Lecture 7. Plato and his Academy

Figure 7.1 Plato

The most important Greek philosopher Plato (427 B.C. - 347 B.C.) lived in Greece and is known as one of the greatest philosophers of all time. He founded the Academy in Athens, an institution devoted to research and instruction in philosophy and sciences. His works on philosophy, politics and mathematics were very influential and laid the foundations for Euclid’s systematic approach to mathematics. The British mathematician and philoso- pher Alfred North Whitehead (1861-1947) once remarked that “the safest generalization that can be made about the history of western philosophy is that it is all a series of footnotes to Plato.” 1

Plato was the son of wealthy and influential Athenian parents. He was named Aristocles after his grandfather, but his wrestling coach, Ariston of Argos, dubbed him “Platon,” meaning “broad,” on account of his robust figure. Plato was originally a student of Socrates2, and as such was influenced by his teacher’s unjustified death. When the master died, Plato was very upset (he was 30 years old), and he began to write down some of the conversations he had heard from his mentor. Almost everything we know about Socrates came from what Plato wrote. 1Is God a mathematician? Mario Livio, Simon & Schuster Paperbacks, New York-London-Toronto- Sydney, 2010, p. 29. 2Socrates (469 B.C.-399 B.C.) was a Classical Greek philosopher. Credited as one of the founders of Western philosophy.

45 Figure 7.2 Socrates and Plato; The Death of Socrates by Jacques Louis David, 1787

Plato’s Academy After a while, Plato began to write down his own ideas about phi- losophy instead of just Socrates’. He travelled to Egypt and Italy to study with students of Pythagoras. Eventually he returned to Athens and established his own school of philosophy at the Academy (This is the origin of the word “academy” used today). Though the Aca- demic club was exclusive, not open to the public, it did not, during at least Plato’s time, charge fees for membership. Therefore, at the time it was not a “school” in the sense of a clear distinction between teachers and students, or even a formal curriculum. There was, however, a distinction between senior and junior members. The Platonic School lasted nine hundred years until it was closed under the order of Christian emperor Justinian in A.D. 529.

Figure 7.3 Plato’s Academy; the modern Academy of Athens, next to the University of Athens.

46 Plato and Mathematics Plato was not a mathematician, but he made many mathe- matical discoveries. His belief and enthusiasm for the subject and his understanding of the universe encouraged mathematicians to pursue mathematics.

Plato said, Suppose there is a cave and inside there are some men who are chained up along the wall so that they can only see the wall and shadows of what is going on outside the cave. These men may think that the shadows were real. Suppose one day, one of these men escaped to outside world, and saw what real tree, real grass, and real things looked like. If he went back to the cave and told the other men what he had seen, would they believe him ? Plato said that we are like those men in the cave. We believe that we understand the real world but the reality is that we can only see the shadows.

Plato allegory cave

For Plato, human beings live in a world of visible and intelligible things. The visible world is what surrounds us: what we see, we hear and we feel, which is a world of change and uncertainty. It likes the “cave”. The intelligible world is made up of unchanging products of human reason: things arising from reason along such as abstract definition of mathematics, which is the world of reality. It likes the outside world of the “cave”. One of Plato’s goals is to help us understand the real world better. Consequently, one of the tasks is to understand mathematics.

47 Therefore he enforced the idea that mathematical objects should not be thought of as real things, but as ideal objects of the mind. And he made a sharp distinction between pure mathematics, which elevates the mind, and the mere solving of practical problems, which he relegated to the lowly realms of commerce. For students enrolled there, Plato tried both to pass on the heritage of a Socratic style of thinking and to guide their progress through mathematical learning to the achievement of abstract philosophical truth. In his most famous work, The Republic, Plato discussed the education that should be received by the philosopher-kings, the ideal rulers of a state. He wrote: those who have a natural talent for calculation are generally quick-witted at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would have been...... arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up...... the study of geometry (i.e., theoretical not practical) is for drawing the soul towards truth. In Gorgias, a Socratic dialogues written by him, Plato wrote: “Geometric equality is of great importance among gods and men.” In the dialogue Timaeus, Plato wrote “the creator god uses mathematics to fashion the world,” and in the Republic, he wrote “knowledge of mathematics is taken to be a crucial step on the pathway to knowing the divine forms.” For him, the mathematical character of the world is simply a consequence of the fact that “God always geometrizes.”

Figure 7.4 Plato’s Academy Grove in Athens, Greece

48 At the Academy, the mathematical part of the education was to consist of five subjects: arithmetic, plane geometry, solid geometry, astronomy and harmonies (music).

Brought in the best mathematicians Plato brought the best mathematicians of his day to teach and do research at the Academy. As a result, almost all of the important mathematical work of the fourth century was done by the friends and pupils of Plato. To name some of his students, we give a short list: √ √ √ • Theodorus3 (proved 3, 5, ··· , 17 are irrational numbers); • Menaechmus4(the first to investigate the ellipse, parabola and hyperbola as sections of a cone); • Aristotle (philosopher who made important contributions by systemizing deductive logic. See Lecture 9); • Euclid (father of geometry. See Lecture 10). • Theaetetus5; • Eudoxus (see Lecture 8); • Archytas 6

First to systematize the rules of the rigorous proof Plato affirmed the desirability of a deductive organization of knowledge. Plato was the first to systematize the rules of rigorous demonstration, and his followers were supposed to have arranged theorems in logical order.

The so-called Platonism is the form of realism that suggests that mathematical entities are abstract, have no causal properties, and are eternal and unchanging. The term Platonism is used because such a view is seen to parallel Plato’s belief in a “World of Ideas”: the everyday world can only imperfectly approximate an unchanging, ultimate reality.

The major problems of mathematical platonism are these: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world,

3Theodorus of Cyrene was a Greek mathematician of the 5th century B.C. He was admired by Plato who mentions him in several of his works. 4Menaechmus, 380-320 B.C., was an ancient Greek mathematician who was known for his friendship with Plato. 5Theaetetus of Athens, 417 B.C.-369 B.C., was a classical Greek mathematician. 6Archytas, 428-347 B.C., was an Ancient Greek mathematician.

49 completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be Ultimate Ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.

Figure 7.5 “No one may enter here (The Academy) who is ignorant of mathematics.”—– Plato’s Academy

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