Early Greek Mathematics

Math 390 Lecture 4

Dr. Janssen

Introduction

Context

Greek life circa 600 BCE
Learned not to accept answers handed down from ancient times
World was knowable by rational inquiry
Idea of mathematical proof comes from the Greeks
Sources are copies of copies of copies and typically date around 1000 CE (originals: 300 CE)

Greek Numeration

Greek Symbol	Roman	Greek Symbol	Roman
\(\alpha\)	1	\(\iota\)	10
\(\beta\)	2	\(\kappa\)	20
\(\gamma\)	3	\(\lambda\)	30
\(\delta\)	4	\(\mu\)	40
\(\epsilon\)	5	\(\nu\)	50
\(\digamma\)	6	\(\xi\)	60
\(\zeta\)	7	\(o\)	70
\(\eta\)	8	\(\pi\)	80
\(\theta\)	9	koppa	90

Early Greek Mathematicians

Thales of Miletus (624-547 BCE)

Earliest Greek mathematician
Prediction of a solar eclipse in 585 BCE
SAS application to measuring distance to a ship at sea
Proved the base angles of an isosceles triangle are equal
Started science as we know it

Pythagoras (572-497 BCE)

Settled in Crotona in 530 BCE
More mystic than rational thinker; commanded great respect from his followers
“Number was the substance of all things”
Interested in different types of numbers (e.g., triangular, etc.)

Pythagorean Theorem

Known in other cultures long before Pythagoras
Pythagoreans assumed that everything could be counted, including lengths.
Thus, one needs a measure, which becomes the (indivisible) unit length.
Pythagoreans assumed such a measure existed for the side lengths and hypotenuse of a right triangle.
430 BCE: Side length and hypotenuse are incommensurable.

How?

A hint from Aristotle

If the side and diagonal (of a unit square) are assumed commensurable, then one may deduce that odd numbers equal even numbers.

Assume that the side BD and diagonal DH are commensurable, that is, each is represented by the number of times it is measured by their common measure. It may be assumed that at least one of these numbers is odd, otherwise there would be a larger common measure. Then the squares DBHI and AGFE on the side and diagonal, respectively, represent square numbers. The latter square is clearly double the former, so it represents an even square number. Therefore its side AG = DH also represents an even number and the square AGFE is a multiple of four. Since DBHI is half of AGFE, it must be a multiple of 2; that is, it represents an even square, hence its side BD must also be even. But this contradicts the original assumptions that one of DH, BD, must be odd. Therefore, the two lines are incommensurable.

Note this implies the existence of proof ingrained into the Greek conception of mathematics.

Famous Geometry Problems

Squaring the Circle
Doubling the Cube
These were attempted by Hippocrates of Chios (mid-fifth century BCE) and seemingly recorded in a textbook on geometry

Plato (429-347 BCE)

Academy founded in Athens around 385 BCE
Legend: “Let no one ignorant of geometry enter here.”
Mathematical education of philosopher-kings: arithmetic, plane geometry, solid geometry, astronomy, and harmonics (music)

Key figures

Pythagoras and his followers
Plato
Aristotle

The Pythagoreans

Pythagoras

580-500 BCE
Born at Samos
Some say he studied under Thales, unlikely given age difference
Traveled to Egypt, Babylon, and possibly India
Contemporary of Buddha, Confucius, and Laozi
Said to have coined the words philosophy and mathematics

Pythagoreans

Founded after travels on the island of Croton, SE coast of Italy
Knowledge and property held in common; attribution given to the group (but really the master)
Mathematical interest moved beyond the exigencies of daily life toward a love of wisdom
Motto: “All is number”

Number Mysticism

Odd numbers have male attributes, even numbers female
‘There is divinity in odd numbers’
1: The generator of numbers and the number of reason
2: First even/female number; the number of opinion
3: the first true male number, the number of harmony, composed of unity and diversity
4: number of justice or retribution, indicating the squaring of accounts
5: the number of marriage, the union of the first true male/female numbers
6: the number of creation
10, the tetractys: the holiest of numbers, representing the number of the universe, including the sum of all possible geometric dimensions

Figurate Numbers

10, the holy tetractys, is an example of a triangular number
The pentagonal numbers are given by \(1+4+ 7 + \cdots + (3n-2) = \frac{n(3n-1)}{2}\)
The hexagonal numbers were derived from the sequence \(1+5+9+\cdots + (4n-3) = 2n^2 - n\)
And so on for larger polygonal numbers (and polyhedral numbers)

All things which can be known have number; for it is not possible that without number anything can be either conceived or known.

[The tetractys was] great, all-powerful and all-producing, the beginning and the guide of the divine as of the terrestrial life.

–Philolaus (died in approx.~390 BCE)

Arithmetic and Cosmology

What is a number?

Mesopotamia: number applied to spatial extension
Egypt: natural numbers and unit fractions
Babylonians: field of rational fractions
In Greece, “number” meant (positive) integer; fractions of integers described the relationship of ratio

When lengths of vibrating strings could be expressed as ratios of simple whole numbers, the results were harmonious
2:1 is an octave, 2:3 is the fifth, 3:4 is the fourth
Extrapolated that the heavenly bodies emitted harmonious tones: the “harmony of the spheres”