Plato
427 BC – 347 BC

I believe it is important to note how we know what we know about Plato. The knowledge we have on Plato comes from letters, supposedly written by him. It is important that we consider the possibilities: 1.) that the letters were written by him and are, therefore accurate of his life, 2.) that the letters were written by someone who had accurate knowledge of his life, and 3.) that the letters were made up by someone as pure fiction. Hopefully, the last of which is not the case.

It is almost certain that Plato was friends with Socrates. In fact, the death of Socrates had a profound affect on Plato; Plato decided to no longer pursue a career in politics in Athens. When Socrates died, Plato decided to travel to Egypt, Sicily, and Italy. He introduced the Greeks to the water clock, which he learned of in Egypt. He learned of the work of Pythagoras while in Italy. This made him begin to appreciate the value of mathematics. The things he learned from studying Pythagoras’s work helped him form his idea that “the reality which scientific thought is seeking must be expressible in mathematical terms, mathematics being the most precise and definite kind of thinking of which we are capable.”

After serving in the military (and not for the first time), Plato returned to Athens and founded his Academy (about 387 BC). This academy was an institution dedicated to research and instruction of philosophy and the sciences. Through the academy, Plato had hopes of training young men there who would eventually become statesmen. He thought that, through his own training, these men would improve the political leadership of Greece once they became involved in the politics.

Plato considered mathematical objects as perfect forms. For example, a line “is an object having length but no breadth.” No matter how thin we make a line, it will not be this perfect mathematical form, because it will always have breadth. He discusses how objects in the real world try to be like their perfect forms but never quite reach it. “The instance taken there is the mathematical relation of equality, and the contrast is drawn between the absolute equality we think of in mathematics and the rough, approximate equality which is what we have to be content with in dealing with objects with our senses.” Plato did not make any important mathematical discoveries, but he believed that mathematics provides the finest training for the mind. This belief was important to the development of mathematics.

In fact, over the door of his academy, the sentence “Let no one unversed in geometry enter here.” was written. Within the academy, Plato concentrated on the idea of proof, and insisted on clear and concise definitions and hypotheses. This is what laid the foundations of Euclid’s systematic approach to mathematics.

The most important mathematical work of the 4th century was done by friends or students of Plato. For example, the first students of conic sections and the creator of solid geometry were members of Plato’s academy. Plato’s name is also connected with Platonic solids.

It is interesting to note what Plato believed the first ten years of a person’s education should consist of: arithmetic, plane and solid geometry, astronomy, and harmonics.

References:

Development and History. 3rd ed. New York: W.H. Freeman and Company,
1993. 6-19.

National Council of Teachers of Mathematics, 1969.