we adopted it from the Arabs during the Middle Ages. This adoption was significant as it allowed us to represent any number using a small set of symbols (in our case 10 symbols including zero). The sexagesimal system left two legacies; time calculation (with minutes consisting of 60 seconds and hours consisting of 60 minutes) and coordinate calculation for navigation (where each degree is divided into 60 minutes and 60 seconds).
Moving on to Egypt it also played a notable role in numerical systems. Our understanding of mathematics primarily comes from the Papyrus Rhind—a document over two meters long—containing around eighty problems along, with operation tables and their solutions. These were aimed at scribes or priests 4,000 years ago.
The ancient Egyptians had a system based on ten but it was not positional. They were unaware of the concept of zero and primarily used calculations for astrological purposes as well as for constructing temples, tombs and measuring agricultural land. However they did not develop a system that went beyond practical necessities. It was the philosophers who took this advancement further.
Origins of Abstract Mathematical Concepts
Although the Sumerian Babylonians might have known how to generate triples they did not comprehend the underlying theorem behind these numbers. It was the mathematicians Thales (7th 6th century B.C.) and Pythagoras (6th 5th century B.C.) who laid the groundwork for abstract mathematical thinking influenced by Mesopotamian and Egyptian traditions. Euclid (4th 3rd century B.C.) solidified this foundation with his work called "The Elements." This comprehensive text not includes a proof of Pythagoras theorem still taught in schools today but also served as a fundamental resource for mathematical learning over millennia.
The Emergence of Geometry and Arithmetic
Geometry and arithmetic were born in the region between Asia Minor (Turkey) and Attica. Within three centuries in this area deductive thinking emerged alongside geometric axioms concepts like infinity, irrational numbers and the belief that everything, in nature could be measured. Around 300 B.C. mathematics emerged as the approximation to science during that time.
The Impact of Greek Influence on Mathematics
Mathematical knowledge played a crucial role in connecting ancient and modern thinking. This knowledge spread to India through Alexander the Greats conquests, which introduced culture to the region. According to John Barrow, notation, effective computation methods and the use of zero were essential components of a successful numbering system. These three elements were combined in ancient India. Hindu scholars further advanced this knowledge inherited from sources and perfected the positional decimal system. Eventually this system made its way to Europe in the century via Arab influences and once again due to Fibonaccis contributions. However there was resistance from institutions like the Church that initially sought to prohibit these "pagan" numbers and even reject the concept of zero.
Mathematics; A Fresh Perspective on Understanding the World
Despite opposition Western society eventually embraced viewing the world through a mathematical lens similar to what Thales, Pythagoras, Euclid and Alexandrians had done before them. This shift had reaching consequences. In music theory for example Pythagoras was among those who discovered relationships between sounds – an innovative concept, at that time.
The introduction of numerals had a profound impact on the economy during medieval times. It played a role, in the development of double entry bookkeeping and banking. In the 1400s the rediscovery of Hellenistic writings opened up new possibilities for geographers to accurately measure the planet navigators to explore uncharted lands and astronomers to better understand celestial movements. This knowledge ultimately led to Galileo Galilei (1564 1642) formulating an approach based on his belief that the universe and nature can be described using mathematical language, specifically numbers.
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