Mathematics
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This is the Sleepy History of Mathematics, narrated by Simon Mattox, written by Alexandra Turney. At least once in your life, you've probably tried counting sheep to fall asleep. Three. And then at some point, hopefully, you drift off. Counting is one of the first things we learn as children. It's the most basic element of that tricky subject: mathematics.

Tonight, we'll explore the very beginnings of mathematics and discover how it developed over the centuries. And among many other things, we'll learn how the work of an Indian mathematician could help to unravel the secrets of the universe. So, just relax and let your mind drift as we explore the sleepy history of mathematics.

When you think of the origins of mathematics, you probably picture numbers on an ancient manuscript, papyrus perhaps. But in fact, the earliest evidence we have may be a bone, a tool known as the Aishango bone. This mammal bone is about 10 centimeters long. It was found near a lake in the Democratic Republic of Congo.

and it's believed to be at least 20,000 years old. The bone is covered in engravings, 168 lines, which seem to be ordered in particular groups and patterns. There are different theories about these engravings and exactly what they mean. Some have suggested that the bone may have been used as a calendar. Others think the lines are tally marks,

But, according to some experts, the bone could be considered the world's first mathematical tool. The people who used it more than 20,000 years ago may have had some knowledge of prime numbers. Perhaps the carvings on the bone were used to perform simple calculations. It's a tantalizing thought.

that mathematics began in Africa all those years ago. Just imagine it: people sitting by a fire, studying the intricate carvings on a bone that once belonged to a baboon. However, while the Ishango bone is an intriguing artifact, it's a little too ambiguous to convince some experts.

So, as we trace the beginnings of mathematics, we have to look to another part of the world, Mesopotamia, or present-day Iraq, for stronger evidence. Part of Mesopotamia was known as Babylonia. The people living here around the 2nd century BCE came up with Babylonian mathematics, the earliest recognized system of mathematics in the world.

As is often the case, how early is up for debate. Some early accounting devices from Mesopotamia date back to around the 5th century BCE. But some of the clearest examples of mathematical thought are clay tablets from about 2000 BCE. Despite their age, the clay tablets are remarkably well preserved. We have hundreds of them.

detailed records of Babylonian mathematical knowledge. The Babylonians wrote in cuneiform script, making carvings in the clay while it was still wet. Then, the clay tablets were baked in an oven or left out in the sun. The tablets were hard and durable, allowing them to stand the test of time. From this evidence, we know that Babylonian mathematics was sophisticated,

The tablets reveal not just simple calculations or accounting records, but evidence of more advanced knowledge. The Babylonians used quadratic procedures which resembled algebra, and they had a good understanding of geometry. They knew the Pythagorean theorem before Pythagoras, and they even applied their mathematical knowledge to the skies. In the British Museum in London,

There's a cuneiform tablet which, to the untrained eye, looks a bit like a loaf of bread made from stone. It's covered in layered, triangular carvings. If you're able to read those carvings, you can learn the following: how to estimate the area under a curve by drawing a trapezoid. Now, this trapezoid is a concept that exists in abstract mathematical space,

and it can be used to make astronomical calculations. Amazingly, this Babylonian tablet allowed the user to study the movements of Jupiter. They could work out how far the planet had traveled in an interval of time. It's not clear exactly how old this particular tablet is or who the author was,

but it was found near a temple to Marduk, the Babylonian deity who was associated with the planet Jupiter. So we can imagine that the author of the tablets most likely had a connection to the temple. They were a mathematician, an astronomer, and perhaps a priest as well. As we explore the history of mathematics, we'll see that this is a common theme.

There are overlaps with many other areas of knowledge. Often, mathematicians have also been astronomers, physicists, and philosophers. They've recognized just how interconnected our universe is, and how a discovery can shine light on other fields of knowledge. So far, we've seen how the Babylonians were sophisticated mathematicians.

They were truly ahead of their time. Some of the concepts they used wouldn't appear in Europe until the 14th century. After the Babylonians, one of the next civilizations to make significant breakthroughs was the ancient Greeks. But first, let's take a brief look at a slightly earlier civilization, the ancient Egyptians. They too had advanced knowledge in many different fields.

And, thanks to papyrus records, we have some evidence. From around 3000 BCE to 300 BCE, the ancient Egyptians used mathematics for a range of practical, everyday situations. Maths was necessary for simple accounting and trade. People needed to be able to work with fractions and to use multiplication and division.

