It was a familiar question: “Why are we being forced to do this?”

The topic was trigonometry, the students an engaged bunch who nonetheless were struggling to see where the cosine rule might apply in the futures they were imagining for themselves.

I was about to give my standard- issue answer: “You might need this if [insert tenuous-sounding connection to architecture and design].” But it suddenly sounded unsatisfactory.

I opted for a different tack. I’d just been asked to present some thoughts on the connections between mathematics and classical civilisation. So, instead, I told them that, if they wanted a more in-depth answer, they should come along that lunchtime.

Now I just had to come up with something.

A colleague offered an easy way through: “Just talk about Pythagoras.” In short, put on a toga and ramble about triangles. But what I wanted to explore was how we have lost something of the mathematical awe and wonder that was absolutely central to the development of classical thought. And, if I could restore some of that, I might have a better chance of my students being captivated by a subject and appreciating that it might do more than help them calculate their future wages.

In fact, Pythagoras is a good place to start, because he represents the point I wanted to make very well. It is thought that he got his nickname because it was said that “he spoke the truth no less than the Pythia” - that is, the high priest at the oracle of Delphi. In other words, the real Pythagoras was less a numbers guy than a wisdom guy.

More than a theorem

He was educated in Egypt and learned much of his mathematics there. When he returned to Greece, he formed a religious community heavily influenced by these travels. His followers saw mathematics as a means of exploring “celestial harmony”. Mathematical understanding and what we might call spiritual philosophy went hand in hand.

The Egyptians who influenced Pythagoras had settled along the Nile Valley from around 6,000BC. They used a version of base 10 (the decimal system) for their arithmetic and, as they began to trade, developed a system of representing fractions using parts of the Eye of Horus, an Ancient Egyptian symbol of royal power and good health.

But their crowning achievement was architectural. The soaring pyramids - with their perfect right-angled triangles - were created using a system of knotted ropes that show that the Ancient Egyptians clearly understood the principle that has now come to be known as Pythagoras’ theorem. Yet the placement of pyramids was also informed by astronomical observations. These would have been taken by people who were hybrid mathematician-priests, astrologers who believed that the movements of the stars communicated a divine message.

So the pyramids were physical manifestations of religious beliefs, with one theory postulating that they were a kind of resurrection machine, which launched a dead pharaoh into the turning stars above, and thus into the realm of the gods.

The Egyptians themselves had inherited much of their mathematics from the earlier Sumer-Mesopotamian era - what is often called “the cradle of civilisation”. It is here that we see the beginnings of writing, the wheel, the arch and the plough.

And we have excellent evidence (in clay cuneiform tablets) of their use of arithmetic. This is the bureaucratic mathematics of statecraft: measuring fields and collecting taxes. And, sadly, this is the dimension of the subject that almost exclusively dominates how the subject is presented and taught now. It is a tool for calculating percentages, for adding up money, for working out lengths of sides and who owes what.

But the Sumerians were doing far more than this. Like the Egyptians later, they built upwards as a means of expressing their beliefs.

The ziggurats they constructed would have been awe-inspiring. The fate of one such towering structure has been passed down to us in the story of the Tower of Babel, set down in the Book of Genesis. What real-life events might be behind this we don’t know, but the message of the myth remains: the Babylonians wanted to make a tower that reached right up to the heavens, so that they might make a name for themselves. God sees them aiming at this and panics: “If as one people speaking the same language they have begun to do this, then nothing they plan to do will be impossible for them” (Genesis xi, 6). God scatters them, punishing their attempt to build a stairway to heaven.

All of this informs the sect that Pythagoras forms on his return to Greece. Mathematics is not just the arithmetic of bureaucracy, but the underlying language of the universe, which can open a path to the divine. Plato was heavily influence by Pythagoras, and was himself a renowned teacher of mathematics - very likely instructing Eudoxus of Cnidus, the greatest mathematician in classical Greece, who contributed widely to Euclid’s Elements.

For Plato, philosophy and mathematics were common pursuits. In one of the most famous sections of his masterful Republic, Plato outlines his thoughts on how humankind can discover truth. He invites us to imagine a cave in which we are chained. There is a fire lit behind us, and what we see in our unenlightened state isn’t true reality, but the mere flickering of the shadows of what is real projected on to the wall of the cave in front of us. These shadows are a constructed reality, which the prisoners have no way of critiquing.

