Lecture 3. Thales, the Originator of the Deductive Method

Figure 3.1 Thales of Miletus

Thales Thales lived around 624 B.C.-547 B.C. Most consider him to be Milesian (Miletus, Asia Minor, and now Turkey). He came from a distinguished family. Thales was the ﬁrst known Greek philosopher, scientist, mathematician, and also founder of the Ionian school of ancient Greek thinkers. Thales of Miletus was considered by Plato to be the ﬁrst of the Seven Wise Men of ancient Greece, 1 the only mathematician to be thus honored.

Very long time ago, it was believed that there were many gods; when they were pleased, they treat mankind well: good weather and peace, otherwise, they caused drought, wars, pestilence and other disaster and epidemics. For example, there were gods of love, of war, of the sun, of the oceans, of rain and thunderstorms. These were no way that people could understand these inscrutable gods and there was nothing people can do about it except waiting for the god’s mercy. But with Thales of Miletus, that began to change. Thales

1Seven Wise Men: Although such listings diﬀered widely, a usual one included Bias, Chillon, Cleobulus, Periander, Pittacus, and Solon.

18 developed the following point of views: natural follows consistent principles that could be deciphered. And so begun the long process of replacing the notion of the reign of gods with the concept of a universe that is governed by laws of nature, and created according to a blueprint we could someday learn to read. 2 From him, the idea of mathematical proof began to emerge. As pointed out by Aristotle, “to Thales the primary question was not what do we know, but how do we know it.”

Thales went to Egypt to study science and mathematics. When he came back to Miletus, his unusual mathematical abilities surprised his contemporaries; perhaps he had reﬁned the Egyptian methods.

The Ionian School The ﬁrst of the classical Greek schools, the Ionian School, was founded by Thales in Miletus. We do not know detailed information about this school due to the lack of records, but we do know that Thales did have two pupils: the philosophers Anaximander (c.610-c.547 B.C.) 3 and Anaximenes (550-480 B.C.). Anaxagoras (500-428 B.C.) belonged to his school (see Chapter 6). Also, Pythagoras (585-500 B.C.) learned mathematics from Thales (see Chapter 4).

Thales said: “I will be suﬃciently rewarded if when telling it to others you will not claim the discovery as your own, but will say it was mine.”4

Philosophy and water Thales’ most important contribution to western civilization was his attempt to give rational explanations for physical phenomena. It was Thales’ belief that there were explanations for all phenomena in the universe.

What is the origin of the world? This is the first question of Greek philosophy. Thales thought it is water, and he is often credited with trying to explain everything in terms of water (either everything is made of water, or everything came from water). He said: “Water is the principle, or the element, of things. All things are water.”5 Therefore, water is the origin of all things, and all things ultimately are water. The diversity of the world that we can see is the result of the modification of water to appear as something other than water. Somehow this process of modification (solidification and liquidification) causes earthquakes, whirlwinds and the movement of the stars. Later Aristotle attributed the following four views to Thales: 2Stephen Hawking and L. Mlodinow, The Grand Design, Bantom Books, New York, 2010, p.17. 3see Figure 4.2 in Chapter 4. 4Quoted in H Eves in Mathematical Circles (Boston 1969). 5Quoted in Plutarch, Placita Philosophorum.

19 • The earth rests on water. • Water is the archˆe(principle) of all things. • The magnet has a soul. • All things are full of gods.

According to what Aristotle and other philosophers said about Thales’ view, the rea- soning process behind his conclusion that water is the ﬁrst material principle seems to be unknown, so that it becomes a matter of conjecture. Aristotle hypothesizes that the reason that Thales postulates that water is the ﬁrst material principle is because water and heat are the basis of all life, and water is the basis of heat. Aristotle says that Thales arrived at his view “perhaps from seeing that the nutriment of all things is moist, and that heat itself is generated from the moist and kept alive by it. He got his notion from this fact, and from the fact that the seeds of all things have a moist nature, and that water is the origin of the nature of moist things.”

An astronomer Thales had predicted an eclipse of the Sun in 585 B.C., reported by the historian Herodotus (484-425 B.C.) in his History. There was a war between Lydians and Medes, which endured for five years. Thales predicted that, in the sixth year, a solar eclipse would occur on 28 May, 585 B.C. On that day, quoted from History, “Day was all of a sudden changed into night. This event had been foretold by Thales, the Milesian, who forewarned the Ionians of it, fixing for it the very year in which it took place. The Medes and Lydians, when they observed the change, ceased fighting, and were alike anxious to have terms of peace agreed on.” Thales’ prediction spontaneously halted a battle in progress.

Figure 3.2 Solar eclipse

20 Thales was extremely focused on astronomy. Plato told a story of how Thales was once walking and at the same time studying the stars when he fell into a well. A Thracian slave girl mocked him for being able to see stars in the sky but unable to see what was at his feet. This anecdote was often quoted in antiquity to illustrate the impractical nature of scholars.

Wisdom and fortune Thales was a poor man in spite of his wisdom. He was questioned about the value of his philosophy: why his wisdom is good if it cannot pay the rent?

“He was reproached for his poverty, which was supposed to show that philosophy is of no use. According to the story, he knew by his skills in the stars while it was yet winter that there would be a great harvest of olives in the coming year; so, having a little money, he gave deposits for the use of all olive-presses in Chios and Miletus, which he hired at a low price because no one bid against him.

When the harvest time came, and many were wanted all at once and of a sudden, he let them out at any rate which he pleased, and made a quantity of money. Thus he showed the world that philosophers can be rich if they like, but that their ambition is of another sort.”

—– [from Politics, Aristotle]

Thales and the mule Another favorite story is related by Aesop. It appears that while transporting salt which was loaded on Thales’s mules, one of the animals slipped in a stream. The mule’s load of salt was slightly dissolved by the water so that it’s load became lightened. The beast deliberately repeated the trick to get it’s load lighter. Thales came up with a plan to break the mule of this bad habit. The mule was loaded with sponges and rags, which when the mule rolled over, absorbed the water and made the load heavier. This eventually cured the mule of it’s troublesome habit.6

Figure 3.3 Calculation of the height of a pyramid

6cf. The History of Mathematics, David M. Burton, McGraw - Hill, 2007, p. 88.

21 Thales and mathematics Thales knew how to determine the height of a pyramid from the length of its shadow, and he was able to calculate the distance of a ship at sea from observations taken at two points on land.

Thales has been credited with the discovery of ﬁve propositions:

• A circle is bisected by its diameter.

• The angles at the bases of any isosceles triangle are equal.

• If two straight lines cut one another, the opposite angles are equal.

• (ASA Theorem) If two angles and the included sides of two triangles are correspond- ingly equal, the two triangles are congruent.

• A triangle inscribed in a semicircle has a right angle.

Figure 3.4 ASA Theorem

Thales proved that an isosceles triangle has two equal angles. When studying circles, Thelas was not trying to work out a numerical answer to a particular circle (as the Babylo- nians or the Egyptians had done before him). Instead, Thela wanted to make statement for “every circle in the world.” In other words, he would study a whole class of objects. This was a great new idea and this is why he is called the ﬁrst mathematician in history.

Because there is a continuous line from Egyptian to Greek mathematics, all the above mathematical facts may have been known to the Egyptians. However Thales is usually designated as the father of geometry, or the ﬁrst mathematician. At the very least, Thales did contribute to the organization of geometry; and he implemented the deductive method to give rigorous proofs for mathematical theorems.