Geometry, on the other hand, could be helpful for architectural engineering, constructing a pyramid, for example. The most famous example of ancient Egyptian mathematics is a text dating to around 1550 BCE known as the Rhind Papyrus. Written in hieratic script,

The papyrus contains a range of problems involving divisions, multiplications, fractions, and geometry. These problems are often represented by plots of land, beer, or loaves of bread. For instance, 100 loaves of bread need to be divided among 10 men. Seven of the men get a single share,

The other three men, a boatman, a foreman, and a doorkeeper, each get a double share. The task is to express these share amounts as fractions. The author of the Ryan Papyrus was a scribe named Amos. He and other scribes may have written many other texts like these, early mathematics textbooks designed to help and educate their readers.

But unfortunately, papyrus isn't as tough as the clay tablets used by the Babylonians. We've only been left with a few surviving texts. And from what we can tell, the ancient Egyptians seem to have been more interested in practical applications of mathematics rather than exploring abstract concepts. The next big leap forward came with another civilization: the ancient Greeks.

Ancient Greece is often considered the true birthplace of mathematics. The English word "mathematics" derives from the Greek name, which in turn comes from a word meaning "learning" in a more general sense. In ancient Greece, certain forms of learning were particularly valued, such as geometry and arithmetic. And some of the most famous names in maths history were Greek,

Euclid, Pythagoras, and Archimedes, to name just a few. But what was so special about these mathematicians and their discoveries? Well, before the Greeks, people solved mathematical problems with inductive reasoning. They observed patterns to establish the rules or draw a conclusion. The Greeks, on the other hand, used deductive reasoning.

They started from a premise and used logic and known facts to reach conclusions. Here's a simple example of deductive reasoning. All numbers ending in 0 or 5 can be divided by 5. The number 35 ends with a 5. Therefore, this number must be divisible by 5.

The Greeks also used mathematical rigor to prove these conclusions. With mathematical rigor, there's a strict and thorough approach with no space for ambiguity. Through this new approach, the Greeks transformed mathematics into a logical, systematic discipline. When we compare ancient Greek mathematics to ancient Egyptian mathematics, for instance, we can clearly see the difference.

The Egyptians saw mathematics as a practical tool that could be used for commerce and everyday problem-solving. But in ancient Greece, the discipline of mathematics was elevated to something more abstract. Mathematicians were seeking fundamental truths. Another interesting development in ancient Greece was the fame of individual mathematicians.

On the whole, mathematicians from other ancient civilizations remained anonymous, but we've all heard of the big names in ancient Greece, such as Archimedes. There are a couple of reasons for this. Firstly, mathematics, like other intellectual pursuits, was highly valued in ancient Greek society.

Important work was recorded on papyrus scrolls and manuscripts and preserved in places like the Library of Alexandria. Mathematical knowledge was passed on, copied, and translated by later scholars, ensuring its survival. Also, ancient Greek culture placed more of an emphasis on the achievements of individuals. Their names and their stories were important too,

and certain anecdotes captured people's imaginations. That's how we know about the titans of Greek mathematics. So let's look at just a few examples. Pythagoras of Samos is perhaps best known for the Pythagorean theorem. This can be used to find out the lengths of the sides of a right-angled triangle.

While the theorem was known to earlier civilizations, such as the Babylonians, Pythagoras seems to have been the first to formally prove it. He was also a philosopher, whose teachings included the immortality of the soul. And over time, he became an almost legendary figure. He was associated with a wide range of intriguing beliefs and theories relating to astronomy,

numerology, and music theory. With Pythagoras, it can be hard to separate fact from myth or to identify which teachings were his and which belonged to his disciples. Nonetheless, he's gone down in mathematical history. Another important figure was Euclid of Alexandria. He's often referred to as the father of geometry.

And his mathematical treatise, known as "Elements," became an incredibly influential textbook. Euclid shaped understanding of geometry for centuries to come. Then, of course, there was Archimedes of Syracuse, a brilliant mathematician and also an inventor, physicist, engineer, and astronomer.

He's been described as a great civilization all by himself. Archimedes' mathematical achievements included a pioneering use of infinitesimals. Truly ahead of his time, his work was a precursor to integral calculus. Archimedes also came up with an estimation for the value of pi and methods for calculating areas and volume.

And then there's that famous story, the one about Archimedes leaping out of the bath and shouting, "Eureka!" That's ancient Greek for "I found it." Whether or not the story is true, Archimedes is given credit for discovering the physical law of buoyancy, as if being one of the most extraordinary mathematicians of all time wasn't enough.

