Download Overview of the Classical Period of Greek Mathematics

The Non-Euclidean Revolution
Overview of the Classical Period of Greek Mathematics
Ancient Greek civilization lasted from approximately 2800 BCE through 600 AD. There
were two main periods of mathematical development:
Classical Period: 600 BCE - 300 BCE
Alexandrian Period: 300 BCE - 600 AD
Euclid's Elements appeared in about 300 BCE. It is a compilation and reorganization of
much of the mathematics developed during the Classical Period.
Key issues: During the Classical Period, mathematicians and philosophers wrestled with several
key issues. Among these are:
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The nature of space: is it discrete or continuous?
Which is more fundamental: number (arithmetic) or space (geometry)?
What is the standard of proof in mathematics (intuition, experiment, special cases, logic
& deduction, etc.), and how should mathematics be organized?
What is the relation of mathematics to the world of practical affairs? Who should study
mathematics and why?
Schools of the Classical Period:
Ionian School: founded by Thales of Miletus (c.640-c.546 BCE)
 “Introduced” the abstraction of space and shapes.
 “Introduced” belief in the rationality of the universe.
 First uses of the deductive method ― geometry as a mental activity, with practical
applications.
Pythagorean Society: founded by Pythagoras (c.570-c.495 BCE)
 A mystical, philosophical, and scientific “brotherhood”; in some respects a secret society.
 Number was considered fundamental to all reality, and numbers were endowed with
mystical aspects.
 Knew the “Pythagorean theorem.”
 Discovered incommensurable lengths, or in modern terms, irrational numbers ― which
caused an intellectual crisis (Hippasos of Metapontion).
 Deductive proofs and generalizations based on special cases were used to establish
results.
Eleatic School: (Parmenides & Zeno (born c.495-480 BCE))
 Primarily a philosophic school.
 Zeno known for four paradoxes, intended to refute two possible positions on the nature of
motion and space: that they are continuous, or that they are discrete.
Sophist School:
 The first school of Athens during its ascendancy following the defeat of the Persians in
479 BCE.
 Much mathematical work focused on the three great construction problems: doubling the
cube, squaring the circle, and trisecting an angle.
 Hippias of Elis, Hippocrates of Chios and quadrature of certain “lunes.”
Platonic School (the Academy): founded by Plato (427-347 BCE)
 Greatly influenced by the Pythagoreans.
 Key doctrine: the “forms” or “ideas,” which represent ultimate reality and truth. Physical
objects are imperfect realizations of the forms.
 Abstract “mathematical” reflection needed to study the forms.
 Deductive organization of knowledge preferred because it gives certain truths, and
because it is not tainted by pedestrian, practical, inductive methods.
 Mathematics viewed as preparation for the study of philosophy and the study of the
(ideal) universe.
 Discovered the conic sections (ellipse, parabola, hyperbola).
School of Eudoxus: (408-355 B.C.E.)
 Eudoxus is the greatest geometer of the Classical Period.
 He developed geometric theory of ratio and similarity (and thereby ended the crisis of
incommensurables), but in so doing he made geometry fundamental (not number), and
separated number (or algebra) and geometry.
 To do this, he realized the need to base a deductive structure on explicit axioms.
 He developed the “method of exhaustion” for computing areas and volumes, anticipating
the integral calculus.
 He proposed a geometric theory (involving spheres within spheres) to explain the motion
of heavenly bodies
School of Aristotle (the Lyceum): founded by Aristotle (384-322 BCE)
 Aristotle emphasized the importance of objects (not forms).
 He had a modern concept of definition and understood the need for undefined terms.
 He founded the science of logic, which he derived from examples in mathematics
(Aristotle’s is the last word on logic until the nineteenth century).
Euclid is influenced by all of the above as he begins to write his Elements...