How, then, might we attain our liberty and experience the true nature of things? To climb out of the cave, beyond the shadows and into the sun, we need to become enlightened. And to do this we need not some divine intervention to reach down and lift us, but to elevate our own minds through learning.

It is through reasoning that we will find our liberty, our full human flourishing. For Plato, this reasoning had two components on an inseparable continuum: philosophy and mathematics. Plato’s worldview - inherited from Pythagoras and with a direct lineage back through Egyptian and Sumerian thought - was that mathematics was a means by which the true nature of the universe could be uncovered. Far from being a ponderous tool for measurement and taxation, it was the foundational language of pure logic, which could reveal the mind of the gods. If humans were going to scale the metaphorical Olympus and become godlike, then mathematical reasoning was the equipment that would assure their progress upwards.

The mathematician G H Hardy wrote: “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.” He might equally have said that the unifying language of the Sumerians, the Egyptians and the Greeks was mathematics.

Like God’s scattering of the Tower of Babel’s builders, however, classical civilisation fell and the West entered into the Dark Ages after the collapse of the Roman Empire. It was left to Indian and Islamic mathematicians to sustain progress (developing al-jabr - algebra - as a way of “reuniting broken parts”). As trade developed with the East, these ideas filtered back west, sparking the Renaissance in Tuscany - the rebirth of knowledge.

Central to this was the displacement of medieval Christian ideas by Neo-Platonism - with its roots all the way back to the ziggurat project of Babel that, through the knowledge of all things, humankind could elevate itself. Godliness was not attained through an act of divine salvation but a building upwards of knowledge through logical reason, science and mathematics.

Nowhere is this idea more beautifully represented than in a coded message left by the greatest Renaissance artist in the most venerated place in the Christian world: the Vatican. On the ceiling of the Sistine Chapel, Michelangelo - part of the secret network of Neo-Platonists - painted The Creation of Adam. In this extraordinary fresco, Adam is represented as younger than God, more muscular than God and - in vertical terms - only slightly below God’s level. The fingertips of the two figures are almost touching.

How then, Michelangelo seems to be suggesting, might this tiny gap between muscular, vigorous humanity and an ageing God be closed? His answer is right there in the painting: the figure of God is reaching out through a structure that, art historians and medics have concluded, is a deliberate representation of the human brain. The pursuit of enlightenment is not through divine salvation, but through science and reason and mathematical investigation.

This was taken very seriously by Renaissance thinkers. They raided Ancient Greece for an alternative to the religion that had dominated Western thought for centuries - had been the fire that had thrown shadows on to the cave wall of the Dark Ages for too long. Johannes Kepler, one of the greatest figures in modern science, would describe himself as a Pythagorean, believing that the universe was founded on mathematical relationships. He considered his calculations about the orbits of distant planets as no less than “thinking God’s thoughts after him”.

Out of the cave

You will struggle to find this on a syllabus. The teaching of mathematics has been impoverished through an almost exclusive focus on the bureaucratic skills of measuring, accounting, adding up.

We should not be promoting astrology, of course, nor expecting pupils to do super-complex astronomical calculations. But, if students are truly going to be inspired in our classrooms, should they not know some of this history, something of the celestial drives that have been absolutely foundational to the development of the subject for thousands of years?

My lunchtime session attracted around 25 students, all of whom - I am happy to report in this wholly unscientific sample - have since expressed a keenness to go deeper into these questions of the long history of mathematics. Their experience of the subject is currently rather too akin to the chained people in the cave, watching a dim shadow of the rich things it really has to offer.

In our times, when solid answers are hard to come by, it’s my contention that expanding the horizon of the kinds of deep and mystical questions that mathematics has set out to answer through history can only be good for students, who have to grind through the rather flat diet of arithmetic we currently offer. We should be telling these stories. In other words, it’s time to put the real Pythagoras back into the curriculum.

Kester Brewin teaches maths in south-east London. He was a consultant for BBC Education, and is the author of a number of books on culture and religion. He tweets @kesterbrewin

This article originally appeared in the 22 August 2019 issue under the headline “Maths students, it’s time to enlighten up”