So far, all the mathematicians we've mentioned have been men. But of course, there were female mathematicians too, like Hypatia of Alexandria. While Hypatia wasn't the first female mathematician, she's one of the best known. She's been remembered as an early feminist icon and, because of her tragic death, a martyr for knowledge.

Hypatia doesn't appear to have introduced her own theories. However, she was an influential teacher and scholar, writing commentaries on important mathematical texts, such as Apollonius' "Conics." Through her work, Hypatia made mathematics more accessible.

She and other scholars helped to ensure the survival of ancient Greek knowledge, laying the foundation for modern mathematics. For the rest of our historical journey, we'll mainly be focusing on Europe. However, it goes without saying that mathematicians from other cultures also made important contributions. India, for example, has a rich tradition of mathematics.

As early as the 7th century, and long before their European counterparts, Indian mathematicians were using zero in calculations, not just as a placeholder, but as a number in its own right. Brahmagupta, the mathematician credited with this innovation, also did pivotal work in algebra, arithmetic, geometry, and trigonometry.

And it was the Indians who first invented the numeral system we use today. Later, we'll find out how this system came to Europe. Over the centuries, Indian mathematical knowledge spread to the Arab world. Ancient Greek texts began to circulate too. In fact, it's Islamic scholars we have to thank for the preservation of mathematical works from ancient Greece.

Baghdad, in particular, was a cultural hub where scholars produced translations and commentaries on Greek texts. This helped to ensure the survival and then diffusion of Greek mathematical knowledge. Islamic scholars also made their own significant contributions to the field. For example, the Persian polymath Al-Khwarizmi wrote a hugely important treatise on algebra.

He was the first to treat it as a separate discipline. And another Persian polymath, Naseer al-Din al-Tusi, made countless contributions to different fields of knowledge, from trigonometry to planetary movements. In the West, these influential figures are often overlooked. But it's worth considering that in mathematics, as in all areas of knowledge,

People aren't working alone. They're influenced by their teachers, by all the books they've read, and all the works that influenced those books. So by the time we get to the European geniuses of mathematics, Leibniz in the 17th century, for instance, there's already a long history of scholarship preceding them. That goes for the ancient Greeks, too.

It's likely that Greek mathematicians were influenced by their not-too-distant neighbors, the Babylonians and the Egyptians. In one way or another, knowledge spreads and then evolves. In the Middle Ages, mathematics was flourishing in countries such as India, Iran, and Syria. But over in Europe, it was a slightly different story.

On the whole, education was limited in medieval Europe. Monasteries were centers of learning, but the focus tended to be on religion rather than science and mathematics. When it came to maths, students were mostly taught arithmetic, which had a practical function. Basic calculations were necessary for trade and commerce, but that was all.

From the 6th century, European knowledge of mathematics was based on the works of Boethius and other Latin writers. People were still studying maths, but there weren't really any significant advances. It wasn't until the 12th century that things slowly began to change. European scholars began to have contact with the Islamic world through their travels in Spain and Sicily.

For the first time, they had access to the full text of Euclid's Elements, among other works. Working from an Arabic translation, an English monk translated Euclid's text into Latin, thus making it more accessible. Other translations soon followed, and people in Europe became increasingly interested in mathematics.

At around the same time lived an Italian mathematician named Leonardo Fibonacci. Yes, the one who gave his name to Fibonacci numbers. As a young man, Fibonacci traveled to Algeria. There, he encountered Hindu-Arabic numerals. Those are the numerals we still use today: 1, 2, 3, and so on.

Pibonacci wrote a book about this numeral system, introducing it to Europe. Prior to this, Europe was still using Roman numerals and calculation tools like the abacus. But the Hindu-Arabic numeral system had many advantages, particularly when it came to calculations and record keeping. It would take a few centuries before Europeans fully converted to this new system

But change was on the horizon. During the Renaissance, in the 15th and 16th centuries, there was a wave of interest in all kinds of knowledge, from science to art. Mathematics was important too, but perhaps to a slightly lesser extent. The next phase of significant mathematical advances was still to come.

Nonetheless, people in Europe saw the value in a solid foundation in mathematics, from arithmetic to algebra. This knowledge was very helpful for trade and commerce. As a business owner, if you understood compound interest, you could make more money. Another interesting new development in the Renaissance was the connection between mathematics and art.

