- How A Mathematical Discovery Tormented the Ancient World -
Orton Academy’s Head of Research Dr. Jane Orton reflects on the mathematical discovery that cost one man his life, inspired Plato and shocked the Greek world.
The Peloponnesian War is over. Athens has been defeated and extreme Spartan-backed oligarchs will soon massacre nearby town Eleusis. It is a mild Mediterranean winter, shortly before the mystery rites and you, a slave-boy, are pulled away from your work in the gymnasium by an eccentric philosopher with dirty feet who looks like a sting ray.
The philosopher, well aware of the fact that you have never studied mathematics before, proceeds to question you about the solution to a mathematical problem: how to double the area of a square. What you do next will be immortalised as part of the dialogue in Plato’s Meno, and used as evidence for innate knowledge, the possibility of life before birth and the immortality of the soul.
On Diagrams and Slave-Boys
Even those of us who studied mathematics at school might be thrown by this: the philosopher is, after all, Socrates, one of the most notorious characters in Athens and perhaps the most famous philosopher in the history of the world. However, Socrates is convinced that the boy already knows the solution, and begins to question him on his beliefs about the problem.
Socrates draws a two ft by two ft square on the ground, then asks the boy questions about it. The boy works out that the area is four feet and that a square with double the area will be eight feet; but incorrectly thinks that the sides will be four feet long. Socrates adds in three identical squares, and the boy sees it makes a single large square of sixteen feet. He discovers through the same method that the sides of an 8 ft square cannot be three feet long. Finally, once the philosopher draws in diagonal lines along each small square, the boy comes to know the solution: use the diagonal of the original square as a side of the new square.
This, we are to understand, proves that we are born with knowledge, and that it only takes the correct method of question-and-answer to bring that knowledge to light. Socrates maintains that he never once teaches the boy anything: he only asks questions (although some philosophers maintain that he ‘cheats’ by feeding the boy information).
Not only this, but the boy’s task is made harder by the fact that the problem requires a diagram for its solution. As the length of the diagonal and the side of the square do not share a common measure - they are incommensurable - the only way for the boy to answer the question is to point to the line on a diagram.
The Terror of the Infinite
In fact, the issue of incommensurability posed a challenge for even the most advanced Greek mathematicians, let alone a slave-boy who was new to the discipline. The problem of incommensurables can be linked to the idea of infinity, which undermined the basic doctrine of the cult of Pythagoras. The Pythagorean who made this known is said to have drowned at sea in a shipwreck: divine retribution for besmirching the cult’s beliefs.
This is not the only time the concept of infinity has caused controversy in the intellectual life of Ancient Greece. The same worry can be found in the famous paradoxes of Zeno, one of which claims that not even Achilles, the world’s fastest runner, could catch up with a tortoise who has a head start. Zeno’s point is that, if space is continuous, it must be divisible into an infinite number of magnitudes. Therefore, to move, Achilles must traverse an infinite number of distances, which is impossible because it involves the completion of an infinite number of tasks. In this way, it is impossible for Achilles to move at all, because even the half way point and the quarter way point and so on are divisible ad infinitum.
Greek fear of the infinite changed the course of mathematical history, with Greek mathematics avoiding completed infinities and limits in Greek mathematical proofs. Geometry overtook arithmetic as the intuitive foundation for mathematics: after all, irrational lengths like the diagonal of a square are clear and finite, unlike arithmetic irrational numbers, which are infinite, counter-intuitive and uncomfortable for those with a fear of infinity.
Plato and Mathematical Beauty
Plato himself was very familiar with the controversies of the mathematics of his day, but he saw its great potential in training the mind towards the highest form of enlightenment. In the ‘divided line’ passage of the Republic, Plato tells us that dianoetic, or mathematical, reasoning has two distinctive qualities: the use of hypotheses and the reliance on imagery. Some scholars have even argued that Plato held geometry in low esteem because of its use of diagrams, which are something of a black sheep in Plato’s epistemology. Yet thousands of years before the birth of modern theoretical physics, Plato made the shocking claim that reality itself is made up of geometrical entities.
Mathematical reasoning for Plato is second only to noēsis, or pure philosophy, in epistemological merit. Noēsis, the highest form of reasoning, would require a grasp on the Form, the perfect idea of the object of inquiry. This grasp of the Form eludes most of us, but in the absence of such knowledge, the dianoetic image or mathematical diagram provides us with a particular instance to study. Dianoetic thought actually ‘hunts’ Forms indirectly through the tools of hypothesis and imagery.
Socrates’ square in the Meno fulfils Plato’s requirements for a dianoetic image: it is a sensible object used as an image of a Form (of the Square). Thus, although the dianoetic image can sometimes have a sensible element, its qualities are primarily intelligible. When Socrates draws his figures on the ground in the Meno, we understand that we are meant to think of the properties of the ideal Form. The diagram also enables the boy to give a finite answer, something that would have been impossible without it. Dianoia thus provides a stepping stone to the Form through the diagram.
When we consider the slave-boy passage in the context of the history of mathematics and Plato’s scheme in the divided line, it becomes clear that Plato is not criticising geometers or their use of diagrams. Rather, in the absence of a grasp on the Forms, we need to use the images as stepping-stones. In this case, the diagram or dianoetic image is the best available tool with which to do so. Geometry, far from being a black sheep, thus becomes something of a hero.
Find Out More
To find out more about Zeno’s paradoxes, Pythagorean esotericism and the catastrophe of irrational numbers and incommensurables, take a look at our course on Mathematics and Philosophy in the Ancient World. It’s ideal for someone who has taken an introductory course in Ancient Philosophy, those who like a good historical mystery or mathematicians wanting to know more about the history of their discipline.
For those wanting to take a deeper look at Plato’s philosophy, take a look at Platonic Dialogue Studies, in which we read dialogues such as Plato’s Meno in English translation, or our philosophy course on Plato, in which we discuss Plato’s philosophical method and epistemology, his theory of Forms and his thoughts on beauty, piety, courage, lying and pleasure.
Finally, for those who want to take on Zeno’s pesky paradoxes, why not try our interdisciplinary course, How to Win an Argument?
These are templates of possible routes of study and can be combined, adapted, or designed from scratch to suit your interests and goals. Dr. Orton will work with you to design a course of private tutorials tailored to your needs, ability and schedule – whether you are undertaking your own research for an independent project, writing a book or simply have a personal interest. Click the link to find out what it’s like to work with her!
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