Friday, January 17, 2020

Pythagoras (570-495 BC), Euclid (~ 300 BC), Archimedes (287-212 BC) – Men of mathematics

Greek mathematicians had a significant influence in not only developing mathematical theory but also in embedding mathematics in the classical and modern curricula. Apart from Pythagoras, who not only set up a school but has strict rules around behaviour and the basis of all knowledge in numbers, they were not learning theorists as such but their influence on what has and is still taught is substantial.

Pythagoras (570-495 BC),

In Raphael’s fresco, Pythagoras is the figure writing in a book in the foreground on the left, surrounded by acolytes. He represents abstract mathematics and, in opposition to Socrates, the idea that learning is about the master transmitting immutable knowledge to their students. Both Plato and Aristotle are wary of Pythagoras, as he is a figure shrouded in myth. What we do know is that he was a teacher with students in something resembling a cult or fraternity, shrouded in secrecy. We also know that he had rules about not eating living things and beans, an early advocate of vegetarianism, along with lists of other rules, such as putting your right shoe on first, not looking backwards and so on. Pythagoreanism is a school with students, perhaps more akin to disciples, but also a school of thought. He gives us the archetype for the charismatic teacher and leader, with followers who engaged in a communal lifestyle.
Pythagoras’s theorem, although well known, may not have originated with him. It was known and used centuries earlier by the Babylonians and Indians. That is not to say that he didn’t introduce to the Greeks. But mathematics is said to have lain at the heart of his system, especially geometry. The number 10 (1+2+3+4) was of mystical significance. Other such as Hippasus moved number theory on to irrational numbers, like the square root of two, expanded into a theory of irrational numbers by Eudoxus. 
He is also famous for having discovered the mathematical nature of musical intervals as having numerical ratios. If number lay behind music, does number lie behind all phenomena? There is speculation that he also applied this idea to the movement of the planets. His status during the Middle Ages and influence on Copernicus, Kepler and Newton have ensured his fame.

Euclid (~ 300 BC),

A parallel figure in the foreground of Raphael’s fresco, on the right is Euclid, from the Greek colony Alexandria in Egypt, by far the most important Greek mathematician, leaning down to demonstrate his mathematical proofs, on what looks like a slate, with callipers, where the students are in discussion, working through the proofs in their heads. Again, this contrast exists between the didactic teaching of a canon and the more learner-centric view of the learner as someone who has to learn by doing and reflection. 
Elements, in 13 books, is his most famous work, where his theorems and, more importantly, proofs were deduced from axioms. Familiar examples include the proof that the angles of a triangle add up to180 degrees and Pythagors’s Theorem. It is this logical rigour that is remarkable, influencing the entire history of mathematics and science. It was used as the main textbook in mathematics for over 2000 years, well into the 20th century and all University students for centuries used this book as part of the quadrivium.
One fascinating feature of Euclid’s Elements, was the first ever algorithm in print, a method to calculate the Greatest Common denominators for any given number, an oft-quoted forerunner for the current age of algorithms.
Beyond this he wrote on the rigour of mathematical proof, conic sections, the geometry of spheres and number theory. In his Phaenomena, Euclid aims at astronomy with a treatment of spherical geometry. This was

Archimedes (287-212 BC)

Eureka! Is the word most associated with him, where he supposedly submerges a Golden Crown in a bath of water, measured the displaces volume. The next step, where he divided the mass of the golden crown by its volume, determined whether it was silver or gold. But the story does not appear in any of Archimedes writings.
His reputation rests on his mathematics but also on the practical application of this mathematics. In addition to explanations of levers and fluid mechanics, he is said to have invented the Archimedes Screw for lifting irrigation water, compound pulleys, and many war machines, including an optical device to focus the sun’s rays on invading Roman ships and a crane and claw for sinking ships and improved catapults.
It is his work on circles, spheres and cylinders, parabolas, centres of gravity, law of the lever, curves, conoids, spheroids and floating bodies, along with that famous number ‘pi’, that has ensured his lasting fame. He also appears to have anticipated modern calculus by using a method of exhaustion, increasing the sides of a polygon towards a complete representation of the circle. Archimedes is arguably the greatest of the Greek mathematicians. In the same period, Eraytosthenes (~250BC) used geometry to estimate the circumference of the earth. He noticed that the sun shone down a well in Aswan at midday. On the same day of the year he also measures the shadow of the sun from a column further north in Alexandria, From this he ingeniously calculated the circumference of the earth. 


Pythagoras, Euclid and Archimedes, along with other Greek mathematicians and astronomers put mathematics, mostly geometry, at the heart of the western educational system. It was an indispensable feature for many of the major Greek thinkers who saw it as the foundation for rigorous thinking about the world. They gave mathematics a status in Western thought that has never waned. Its emphasis on geometry, proof and the need for quantitative rigour lies at the heart of later scientific revolutions.
They also made advances in what we would now call engineering, the practical application of science and mathematics into machines and architecture. Beyond this astronomy also benefited from their mathematical bent.
Mathematics is unarguably a subject that needs to be taught and learnt. It has given us advances in medicine, finance, technology, economics, psychology, astronomy and science However, one could argue that its status as a compulsory subject is exaggerated in terms of supposed needs and transferable skills. Roger Schank argues that we have no real need to teach areas of abstract mathematics to most children, such as algebra, quadratic equations and surds, as they are unlikely to ever be used in the real world. When was the last time you used Pythagoras’s Theorem, if ever? The focus on abstract, as opposed to the basics, problem solving, reading data,  and maths at work and in the real world, has become endemic. There is a sense in which mathematics has a gained status in the curriculum beyond its actual benefits.
The OECD PISA results, who chose as their first target maths, have become a major international attraction for educators, and have sparked off an annual educational ‘international arms race’. Yet maths has never been the sole touchstone for being 'smart' or 'employable'. In one sense, important as the subject is, maths has become a totem in the curriculum, hard to learn, hard to teach and easy to test, in other words the ideal recipe for mass failure.  Additionally, We do not actually live in a more mathematical world. We live in a world where most maths is done by calculators, computers and machines, or a relatively small number of experts. The vast majority of us need little actual maths, other than ‘functional maths’. To funnel all young people into a path that demands a mostly irrelevant, maths curriculum is to turn them off school and learning. This obsession with maths may, mathematically, be the very things that lowers our general educational attainment. In many countries, education policy is rooted in, and firmly targeted at, the PISA results. It is used by politicians as an instrument of convenience.  Both left and right now use the ‘sputnik’ myth to chase their own agendas – more state funding or more privatisation. This, some claim, is a shame, as it may be unhelpful to have yet another dysfunctional, deficit debate in education. 


Kirk, G.S., Raven, J.E. and Schofield, M., 1983. The presocratic philosophers: a critical history with a selcetion of texts. Cambridge University Press.
Stewart, I., 2008. The Story of Mathematics: From Babylonian Numerals to Chaos Theory. Quercus.

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