In Europe, painters in particular took an interest. They realized that a deeper understanding of geometry could help them to create more realistic artworks. Piero della Francesca was an Italian painter and mathematician who used his knowledge of geometric forms to great effect. One of his best-known paintings is "The Baptism of Christ,"

which now hangs in the National Gallery in London. It's a subject that's been portrayed in countless works of art, but there's something special about Piero della Francesca's masterpiece. Even at first glance, it evokes a sense of calm and order. Christ stands in the middle with his palms together as he's baptized by John the Baptist, who stands to the right.

There's a serene atmosphere, and the composition feels perfectly balanced. In part, that's because the painting is also a geometric study. The layout seems to be based on carefully positioned shapes: a square, a semicircle, and an equilateral triangle, which create a harmonious feel.

The influence of geometry and perspective studies can be seen in many works of Renaissance art. Just look at the refined realism of paintings by Leonardo da Vinci, Raphael, and Michelangelo. These artists recognized the wonderful potential of mathematics. It could be practical, it could be abstract, and it could be beautiful too.

The 16th and 17th centuries were the era of the scientific revolution in Europe. Scientists such as Galileo and Isaac Newton transformed our understanding of the universe. Around this time, there was also a mathematical renaissance, with important advances in many different areas. It's no coincidence. Maths and physics went together after all.

the two subjects were deeply intertwined. Isaac Newton's scientific achievements, such as his laws of motion and universal gravitation, were founded on an advanced understanding of mathematics. Also, by expanding on the work of his predecessors, Newton developed the foundations for calculus, the mathematical study of rates of change.

Calculus would later prove to be critical in so many different fields, from engineering to computer science. It's been described as "the most effective instrument for scientific investigation that mathematics has ever produced." But Newton shouldn't get all the credit for calculus. At around the same time, the German polymath Gottfried Leibniz was hard at work.

Independently from Newton, Leibniz also developed the ideas that would form the basis of both differential and integral calculus. While certain mathematicians were indeed exceptional, that's not quite the full picture. Each mathematician, however brilliant, builds on the work of others.

And in some cases, as with Newton and Leibniz, they come to similar conclusions based on their separate studies. So, who gets the credit? In the case of calculus, we could even point to another, less famous Isaac, Isaac Barrow. Barrow was a Cambridge professor whose work also helped to lay the foundations for calculus.

He was a great influence on one of his students, none other than Isaac Newton. In fact, recognizing Newton's potential, Barrow resigned his professorship and allowed Newton to take his place. History has remembered Newton, the polymath with the extraordinary career, while Barrow has been somewhat forgotten.

And once you start looking, the history of mathematics is full of such figures: less famous, perhaps less talented scholars, who nonetheless played a part. Still, some mathematicians really stood out. Leonhard Euler, for example. In the 18th century, Euler was one of the leading mathematicians in Europe.

He was incredibly prolific, producing 866 distinct works and more than 30,000 pages of material. What did Euler specialize in? What didn't he specialize in? Algebra, geometry, calculus, number theory, graph theory, and many different areas of physics, as well as music theory.

But if we have to single out just one of Euler's mathematical achievements, perhaps it's this: he successfully applied mathematics to real-world problems. For instance, take the Euler-Bernoulli beam theory, a model for calculating how beams bend when a load is applied. Understanding how beams behave under pressure is essential for structural engineering.

In order to build a structure on a grand scale, the Eiffel Tower, for example, or a Ferris wheel, beam theory is very helpful. There are countless other examples of Euler's influence and legacy, too many to list. But before we move on, let's consider another interesting topic: mathematical beauty. Euler came up with an equation known as Euler's identity.

It goes like this: e to the power of i times pi plus 1 is equal to zero. To the average person, it may not sound that special, but for mathematicians, it's an equation of remarkable beauty. Euler's identity has been described as exquisite and likened to a Shakespearean sonnet because of its elegant simplicity.

The equation also manages to unite different areas of mathematics in a surprising and satisfying way. The problem with mathematical beauty is that it's difficult to appreciate without some understanding of the subject. For most of us, it's less accessible than other forms of beauty. A painting, for example.

But for those who get it, mathematics offers all kinds of rewards, knowledge, and understanding at a higher level. And as it becomes more advanced, it also becomes truly awe-inspiring. As we leave Euler behind and move on to the 19th century, things begin to get even more complex, both the mathematics and the history.

By this point, there were so many important mathematicians working in such a wide variety of areas that it's hard to know who or what to single out. Also, some branches of mathematics were becoming more and more abstract. There was still a connection with science, of course, and practical applications.

