Lecture 2
Greek Mathematics
The earliest records of peoples in the areas near Greece come
from the Stone age (10,000 B.C.E. to 3,000 B.C.E.)
Greek civilization begins about 3000 B.C.E.
- Minoan civilization develops on the island of Crete (2600 B.C.E.)
- Mycanaen civilization develops on the mainland (1600 B.C.E.)
- "Dark Ages" 1100 B.C.E. - 800 B.C.E.
Some famous Greek philosophers/mathematicians. You can get more
biographical information here
- Thales
(625-546 B.C.E.) Founder of the Ionian School.
- Anaximander (610-547 B.C.E.)
- Anaximenes (585-525 B.C.E.)
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Pythagoras (569-475 B.C.E.) Founder of Pythagorean School.
Student of Thales.
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Anaxagoras (500-428 B.C.E.)
- Parmenides (515-? B.C.E.) Member of the Eleatic School.
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Xeno (495-435? B.C.E.)Member of the Eleatic School.
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Democritus (460-370 B.C.E.)
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Plato (427-347 B.C.E.) Founder of the Academy in Athens.
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Eudoxus (408-355 B.C.E.)
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Aristotle (384-322 B.C.E.) Founder of the Lyceum in Athens.
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Euclid (325-265 B.C.E.)
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Archimedes (287-212 B.C.E.)
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Pappus (290-350 A.D.)
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Theon (335-405 A.D.)
In roughly 775 BCE the Greeks replaced their hieroglyphs with the Phoenecian
alphabet. They wrote on papyrus, so
early writings are not preserved.
Greek history falls into the following periods:
- Archaic period (660-480 BCE)
- Early Classical period (480-450 BCE)
- Golden Age (450-400 BCE)
- Late Classical (400-300 BCE)
- Hellenistic (300 BCE -
Greek mathematics was influenced heavily by Greek philosophy. Greeks first
conceived of mathematics in abstract terms.
To the Pythagoreans, all numbers were whole numbers, or ratios of whole
numbers. The proof that the square root of two is irrational
was known to them, and greatly disturbed them.
Greeks were concerned with the dichotomy between the discrete and the
continuous. There were ongoing dialogues as to whether space and time were
discrete or continuous. Xeno's 4 paradoxes addressed both of these
conceptions. An example of one of his paradoxes can be found here
(The paradox of the arrow).
Greeks were interested in knowing what could be proved using only straightedge
and compass (based on the ideals of a line and a circle).
Three famous geometrical construction problems
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squaring the circle - constructing a square with the same area as a
given circle.
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doubling the cube - constructing a cube with volume equal to twice a given
cube.
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trisecting a given angle . This is known to be impossible using only a
straightedge and a compass, but is possible if you are permitted to make two
marks on the straightedge!
Plato valued abstract ideas and mathematical concepts as examples of the
ideal.
Aristotle, a student of plato, was the greatest scientist of his time. He
considered the foundations of logic and proof, axioms and postulates, and laid
the basis for Euclid's
Elements. He was a tutor to Alexander the Great.
Euclid distilled the geometrical knowledge of his time into a form which was
used, virtually unchanged, into the 1900's. He determined the choice of
axioms, the arrangements of the theorems, as well as the style of the proofs.