The Origins of Greek Mathematics

Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is

• The most impressive of all civilizations,
• The most influential in Western culture,
• The most decisive in founding mathematics as we know it.

Basic facts about the origin of Greek civilization and its mathematics.

• The best estimate is that the Greek civilization dates back to 2800 B.C. -- just about the time of the construction of the great pyramids in Egypt. The Greeks settled in Asia Minor, possibly their original home, in the area of modern Greece, and in southern Italy, Sicily, Crete, Rhodes, Delos, and North Africa.
• About 775 B.C. they changed from a hieroglyphic writing to the Phoenician alphabet. This allowed them to become more literate, or at least more facile in their ability to express conceptual thought.
• The ancient Greek civilization lasted until about 600 B.C.
• The Egyptian and Babylonian influence was greatest in Miletus, a city of Ionia in Asia Minor and the birthplace of Greek philosophy, mathematics and science.
• From the viewpoint of its mathematics, it is best to distinguish between the two periods: the classical period from about 600 B.C. to 300 B.C. and the Alexandrian or Hellenistic period from 300 B.C. to 300 A.D. Indeed, from about 350 B.C. the center of mathematics moved from Athens to Alexandria (in Egypt), the city built by Alexander the Great (358 -323 B.C.). It remained the center of mathematics for a millennium until the library was sacked by the Muslims in about 700 A.D.

The Sources of Greek Mathematics

In actual fact, our direct knowledge of Greek mathematics is less reliable than that of the older Egyptian and Babylonian mathematics, because none of the original manuscripts are extant.

There are two sources:

• Byzantine Greek codices (manuscript books) written 500-1500 years after the Greek works were composed.
• Arabic translations of Greek works and Latin translations of the Arabic versions. (Were there changes to the originals?)
• Moreover, we do not know even if these works were made from the originals. For example, Heron made a number of changes in Euclid's Elements, adding new cases, providing different proofs and converses. Likewise for Theon of Alexandria (400 A. D.).

The Greeks wrote histories of Mathematics:

• Eudemus ( century B.C.), a member of Aristotle's school wrote histories of arithmetic, geometry and astronomy (lost),
• Theophrastus (c. 372-c. 287 B.C.) wrote a history of physics (lost).
• Pappus (late cent A.D.) wrote the Mathematical Collection, an account of classical mathematics from Euclid to Ptolemy (extant).
• Pappus wrote Treasury of Analysis, a collection of the Greek works themselves (lost).
• Proclus (A.D. 410-485) wrote the Commentary, treating Book I of Euclid and contains quotations due to Eudemus (extant).
• various fragments of others.

The Major Schools of Greek Mathematics

• The Ionian School was founded by Thales (c. 643- c. 546 B.C.). Students included Anaximander (c. 610-c. 547 B.C.) and Anaximenes(c. 550-c. 480 B.C.). Thales is sometimes credited with having given the first deductive proofs.
  • The importance of the Ionian School for philosophy and the philosophy of science is however without dispute.

  Thales

  The Major Schools of Greek Mathematics

• The Pythagorean School was founded by Pythagoras in about 585 B.C. More on this later. A brief list of Pythagorean contributions includes:
  1. Philosophy.
  2. The study of proportion.
  3. The study of plane and solid geometry.
  4. Number theory.
  5. The theory of proof.
  6. The discovery of incommensurables.
  

• The Eleatic School from the southern Italian city of Elea was led at one time by Zeno who brought to the fore the contradictions between the discrete and the continuous, the decomposable and indecomposable. Indeed, Zeno directed his arguments against both opposing views of the day that space and time are infinitely divisible; thus motion is continuous and smooth, and the other that space and time are made up of indivisible small intervals, in which case motion is a succession of minute jerks.

  Zeno's Paradoxes Zeno constructed his paradoxes to illustrate that current notions of motion are unclear, that whether one viewed time or space as continuous or discrete, there are contradictions. They are

  Dichotomy. To get to a fixed point one must cover the halfway mark, and then the halfway mark of what remains, etc.

  Achilles. Essentially the same for a moving point.

  Arrow. An object in flight occupies a space equal to itself but that which occupies a space equal to itself is not in motion.

  Stade. Suppose there is a smallest instant of time. Then time must be further divisible!

  

  Now, the idea is this: if there is a smallest instant of time and if the farthest that a block can move in that instant is the length of one block, then if we move the set B to the right that length in the smallest instant and the set C to the left in that instant, then the net shift of the sets B and C is two blocks. Thus there must be a smaller instant of time when the relative shift is just one block.

  The Eleatic School

• Democritus of Abdera (ca. 460-370 B.C.) should also be included with the Eleatics. One of a half a dozen great figures of this era, he was renown for many different abilities. Examples:
  • He was a proponent of the materialistic atomic doctrine.
  • He wrote books on numbers, geometry, tangencies, and irrationals. (His work in geometry was said to be significant.)
  • He discovered that the volumes of a cone and a pyramid are 1/3 the volumes of the respective cylinder and prism.
• The Sophist School ( 480 B.C.) was centered in Athens, just after the final defeat of the Persians.

  Emphasis was given to abstract reasoning and to the goal of using reason to understand the universe.