But increasingly, the fields of maths and science were considered separate. This gave mathematicians the freedom to explore really abstract areas, such as abstract algebra and the beginnings of group theory. Some scholars were interested in pushing the boundaries of mathematics, exploring the limits of what was possible.

Certain discoveries had implications that went beyond mathematics, making people question the very nature of reality. For example, the 19th century saw the exploration of non-Euclidean geometry. You probably remember Euclid from earlier in our history, the ancient Greek mathematician who was considered the father of geometry.

Euclidean geometry is useful for understanding flat spaces, but non-Euclidean geometry describes curved spaces – the Earth, for instance – which is an example of spherical geometry. Non-Euclidean geometry – and, more specifically, Riemannian geometry – paved the way for Einstein's general theory of relativity.

It provided a mathematical framework for space-time, guiding us towards a deeper understanding of the universe. At universities and newly formed mathematical societies across Europe, mathematicians made rapid progress. Historically, mathematics had mainly been an intellectual pursuit, not a reliable source of income.

But by the 20th century, it had become a career path. With more mathematicians doing more research, collaborating, and specializing, it's no wonder that progress accelerated. Some discoveries were so conceptual and so challenging that they were almost impossible for non-mathematicians to grasp. But for those involved in the research, these were exciting times.

And throughout the 20th century, mathematicians took an ambitious approach, tackling some of the toughest problems in the field. There was plenty to keep them busy. In the year 1900, the German mathematician David Hilbert published a list of 23 mathematical problems. These came to be known as Hilbert's Problems.

Number 16, for example, is about the topology of algebraic curves and surfaces, a subject so tricky that even today the problem remains unsolved. Hilbert's list became a kind of research agenda for the 20th century. Those and other problems in mathematics encouraged innovation, and scholars continued to push boundaries.

The German-born French mathematician Alexander Grotendieck was one such pioneer. An eccentric genius, he revolutionized a branch of mathematics known as algebraic geometry. Grotendieck's work created a new foundation for this abstract, incredibly complex subject, and continues to influence mathematicians to this day.

The work of 20th century mathematicians has also had significant real-world consequences, not all of them positive. John von Neumann, a Hungarian mathematician, worked on the Manhattan Project during the Second World War. His expertise helped to improve the efficiency of the atom bomb. Von Neumann has one of the most extraordinary complicated legacies of any mathematician.

Among other things, he pioneered game theory and also helped to create the foundations for artificial intelligence. He was something of a visionary, understanding the incredible potential of computers as early as the 1940s. There are so many other areas of mathematics we could explore, the seemingly infinite discoveries of geniuses across the world throughout history. There are too many to count,

let alone explore in detail. But before we finish, let's take a moment to consider one last figure, a man considered not just one of the greatest mathematicians of the 20th century, but one of the greatest of all time. His name was Srinivasa Ramanujan. He was born in India in 1887, and he died in 1920 at the age of just 32.

but in his short life he achieved so much. To this day, mathematicians are still grappling with the implications of his work. Ramanujan came from a poor family and suffered from ill health throughout his life. As a mathematician, he was almost completely self-taught. While working as a clerk in Madras,

Ramanujan decided to contact an important mathematician at the University of Cambridge, Godfrey H. Hardy. When Hardy received Ramanujan's mathematical theorems, he was astonished. Ramanujan was clearly a genius. It seemed that he'd re-derived a century of European mathematics on his own.

Ramanujan was invited to Cambridge, where he spent three years collaborating with Hardy. During his short career, Ramanujan compiled as many as 4,000 theorems, many of which were completely original, including work on highly composite numbers and the partition function. According to Ramanujan, some of his mathematical insights came to him in dreams and were inspired by the divine.

He once said, "An equation for me has no meaning unless it expresses a thought of God." For modern-day mathematicians, Ramanujan's findings are inspiring, a source of mathematical wonder. And for physicists, Ramanujan might even help to uncover the mysteries of the universe.

To give one example, his work on modular forms has provided a mathematical foundation for string theory. String theory is a framework in theoretical physics. It says that everything in the universe is made up of tiny vibrating strings. And if it's correct, some experts believe it could open up all kinds of exciting possibilities.

including higher dimensional space, parallel universes, and maybe even time travel. Of course, at the moment, these are just theories. But it's a fascinating thought that Ramanujan's work could be the key to unlocking some of these mysteries. The story of mathematics is the story of brilliant minds exploring our limits.

the limits of the human mind and the known universe. We don't know exactly what the future holds, but one thing is certain: through their breathtaking intelligence and infinite ambition, mathematicians will help to take us there.