Included please find a collection of maps of the old Greek world. There is limiting copyright on these pictures.

The Naissance of Greek scholarly communities lies shortly after 600 BC. Among the earliest Greek `scholars' is Thales of Miletus, who lived until 547 BC. Scholars attempt rational explanations of the world around them. Our interest is in the development of precise mathematical reasoning.

Pythagoras of Samos lived between 575 BC and 475 BC. The Pythagorean school, started by him in Croton, was a half religious, half scientific, community whose members are the only early source of information about its founder's work related to numbers and geometry. The membership of this school associated geometry and numbers with signs from the gods. Even today we hear talk of lucky numbers, of six six six, or of a special meaning of the pentagram.

Consider the early Pythagorean belief that all numbers from geometry must be rational numbers, that is, must be numbers that are fractions of whole numbers, like 2/3 and 175/101. We may go through the following speculation to prove it wrong:

• When we think of all rational numbers as points on a straight line, it appears at first obvious that they `fill' the whole line. There simply is no room for more points.
• On second thought, if the above argument were acceptable, then so would the following:
  • Draw a straight line. On it, mark off the whole numbers 0, 1, 2, 3, 4, 5, and so on.
  • There are gaps between the whole numbers on the line. In the consecutive gaps, we can put the numbers 1/2, 3/2, 5/2, 7/2, 9/2, and so on. Now the gaps have width 1/2.
  • Next cut the gaps again in half by inserting the numbers 1/4, 3/4, 5/4, 7/4, 9/4, 11/4, and so on. Now the many small gaps have width 1/4.
  • Next cut the gaps again in half by inserting the numbers 1/8, 3/8, 5/8, 7/8, 9/8, 11/8, 13/8, and so on. Now the many small gaps have width 1/8.
  • When we keep doing this, the whole line will be `filled' with numbers that are fractions with as denominator a power of 2.
  By the earlier argument, the line should be `full.' Obviously it isn't, because numbers like 1/3 are missing. So the earlier argument why all numbers on the line should be rational, is fallacious.
• Next, we construct a number of geometry which is not a rational number. Consider the right triangle whose shorter sides have length 1. By the well-known Pythagorean Theorem, the hypothenuse has length the square root of 2. Let us use the letter h for this length. So h times h equals 2. Suppose, to the contary, that h equals a fraction m/n of positive whole numbers. Then
  • n times h equals m.
  • So also n times n times h times h equals m times m.
  • So n times n times 2 equals m times m.
  Now factor both sides into prime factors. Then the left hand side has an odd number of prime factors 2, while the right hand side has an even number of prime factors 2. Contradiction. So h is not a rational number.

Communities of interchange survived in the Greek world. By about 325 BC, Alexander of Macedonia, known as Alexander the Great, had conquered almost the whole civilized world. In his attempt to remove a despot, he became one himself. Alexander and many of his successors, were great admirers of Greek civilization. Its culture spread, and for a thousand years Greek was the standard for intellectual and international communication.

The earliest popularly known significant work on mathematics is Elements, written by Euclid of Alexandria around 300 BC. The contents revolve around a rigorous introduction to the foundations of geometry, as it had developed in the hellenistic world of his time. What follows is an incomplete list of aspects that make Elements of great historical significance.

• The text survived through history as the primary, and in many aspects only, source on the most rigorous foundations of a significant part of mathematics. For the better part of two millennia it was arguably the most important mathematics text known to exist.
• Contrary to `mathematics' elsewhere in the world, Euclid's Elements is not just a list of truths about numbers and geometry. Instead, a short list of assumed `true' principles is given, in the form of axioms or postulates. From these, less elementary mathematical truths are derived through rigorous proofs. This approach is nowadays universal, and is known as the axiomatic method. In current common practice, we no longer make distinctions between axioms and postulates; we just talk about axioms. The more important statements that we rigorously derive from the axioms, are called theorems. People often talk about theories when they refer to lists of axioms, plus the theorems that follow from them.
• A collection of axioms from which one tries to prove all the other `true' facts, is called an axiomatization. The very look of an axiomatization may tell us a lot about the kinds of statements one might expect to prove from them. One of Euclid's axioms, associated with the parallel postulate, became singled out as possibly derivable from the other axioms. In other words, the parallel postulate was possibly redundant as axiom. This conjecture was not resolved for more than two millennia, partly because the concept of axiomatic method was barely understood.

We will not discuss what makes a rigorous proof, or how exactly rigorous proofs are related to axioms. Arguably the most important ancient thinker on that subject was Aristotle, who was a few decades older than Euclid.

On the World Wide Web, go to any good search engine website, and search for further websites by selecting the words Euclid plus Elements.

A serious part of Greek work survives to this day. One reason is the recognized special value of Hellenistic achievements. Another is a series of lucky coincidences which kept copies of old works in existence until after 1450, when book printing secured their preservation.