  This school had amongst its chief pursuits the use of mathematics to understand the function of the universe.

  At this time many efforts were made to solve the three great problems of antiquity: doubling the cube, squaring the circle, and trisecting an angle -- with just a straight edge and compass.

  One member of this school was Hippias of Elis (ca. 460 B.C.) who discovered the ] trisectrix, which he showed could be used to trisect any angle.

  

  How to draw a trisectrix: Imagine a radial arm (like a minute hand of a clock) rotating at uniform speed about the origin from the vertical position to the horizontal position in some fixed period of time. (That is from 12 O'clock to 3 O'clock.) The tip of the arm makes a quarter circle as shown in red in the picture. Now imagine a horizontal (parallel to the x-axis) arm falling at uniform speed from the top of arm to the origin in exactly the same time. The trisectrix is the intersection of the two arms. The curve traced in black is the trisectrix. As you can see, the trisectrix is a dynamically generated curve. The Platonic School and those subsequent did not accept such curves as sufficiently ``pure" for the purposes of geometric constructions.

  Hippocrates of Chios, though probably a Pythagorean computed the quadrature of certain lunes. (This is the first correct proof of the area of a curvilinear figure.) He also was able to duplicate the cube by finding two mean proportionals (Take a=1 and b=2 in a:x=x:y=y:b. Solution: , the cube root of 2.)

• The Platonic School, the most famous of all was founded by Plato (427-327 B.C.) in 387 B.C. in Athens.

  Pythagorean forerunners of the school, Theodorus of Cyrene and Archytas of Tarentum, through their teachings, produced a strong Pythagorean influence in the entire Platonic school.

  • Members of the school included Menaechmus and his brother Dinostratus and Theaetetus(c. 415-369 B.C.)
  • According to Proclus, Menaechmus was one of those who ``made the whole of geometry more perfect". We know little of the details. He was the teacher of Alexander the great, and when Alexander asked for a shortcut to geometry, he is said to have replied,

    ``O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry these is one road for all."

  • As the inventor of the conics Menaechmus no doubt was aware of many of the now familiar properties of conics, including asymptotes. He was also probably aware of the solution of the duplication of the cube problem by intersecting the parabola and the hyperbola , for which the solution is . For, solving both for x yields

    

  • The academy of Plato was much like a modern university. There were grounds, buildings, students, and formal course taught by Plato and his aides.
  • During the classical period, mathematics and philosophy were favored.
  • Plato was not a mathematician -- but was a strong advocate of all of mathematics.
  • Plato believed that the perfect ideals of physical objects are the reality. The world of ideals and relationships among them is permanent, ageless, incorruptible, and universal.
  • The Platonists are credited with discovery of two methods of proof, the method of analysis and the reductio ad absurdum.
  • Plato affirmed the deductive organization of knowledge, and was first to systematize the rules of rigorous demonstration.
  • The academy was closed by the Christian emperor Justinian in A.D. 529 because it taught ``pagan and perverse learning".

• The School of Eudoxus founded by Eudoxus (c. 408 B.C.), the most famous of all the classical Greek mathematicians and second only to Archimedes.
  • Eudoxus  developed the theory of proportion, partly to account for and study the incommensurables (irrationals).
  • He produced many theorems in plane geometry and furthered the logical organization of proof.
  • He also introduced the notion of magnitude.
  • He gave the first rigorous proof on the quadrature of the circle. (Proposition. The areas of two circles are as the squares of their diameters. )
• The School of Aristotle, called the Lyceum, founded by Aristotle (384-322 B.C.) followed the Platonic school. It had a garden, a lecture room, and an altar to the Muses.

  

  The School of Aristotle

  Aristotle set the philosophy of physics, mathematics, and reality on a foundations that would carry it to modern times.

  He viewed the sciences as being of three types -- theoretical (math physics, logic and metaphysics), productive (the arts), and the practical (ethics, politics).

  He contributed little to mathematics however,

  ...his views on the nature of mathematics and its relations to the physical world were highly influential. Whereas Plato believed that there was an independent, eternally existing world of ideas which constituted the reality of the universe and that mathematical concepts were part of this world, Aristotle favored concrete matter or substance.

  Aristotle regards the notion of definition as a significant aspect of argument. He required that definitions reference to prior objects. The definition, 'A point is that which has no part', would be unacceptable.

  Aristotle  also treats the basic principles of mathematics, distinguishing between axioms and postulates.

  • Axioms include the laws of logic, the law of contradiction, etc.
  • The postulates need not be self-evident, but their truth must be sustained by the results derived from them.
  Euclid uses this distinction.

  Aristotle  explored the relation of the point to the line -- again the problem of the indecomposable and decomposable.

  Aristotle  makes the distinction between potential infinity and actual infinity. He states only the former actually exists, in all regards.

  Aristotle  is credited with the invention of logic, through the syllogism.

  1. The law of contradiction. (not T and F)
  2. The law of the excluded middle. (T or F)
  His logic remained unchallenged until the century. Even Aristotle  regarded logic as an independent subject that should precede science and mathematics.

  Aristotle 's influence has been immeasurably vast.

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