History of Mathematics - From Tally Marks to Algorithms
Mathematics is one of humanity’s oldest and most universal achievements. Its story stretches back tens of thousands of years. It began with tally marks scratched on bones in prehistoric Africa. Today, it powers the algorithms behind artificial intelligence. Understanding where mathematics came from helps us appreciate not just what it is, but why it works the way it does.
One of the most remarkable surviving artefacts is Plimpton 322, a Babylonian clay tablet dated to around 1800 BCE and now held at Columbia University. It lists fifteen rows of numbers that turn out to be Pythagorean triples – sets of whole numbers satisfying a² + b² = c² – over a thousand years before Pythagoras was born. Whether the scribes who made it understood the general theorem or were simply cataloguing useful number patterns for surveying and construction is still debated by historians, but it shows that the mathematical relationships taught in a modern classroom were already being explored in the ancient world, long before anyone wrote down a formal proof.
Prehistoric Mathematics
The earliest evidence of mathematical thinking dates to around 43 000 BCE. The Lebombo bone, found in Africa, has 29 tally marks. It may have been used as a lunar calendar. The Ishango bone (c. 20 000 BCE) is also from Africa. Its markings suggest an early understanding of prime numbers and arithmetic. Counting was humanity’s first mathematical act. It was driven by the need to track animals, seasons, and resources. These early counting systems had no written number symbols at all. Instead, people used marks, notches, and knots to stand for a quantity.
Ancient Civilisations
The table below compares five ancient civilisations and the mathematical ideas each one gave the world.
| Civilisation | Period | Key Contributions |
|---|---|---|
| Babylonians (Mesopotamia) | c. 3000–300 BCE | Base-60 number system, quadratic equations, and Pythagorean triples, used 1000 years before Pythagoras. |
| Egyptians | c. 3000–300 BCE | Fractions, area and volume calculations, and practical geometry for construction. |
| Indians | c. 1500 BCE onwards | Zero as a true number, decimal place value, and early algebra and trigonometry. |
| Chinese | c. 1100 BCE onwards | Negative numbers, the remainder theorem, and Pascal’s triangle, discovered 600 years early. |
| Greeks | c. 600–200 BCE | Formal proof, deductive geometry, number theory, and the irrationality of √2. |
Ancient Greece – The Birth of Proof
The Greeks transformed mathematics from a practical tool into a theoretical discipline. Thales of Miletus (c. 624–546 BCE) is credited with the first mathematical proofs. Pythagoras (c. 570–495 BCE) founded a school that treated numbers as sacred. Euclid (c. 300 BCE) wrote the Elements. It was 13 books of geometry, built entirely from just five simple axioms. It is arguably the most influential textbook ever written. Archimedes calculated π to remarkable accuracy. He also anticipated integral calculus by nearly 1900 years.
The Islamic Golden Age
Between the 8th and 13th centuries CE, Islamic scholars preserved Greek knowledge and made transformative advances of their own. Al-Khwarizmi (c. 780–850 CE) wrote a book called Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala. Its title gave us the word algebra. His own name later gave us the word algorithm. Al-Battani advanced the field of trigonometry. Omar Khayyam found a way to solve cubic equations using geometry.
The European Renaissance and Beyond
From the 12th century onwards, European mathematicians built on these earlier ideas. The table below highlights seven of the most important names.
| Mathematician | Period | Achievement |
|---|---|---|
| Fibonacci | c. 1170–1250 | Introduced Hindu-Arabic numerals to Europe and described the Fibonacci sequence. |
| Descartes | 1596–1650 | Invented coordinate geometry, also called the Cartesian plane. |
| Newton & Leibniz | 1643–1727 / 1646–1716 | Independently invented calculus at around the same time. |
| Euler | 1707–1783 | Founded graph theory and introduced everyday notation such as π, e, i, Σ, and f(x). |
| Gauss | 1777–1855 | Advanced number theory, statistics, and non-Euclidean geometry. |
| Cantor | 1845–1918 | Created the theory of infinite sets, showing infinity comes in different sizes. |
| Turing | 1912–1954 | Laid the foundations of computing and computability theory. |
The Story of Zero
Zero is one of humanity’s most profound mathematical inventions. The Babylonians had a placeholder symbol for “no value,” but they never treated it as a real number. Brahmagupta (628 CE) was the first person to define zero as a true number, complete with rules for arithmetic. From India, zero travelled to the Arab world and then on to Europe, where it was initially met with resistance. Without zero, place value notation would be impossible. And without place value, modern arithmetic, computing, and science could not exist either.
Modern Mathematics
The 20th century saw mathematics split into hundreds of specialised fields. At the same time, it became indispensable to science and technology. Cryptography protects internet transactions. Statistics drives medical research. Topology is used in quantum physics. Graph theory underlies social networks and GPS routing. Mathematics is no longer just about numbers. It is the fundamental language of the modern world.
Interesting Facts
- The word mathematics comes from the Greek word mathema. It means “that which is learnt.”
- Ancient Babylonians could solve quadratic equations, but they had no symbolic algebra. Every step was written out in words.
- Euclid’s Elements was the second most printed book in the Western world after the Bible. It held that position right up to the early 20th century.
- Euler produced more mathematics than any other mathematician in history. He kept working even after going completely blind in 1771.
- The plus (+) and minus (−) signs we use today did not appear in European texts until the late 1400s.
- The word algorithm comes directly from Al-Khwarizmi’s own name, carried into Latin and then into English.
Key Takeaways
- Mathematics began with counting and grew through thousands of years of contributions from dozens of civilisations.
- The Greeks introduced the idea of formal proof – a cornerstone of modern mathematics.
- Islamic scholars preserved ancient knowledge and invented algebra during Europe’s Middle Ages.
- Zero, calculus, and set theory each transformed what mathematics could do.