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.
 

The Greeks wrote histories of Mathematics:

•    Eudemus ( century B.C.), a member of Aristotle's school wrote historiesof arithmetic, geometry and astronomy
 

•    Theophrastus (c. 372-c. 287 B.C.) wrote a history of physics
 

•    Pappus (late cent A.D.) wrote the Mathematical Collection, an account of classical mathematics from Euclid to Ptolemy
 

•    Pappus wrote Treasury of Analysis, a collection of the Greek works themselves.
 

•    Proclus (A.D. 410-485) wrote the Commentary, treating Book I of Euclid and contains quotations due to Eudemus
 

The Major Schools of Greek Mathematics

 

•    The Ionian School was founded by Thales (c. 643- c. 546 B.C.). Thales is sometimes credited with having given the first deductive proofs
 

o    The importance of the Ionian School for philosophy and the philosophy of science is however without dispute.
 

•    The Pythagorean School was founded by Pythagoras in about 585 B.C. More on this later. A brief list of Pythagorean contributions includes:
 

o    Philosophy.
 

o    The study of proportion.
 

o    The study of plane and solid geometry.
 

o    Number theory.
 

o    The theory of proof.
 

o    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.
 

 •    The Sophist School ( 480 B.C.) was centered in Athens, just after the final defeat of the Persians.
 

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

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

o    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.
 

o    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.
 

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

o    Members of the school included Menaechmusand his brother Dinostratusand Theaetetus(c. 415-369 B.C.)

o    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”
 

o    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

o    The academy of Plato was much like a modern university. There were grounds, buildings, students, and formal course taught by Plato and his aides.
 

o    During the classical period, mathematics and philosophy were favored.
 

o    Plato was not a mathematician -- but was a strong advocate of all of mathematics.
 

o    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.
 

o    The Platonists are credited with discovery of two methods of proof, the method of analysis  and the reductio ad absurdum.
 

o    Plato affirmed the deductive organization of knowledge, and was first to systematize the rules of rigorous demonstration.
 

o    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.
 

o    Eudoxus  developed the theory of proportion, partly to account for and study the incommensurables (irrationals).
 

o    He produced many theorems in plane geometry and furthered the logical organization of proof.
 

o    He also introduced the notion of magnitude.
 

o    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.
 

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

o    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 definitionas 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.


GAYAN SHYAMAL

2012/2013